European Mathematical Society - 00 General
https://euro-math-soc.eu/msc/00-general
enLumen Naturae
https://euro-math-soc.eu/review/lumen-naturae
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Matilde Marcolli is currently a mathematics professor at CalTech, working in the division of physics, mathematics, and astronomy. She is also interested in information theory and computational linguistics. Therefore, it is not a surprise to find a lot of mathematics, (particle) physics, and cosmology in this book, which she discusses in some detail. Less obvious is that this is also a book about art (mostly modern and often abstract paintings) and that she makes a remarkable (and rather successful) effort to link art and science and to point to parallels between both, sometimes by exploring the common underlying philosophical ideas.</p>
<p>
However, in the introduction, she warns the reader that she is not an art critic or an art historian. And yet, it is a book about art, and it is also a book about mathematics and physics, but it is not a science book either. This book project grew out of a series of lectures that she has given in which she has explored the ideas that are bundled and extended in this book project. Its purpose is to explain modern science to the artists and to enlighten the art for scientists. It should prevent utterances from ignorant colleagues scientists like "a child can do this" when judging an abstract painting. It is unwise to give such an opinion if one is ignorant about the artistic background.</p>
<p>
The title of the book <em>Lumen Naturae</em> captures that common idea in art and science where insight is obtained from exploring nature, its physics and the structure of matter. A form of Enlightenment that can be captured by both the artist and the scientist. It is the elegance of the structures, the beauty of physical laws, the understanding of space and time that is revealed to us by progressive insight. All that is there for us to explore, and that does not come from a superior being or transcendence. Artists and scientists are both spearheads pulling society forward to the next level.</p>
<p>
To convey these ideas to the reader, Marcolli does not shy away from explaining detailing mathematical concepts. Her defence is that if desired, the reader can jump over the details and still catch the main idea, while the more savvy reader will only be stimulated to explore the quite extensive literature through which she guides the interested reader who is willing to dig somewhat into the concepts or to do some extra reading to understand the formulas or concepts that are mentioned here. On the other hand, she links the mathematics being discussed to many paintings that are reproduced in colour but on a relatively small scale (it is not a picture book). This link is obviously her (personal) interpretation which may differ from the reader's. However, unlike the scientific elements of this book, these paintings and the artists are not discussed in great detail. There is little or no historical context and there are no biographies of the artists. Here again she refers to the literature for the details.</p>
<p>
So what exactly does Marcolli discuss to illustrate her objectives outlined above? Besides an introductory chapter, there are ten more chapters of variable lengths in which she brings a coherent story about some topic. The first one is somewhat unexpectedly about still life paintings. She chose this topic to illustrate the spacetime model. I will discuss this in somewhat more detail although it is a chapter with almost no mathematical discussion. Still life painting has survived from the Flemish painters of the 17th century till the cubist and dadaist artists of the 20th century. Time can be introduced in the scene by putting together fruits or flowers from different seasons on the same scene just like cubists tried to catch different viewpoints and different instances in the same picture or a music instrument, placed amidst food is an indication of time, since music can only be experienced as time passes. And there is of course the symbolic vanitas that places time in perspective of a human lifetime, and seashells on a table have rings that refer to the time it took to grow into their particular shape. Books and scientific instruments may refer to the flow of knowledge as time evolves. In some paintings, the objects depicted become abstractions, detached from their environment just like science is more and more relying on abstraction. With some imagination, the elegant curves in paintings by Cézanne can be seen as spacetime curved by gravity, while surrealists like for example Dalí depict the relativity and flexibility of time more explicitly as a melting clock.</p>
<p>
As I said, there is little mathematics in that first chapter, but the theme of space and time is a recurrent one, repeated in several of the subsequent chapters. For example chapter three, discusses space, form and structure. Structure is illustrated with the mathematical number system that starts with integers, then rationals, reals, complex numbers,... and with Klein's Erlangen geometry project. There are many meanings in mathematics connected to "space": from vector space, to topological space, and from projective geometry to the piecewise linear space of computer graphics, there are knots, graphs and networks, metric and normed spaces, fractals, measure theory and tilings. In other words, this chapter is like a selected survey of mathematics where art is mentioned, but it is not the main topic.</p>
<p>
In the next two chapters she discusses entropy, randomness, information, and complexity. This gives her the occasion to give some critique on the pamphlet by Rudolf Arnheim from 1971 in which he considers the notions of order and disorder in modern art. Indeed some abstract paintings seem to place objects randomly in space or they look rather chaotic (think of Jackson Pollock), but there are many kinds of randomness, and even in chaos there is some information to be found. The seemingly random cosmic background radiation gives important information about how our universe was formed, and stochastic processes like random walks follow precise rules.</p>
<p>
The void may be represented by painting a black square or even a white square on a white background. In any case, 20th century avant-garde postwar artists were looking for a way to represent it, just like physicists tried to understand empty space, which turned out to be not empty at all since they are facing phenomena like dark matter and dark energy. Empty space definitely has some shape because gravity is defining its geometry. It is also very dynamical with particles popping in and out of existence as quantum mechanics shows (this is a pretext for Marcolli to tell us more about Feynman diagrams, quantum field theory and quantum gravity, and the different interpretations of quantum theory).</p>
<p>
Then there are some shorter chapters of a more theoretical nature. One on the geometry of numbers (Von Neumann definition of natural numbers, primes, their randomness, Ulam spirals and the zeta function); one on laws of physics (the standard model of particle physics); and one on the shape of our universe (from the Kepler's laws of the solar system to cosmic topology). The book ends with two chapters that are somewhat different from the previous ones. There is a larger chapter about a 20th century avant-garde futurist movement with an anarchist and socialist undertone, and there is a shorter chapter about illustrations in books, which leads to her personal watercolor paintings over pages in her mathematical notebook.</p>
<p>
The futurist chapter starts with the train as the power engine bringing society to a new destination, Similarly networks of roads connect people over larger distances, while networks of electrification and other services and provisions make that people clump together in cities. Mechanisation and specialisation should give individuals more time for leisure and entertainment. The importance of science is realised and being a scientist becomes a respected profession. Gradually the human body is also conceived as a mechanical machine or factory that can be studied by physics and chemistry. Science rules and there is no limit to what can be achieved, including the conquest of space and understanding the cosmos.</p>
<p>
It may be clear from this review that in this book Marcolli can (and does) connect mathematical and physical laws to many paintings but the main content of the book and most detailed information is given about the former. I can imagine that the reader-scientist will enjoy to see how many paintings can be linked to abstract theory and that the same kind of philosophy, the same ideas, and even the same emotional sensation can be associated with a theory or a set of formulas as with some visualisation produced by a creative artist. An artist (or any other reader) who is not skilled in the theoretical aspects that are forwarded by Marcolli, may have a harder time to understand all the formulas, but he or she can well be inspired to look up some literature, and here Marcolli does an excellent job in guiding all kind of readers through the extensive list of references after every chapter. The book is carefully edited, but the glossy pages, necessary to give numerous colour reproductions of the paintings, makes it heavy to hold while reading. I spotted a few minor typos. For example page 261: "modifi ed" and "defi ned" and p. 316: "lass" instead of "less", but these are really minor and do not harm an otherwise excellent typography. The pictures are always close to the reference in the text, which saves a lot of paging back and forth. Also the index is extensive and carefully compiled to easily retrieve the many subjects that were discussed in the text.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Marcolli explains many mathematical, physical, and cosmological issues including the formulas and technical details. These are often connected to topics like shape, space, matter, void, randomness, structure, or time. She links this to modern art (almost all paintings) which reflect the same or a related idea and sometimes with the same level of abstraction. The title refers to the fact that scientists and artists alike get their insights by studying the above mentioned issues in the world we live in. This enlightenment is natural and is provided to us by the universe we live in.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/matilde-marcoll" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Matilde Marcoll</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/mit-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MIT Press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2020</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-262-04390-8 (hbk), 978-0-262-355831-6 (ebk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 44.95 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">388</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://mitpress.mit.edu/books/lumen-naturae" title="Link to web page">https://mitpress.mit.edu/books/lumen-naturae</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A30</a></li></ul></span>Mon, 31 Aug 2020 11:13:23 +0000Adhemar Bultheel51087 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/lumen-naturae#commentsSleight of Mind
https://euro-math-soc.eu/review/sleight-mind
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Matt Cook is an economist, a composer, a storyteller (as the author of thrillers), and he performs as a magician. Several magical tricks rely on creating an intuitive expectation and then come up with a totally different result. This creates amazement and unbelief in the audience. This is also very much the effect of a paradox. Given that Cook is not a professional mathematician himself, it comes as a surprise to find rather much abstraction and mathematics in this book.</p>
<p>Logical paradoxes are often found in popular science books discussing mathematics, games, and puzzles. Many of these "popular" paradoxes you can also find in this book but there are many more. Although this book is written for a general public, it is not leisure reading, since the discussion of the paradoxes goes in depth and that requires precise definitions and sometimes it touches upon the foundations of logic, mathematics, probability, or whatever topic the paradox is about.</p>
<p>The different topics are arranged in different chapters and the format is always similar. There is a general introduction to the subject, and that involves the definition of the concepts that are required for the discussion of the paradoxes to follow. These are precise but the selected terms and the associated technicalities are restricted to a minimum. Only what is essential is defined and only as precise as needed. For example in the chapter on probability it is defined what a probability space is, and that involves a sample space, a sigma-algebra, and a probability function, which are described by words, rather than formulas. Of course it is also explained how a random variable and its density function are defined and the Bayes theorem is introduced (this inevitably results in a formula). So, there are some formulas, but they are suppressed as much as possible, describing the definitions mostly in words and by using examples. I guess this is intended not to shy away the non-mathematician, but if you are a mathematician, then, given the intended rigour, it feels a bit awkward and verbose. Of course some formulas cannot be avoided, for example to illustrate what is in the Principia Mathematica of Whitehead and Russell a formula here and there is unavoidable.</p>
<p>When Cook comes to the many examples of paradoxes, it assumes an attentive reader because the lack of formulas requires sometimes complicated sentences that are often almost philosophical. Also here, a returning format is used. First the paradox is formulated, wherever possible, mentioning its origin. Cook usually tells a story to make the paradox concrete for the reader, rather than formulating it in its mathematical or abstract form. Then the opposing explanations (often there are only two) are formulated. The main discussion then explains why one is wrong and the other is correct. Sometimes there are more possibilities and more than one explanation is possible depending on how some components are defined or interpreted, which happens when the problem is ill-posed or under-defined.</p>
<p>Let me give some examples that illustrate the types of paradoxes and the depth of the discussion. A first chapter is dealing with infinity, which is not the simplest one to start with, but it is also the underlying concept in some subsequent chapters. It is clearly a concept that has caused a lot of confusion throughout the history of mathematics and logic. First we are instructed about bijections and countable sets, Cantor's diagonalization process, the cardinals $\aleph_k$, and the continuity hypothesis. Then the paradoxes can be explained: Hilbert's Hotel, Stewart's HyperWebster Dictionary, and many more. After introducing some additional group theory also the Banach-Tarsky theorem is explained in some detail. Not really a proof, but still the reader is given some idea of why this seemingly impossible result holds. Zeno's paradoxes of motion are of course somewhat related to the concept infinity, and so these are discussed making use of what was obtained in the previous chapter. Thomson's lamp is also related. If a lamp is alternately switched on and off at time instances $1−2^{−n}$, then deciding whether at time $t=1$ the lamp will be on or off is impossible.</p>
<p>With chapter four, probability is introduced. The Simpson paradox and the Monty Hall problem are probably the best known but there are others that allow much more variations and require much more discussion. In the chapter on voting systems we are introduced to social choice theory and Arrow's impossibility theorem. This is not completely unrelated to the topic of game theory which plays a role in, for example, price setting in a economic system. The Braess paradox is the unexpected result that by adding an extra road to a traffic system, the traffic may be slowed down.</p>
<p>With self-reference we are back to the foundations of mathematics with axiomatic set theory, and, among others, the paradoxes of Russell (the set of al sets that are not a member of themselves) and the liar (I am always lying). Inevitably this leads to Gödel's incompleteness theorems, a theory of types, the ZFC axiomatic system, etc. Also the unexpected hanging is a tough paradox discussed here. Somewhat in the same style is the chapter on induction, where some elements of formal logic are introduced.</p>
<p>A chapter involving geometry has curves, areas, and volumes with fractal dimension. There is not really a paradox here, but the fact that a dimension can be a fraction and need not be integer is considered to be paradoxical. But there are other simpler geometric examples. In many calculus books, we find the hard-to-believe fact that we can create an infinitely large overhang by stacking bricks if brick $k$ (numbered from top to bottom) overhangs the underlying one by $1/(k+1)$. This is an example where the mathematical fact that $\sum_{k=}^\infty 1/k$ diverges is replaced by a "story" of stacking overhanging bricks. Some typical mathematical beginners errors can also give some unexpected results, dividing by zero for example, or summing divergent series.</p>
<p>Finally Matt Cook has invited some colleagues to discuss paradoxes from physics. With statistical mechanics, the reader learns about entropy, Maxwell's Demon, and other classics such as the Brownian Ratchet driven by Brownian motion, and the Feynman's sprinkler problem. The unexpected results of special relativity are well known, and quantum physics is still difficult to understand in all its consequences and different interpretations are still discussed today.</p>
<p>In the final chapter the age-old question whether mathematics is discovered or invented is tackled. As one might expect, the answer is not exclusive for one or the other.</p>
<p>Mind, the paradoxes that are mentioned in this survey, are only few and exemplary for the many examples that can be found in this book (there are over 75). I can imagine that for readers who are totally mathematically illiterate, some steps may be hard, if these use terminology or arguments that are taken for granted. Nevertheless also those are considered potential readers because there is a short addendum introducing some very elementary mathematical notation. Cook also added a rather extensive bibliography, but many of the references are papers where the paradoxes were originally formulated, or papers discussing the solution. Thus not really the popularizing kind of literature for further reading. The index though is well stuffed and useful, since there is sometimes cross referencing across the chapters.</p>
<p>I could spot a typo in the discussion of the Banach-Tarsky theorem. When discussing successions of irrational spherical rotations left, right, up, down, denoted as L,R,U,D, strings of these letters are formed to denote points on a sphere. Uniqueness requires eliminating the succession of opposite rotations (free group). Thus UD, DU, LR, or RL are not allowed in a string. However in the table page 25 appears the string DUL which is not allowed.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Several paradoxes are analysed in depth. Some are well known others are less familiar: Zeno's paradoxes, Monty Hall problem, Banach-Tarsky theorem, paradoxes related to voting systems, self reference, but also statistical mechanics, special relativity and quantum physics and many more pass the review. The finale is a discussion of the ultimate question: Is mathematics invented or discovered?</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/matt-cook" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Matt Cook</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/mit-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MIT Press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2020</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780262043465 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 34.95 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">368</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://mitpress.mit.edu/books/sleight-mind" title="Link to web page">https://mitpress.mit.edu/books/sleight-mind</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A15</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/81p05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81P05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/63a10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">63A10</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/70-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/83-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-01</a></li></ul></span>Wed, 01 Apr 2020 11:54:56 +0000Adhemar Bultheel50644 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/sleight-mind#commentsThe Best Writing on Mathematics 2019
https://euro-math-soc.eu/review/best-writing-mathematics-2019
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is the 10th volume in this series reprinting every year a collection of diverse texts on mathematics (i.e., not necessarily mathematical papers) that are accessible to a broad public. I have been reviewing these books since 2012, and I have repeatedly explained the idea behind the concept and the kind of papers that are selected in my reviews. These ideas have not changed in this anniversary volume, so I will not repeat them here. If you are not familiar with the concept of the series, you can look it up and read all about it in the previous reviews <a href="/review/best-writing-mathematics-2012">2012</a>, <a href="/review/best-writing-mathematics-2013">2013</a>, <a href="/review/best-writing-mathematics-2014">2014</a>, <a href="/review/best-writing-mathematics-2015">2015</a>, <a href="/review/best-writing-mathematics-2016">2016</a>, <a href="/review/best-writing-mathematics-2017">2017</a>, <a href="/review/best-writing-mathematics-2018">2018</a>.</p>
<p>
This volume reprints 18 papers almost all originally published in 2018. The fact that the subjects of the papers are usually crossing the boundary between two or more domains is one of their interesting features. It is remarkable how smoothly the sequence of papers in this book migrates from one subject into the next, due to a careful selection and collation strategy of the editor.</p>
<p>
For example the first paper links geometry to gerrymandering. The latter is a manipulative subdivision of the sets of voters in a the-winner-takes-it-all system to enforce some outcome of the voting. Finding a fair subdivision is a combinatorial problem that can only be solved in a feasible time using Markov Chain Monte Carlo techniques. This smoothly connects to the next paper about a problem from the <em>Scottish Book</em>, a legendary diary from Polish mathematicians meeting in Lviv (Poland) in the 1930's. The problem posed by Hugo Steinhaus in there gave rise to the ham-sandwich theorem, which is also about a problem of fair partitioning. In two dimensions the problem reduces to cutting a pizza and all of its ingredients distributed on top into fair parts.</p>
<p>
Politicians may be interested in gerrymandering and perhaps even in fair distribution, but they may also have something to say on the educational system, and in how to distribute different subjects that children have to learn over a limited education time. In that respect it is important to know if mathematics learns children how to think. Some claim that this can also be learned by studying languages (like Greek and Latin), computer science, or even by solving brain teasers and puzzles. After a careful analysis of this question in relation with different mathematical subjects, the authors of the next paper, conclude with some recommendations on how to teach calculus.</p>
<p>
Speaking of puzzles, the next paper deals with the Rubik's cube and all its generalizations that were realized practically or that were studied on an abstract mathematical basis. Three-dimensional geometry of the cubes brings the reader to the next paper discussing 3D objects that when viewed from different viewpoints create some optical illusions. This optical paradox is geometrically analysed and ingeniously illustrated using a picture of the object simultaneously with its reflection in a mirror representing the alternative viewpoint. The mirror is a perfect link to the detection of mirror symmetry in string theory, which became an important subject in both theoretical physics and algebraic geometry.</p>
<p>
The illustrations in the texts are grey-scale, but when in the original text they were in colour, then sometimes the caption of a grey-scale image refers a line or area of a certain colour. To mitigate this, colour versions of the illustrations of all the papers are collected at this point of the book. This somewhat hides the abrupt switch to more computer related papers that now follow. The first of these more computational type texts is about the application of a so called probabilistic abacus to find the probability that some event will happen. This computational mechanism was invented by A. Engel in 1975. It simulates a finite game played on a graph based on chip-firing. This computational technique is now known as Engel's algorithm.</p>
<p>
Computers play also an increasing role in the analysis and classification of integer sequences. The on-line encyclopedia (<a href="https://oeis.org/" target="_blank">OEIS</a>) started by Neil Sloane in 1996 had 100k entries in 2004. Sloane's paper in this collection is listing some fascinating examples among which an (in 2018) recent entry 250000. At the time of writing this review (Jan 2020) the OEIS has 331811 entries and counting. If anything is related to computers nowadays, then it is certainly big data. That topic made a bliz career in research funding and was promptly turned into a buzz word. The next paper briefly discusses examples of well known big date problems: from search engines to health care to recommender systems to farming, and I am sure we haven't seen the last of it</p>
<p>
What can be computed or even what can be decided is a fundamental question to ask in computer science as well as in mathematics (cfr. the halting problem and Gödel's incompleteness theorem). The next paper explains that deciding whether all materials have a spectral gap (i.e. the gap between the energy of the ground state and the first excited state) is proved to be impossible, using Turing machines and ideas from plane tilings. Computer generated proofs and verifying proofs by computers become more and more common practice. That is illustrated with some historical examples in a paper that is wondering how we should proceed for the future.</p>
<p>
Quantum physics and the quest for a theory of everything has divided physics research. The pure mathematical labyrinth in which theoretical physics has evolved as opposed to the classical empirical physics is not completely unrelated to mathematical models that have been designed for other scientific disciplines. The phenomena one wants to study are simplified to models that isolate some interesting characteristics. Given such a model (as a set of equations and constraints), also solving the models analytically or computationally, may require further simplifications to become feasible. Computed results are validated and when not matching with reality, the model may need adaptation. Is not mathematics of modelling here a kind of empirical science. This brings us on the verge of philosophy about mathematics. More philosophy is in a paper asking what it means that 2+3=5 (what is meant is adding of numbers, not counting quantities), Do the numbers 2 and 3 actually exist? We assume they do, since it is so obvious. But why then prove Fermat's last theorem while it is so obvious that it must hold? More on philosophy, in particular about the link to the history of mathematics is illustrated in a paper about Gregory's notion of infinitesimals and continuity as compared to the Weierstrass approach of epsilon-delta definitions. Some purists think infinitesimals are evil, others consider it a blessing to work with. The authors however conclude that eventually, after closer analysis, the two historical approaches are not that different.</p>
<p>
We humans do not like chaos. We try to make sense of things and are constantly looking for patterns. The Kolmogorov complexity corresponds to finding the shortest program that can describe some (mathematical) object like for example a sequence. This links back to the previously discussed problem of computability or decidability. The seemingly complex problem to describe "the smallest number that cannot be described by less that 15 words" is trivial and yet impossible to grasp. Just like an infinitesimal, something smaller than anything finite and yet not zero is difficult to conceive, and still easy to describe and work with.</p>
<p>
What we believe to be true and what actually is true is, with the constant exposure to information, an important issue in an epoch of fake news. Statistics is in this respect a seemingly scientific tool to sustain some fact, but unfortunately, it is easily misused. A paper discussing this ethical issue gives some recommendations about this like: be open about data and methods, be aware of the limitations of statistics, be open for criticism, etc. and I would like to add to that: be careful about causality claims.</p>
<p>
The two remaining contributions are diverse. One is a plea to return to the original idea of Fields when he installed the Fields Medal. Should one recognize brilliant mathematicians who accomplished something big in mathematics and thus are already "established", or should one celebrate a mathematician who is pioneering a new field in mathematics? The original idea was to stimulate (international) collaboration, not competition. Since the Fields Medal got the status of a mathematical Nobel Prize around the 1960's, that original idea is violated and it became the subject of competition. The last paper is about an Eulogy delivered by Melvyn Nathanson for Paul Erdős in 1996 shortly after Erdős passed away, and some considerations Nathanson has to add now (in 2018). The paradox of Erdős is that he was enormously prolific and versatile, even creating new fields and yet he never embraced the new mathematical domains of the twentieth century. How could he publish such important theorems and yet know relatively little?</p>
<p>
I should also mention the list of interesting books that appeared in 2018 and that get some recommendation from Pitici. As in previous volumes there is also a long list of papers that could have been selected as well for this collection (but they were not) and of other writings such as reviews of books and essays, teaching notes, and special journal issues. Thus this book is again a most interesting collection of mathematics related papers of the usual quality.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is volume 10 of Picici's annual harvest of papers on diverse topics related to mathematics that are collected from different journals and books. The contributions relate mathematics to philosophy, history, education, communication, computer science, games, puzzles, statistics, etc. Most of them were published in 2018 and are written for a generally interested readership.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/mircea-pitici" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mircea Pitici</a></li><li class="vocabulary-links field-item odd"><a href="/author/ed-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(ed.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691198675 (hbk), 9780691198354 (pbk), 9780691197944 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even"> £ 66.00 (hbk), £ 20.00 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">287</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/books/hardcover/9780691198675/the-best-writing-on-mathematics-2019" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691198675/the-best-writing-on-mathematics-2019</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00b15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00b15</a></li></ul></span>Fri, 31 Jan 2020 10:37:23 +0000Adhemar Bultheel50361 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/best-writing-mathematics-2019#commentsA Brain for Numbers
https://euro-math-soc.eu/review/brain-numbers
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Humans have become the dominant species on this Earth, and that is partially because humans are numerate and and are able to do calculations. But are humans the only life form that has a sense of numbers? Why are some people better with numbers than others? Does it require a particular kind of brain to become a (good) mathematician? This book is not about mathematics, but it gives some partial answers to the previous questions. Nieder gives an extensive account of what has been learned from experiments about how the human (and animal) brain deals with numbers.</p>
<p>
To be able to give some answers, it requires first to define exactly what is meant by the concept number. So a first part of the book is required to define cardinal numbers as objective properties of a set. Moreover there is an intrinsic ordering (ordinal numbers) and numbers can be represented in many different ways: visually by dots (of the same or different size) or different objects or by symbols, or by sounds or by tactile input. All these require different brain activity.</p>
<p>
In a second part, numerous experiments are reported to illustrate that in view of the Darwinian theory there must have been a common ancestor in the evolutionary tree. These experiments show indeed that insects, fishes, birds, and humans have some notion of (small) numbers since some instinctive concept has been observed in all living creatures. The particular reptiles of the test were an exception to this general rule, but not all the different kind of reptiles were tested. Of course, not every life form had the same ability to discriminate between quantities. Nevertheless, there must be an evolutionary advantage for survival to have an approximate idea of quantities. This instinctive knowledge is also observed in human babies. Of all the experiments resulted the well known Weber's law of psychology. It says that the change in stimulus (e.g., the number of dots presented) is noticed when it is above a certain percentage and Weber's student Fechner added to this that the perception of that change in stimulus is logarithmic: $dp=k\ln (S/S_0)$ where $p$ stands for perception and $S$ for stimulus.</p>
<p>
To locate the numerical brain activity, Nieder describes the structure of the human brain in part three. All kinds of experiments enable us to locate the parts of the brain that are active when the subject is exposed to a number and even the activity of neurons can be measured. These experiments confirm that there is some innate number instinct.</p>
<p>
Homo sapiens differs from other animals by its ability to deal with numbers in symbolic form. This is the subject of part four. Here Nieder explains the origin and history of our notation and symbolic representation of numbers, and how we can learn animals to connect numbers to symbols. These symbolic representations are essential when one wants to do calculations that go beyond the small numbers. Where in the brain does calculation take place? Surprisingly, again there is some notion of approximate addition and subtraction of small numbers present in animals and even in babies. Professional mathematicians do not have a different brain and even mathematical prodigies do not have a brain that differs physiologically as has been found in postmortem determination. It may also come as a surprise that symbolic number representation is not necessarily connected with the way we process language. Another surprise is the strange connection between numbers and space. It seems like we have mentally an innate idea of a horizontal number line with numbers growing from left to right, which corresponds to our order of reading text.</p>
<p>
Part five deals with how children evolve from saying one, two, three,... as a sequence of meaningless words to consciously associating these words with abstract numbers and how they learn to calculate. Some people suffer from dyslexia, others from dyscalculia, and it has been observed that dyscalculia affects life in a way that is worse than being illiterate. It has been investigated if that may have genetic causes, but genes seem to be only partially responsible if a person has difficulty to calculate.</p>
<p>
The last part is about how the brain behaves when the number concept deviates from empirical reality. For example the concept of zero. It is a major step to accepted it as a number. A number can be visualized by showing a number of objects or dots on a screen. It is however difficult to represent "nothing", but nevertheless there is some sense of zero present in babies as it is illustrated with experiments. Zero is an important step towards an abstraction that leaves experimental reality. It opens the gate to negative numbers which can only be represented symbolically. Imagine five dots disappearing behind a screen, then seeing two dots leaving. When the screen comes down, one is expecting to see three dots. However when two dots are hiding and five emerge, there is no visual expectation about what to see when the screen comes down. A minus three can only be imagined symbolically.</p>
<p>
This brief summary of the content illustrates what one has to expect from this book. There is no mathematical model of the brain. There are actually not really mathematics, but there is an extensive, yet very accessible, description of what happens in the brain when we catch some idea of quantities and numbers from sensory perception, how it is stored and what happens when we calculate. That is at the very lowest and basic level that can hardly be called mathematics, but it is a first and an essential ingredient to start doing mathematics. Nieder describes numerous experiments on animals and humans that reveal some surprising facts. He is not only citing the latest generally accepted results, but he also describes the historical evolution, illustrating how more and better insight was obtained. Whether it comes to the mathematical concepts of numbers, or the history of the symbolic notation, or the biological structure of the brain, he takes the time to explain all the necessary concepts so that any layman can follow the details of the experiments and the conclusions. Thus the reader will not learn how to become a better mathematician, a computer scientist will not learn how to make models for artificial intelligence, and educators will not learn how to teach mathematics properly. It is however an intriguing story about the magnificent engine of the human brain which allows us to deal with numbers. Numbers are an important, yet only a small part of the overall human intelligence. We know already many, sometimes surprising, things, about our brain, but basically it is still a big mystery and extremely hard to be modelled by any artificial (so called intelligent) software.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This book reports on what is known about how human (and animal) brains deal with numbers and calculations. How much is genetic? How much is trained? How much is instinct? Have mathematicians a different brain? Answers to this kind of questions are provided.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/andreas-nieder" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Andreas Nieder</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/mit-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MIT Press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780262042789 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 28.00(hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">392</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://mitpress.mit.edu/books/brain-numbers" title="Link to web page">https://mitpress.mit.edu/books/brain-numbers</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A30</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97f20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F20</a></li></ul></span>Fri, 20 Dec 2019 14:56:13 +0000Adhemar Bultheel50115 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/brain-numbers#commentsFUNdamental mathematics
https://euro-math-soc.eu/review/fundamental-mathematics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Comedy and mathematics are to most "ordinary" people concepts that belong to two different well separated worlds. Everybody can appreciate a good joke and a laugh from time to time, while mathematics to many is just blood sweat and tears. Of course some teachers and many authors of popular math books have used puns and humour to give the mathematics some sugar coating. However, their main subject is nevertheless serious mathematics. There do exist humorous novels about mathematicians. Two examples are <em>Goldman's Theorem</em> by Ron J. Stern (2008) and <a href="/review/mathematicians-shiva" target="_blank"><em>A mathematician's shiva</em></a> by Stuart Rojstaczer (2014). And there is Jorge Cham, cartoonist and creator of <em>PhD Comics</em> who wrote <em>We have no idea</em> (2017) together with Daniel Whiteson about still open physical-mathematical problems. It has text that is richly illustrated with cartoons. The text has some slapstick kind of humour that approaches a bit the style of the book under review. But in fact I do not know of any book that is similar to <em>FUNdamental mathematics</em>, (which does not mean that such books do not exist). The capitalized FUN in the title is essential and it is further specified by the subtitle <em>a voyage into the quirky universe of maths and jokes</em>. Think of Monty Python brought by a stand-up comedian that goes on for over 300 pages, and who gets his inspiration in mathematics. There is indeed mathematics in this book but there are even more absurd jokes about mathematics, mathematicians, and about almost anything that the author has ever experienced. It is sometimes hard to know where exactly is the boundary between the mathematics and the nonsensical joke. There exist rules that stand-up comedy or sitcoms should have 4-6 laughs per minute. Eelbode applies a similar rule in this book with a joke every few paragraphs. To avoid an overdose, the reader should consume the book in limited portions. It's like with alcoholic beverages: read, but read wisely. As the author himself advises in his introduction, it should be administered a few pages at a time.</p>
<p>
So if we sieve out the mathematical content, what are then the subjects that are covered? Well, there are quite a few and here is a grasp of some of them. We find the Hairy Ball Theorem, Fermat's last theorem, the parallel axiom, the abc-conjecture, Hilbert's hotel, Russell's paradox and Gödel's incompleteness theorems, the kissing number, conditional probability, the travelling salesman problem and P vs NP, topology, game and graph theory, the cube and sphere in high dimensions, the Gamma function, etc. This list is not exhaustive and some of these topics are (in between the jokes) somewhat seriously discussed, while others are more briefly mentioned and considered too advanced to go into details.</p>
<p>
Some of the quirky characteristics of the book are that every chapter ends with a suggested music playlist. Music that can be listened to while reading the book or perhaps that the author was listening to while writing it. Hence it requires a taste that is somewhat similar to the author's preferences since it contains heavy metal with names like Iron Maiden and Metallica, but also dance, rap and plain rock. The last chapter has eleven exercises that look like mathematical multiple choice problems. They are followed by solutions that are again a kind of jokes that comment on every possible choice.</p>
<p>
Another fun-element is that there are a lot of numbered, formal looking definitions, but almost none of these define something mathematical. I give an example of a short one (definition 18) "sandals: A rather special kind of footwear, often worn by mathematicians and people who believe that this will help to reduce their ecological footprint". Other definitions can be very long and they can take up several pages. Very occasionally, there is a theorem with a proof, but again this is not really mathematics. For example (theorem 2): "Music festivals are miniature copies of India" or (theorem 3): "Polar bears are colour blind". The proofs take about two pages each. There are also many cartoon illustrations that are often mathematically informative, or sometimes just fun.</p>
<p>
There are funny "scenes from the life of a mathematician" when he/she is trying to publish a paper, or of the behaviour of students during a math exam, or the adventure of attending a mathematical conference. I quote a paragraph from the latter to illustrate the language used:</p>
<blockquote><p>
In an honourable attempt to reduce the travel costs — there is only so much I can do with that bench fee — I usually have an itinerary involving too many flight legs and not enough armrests. As a result, I am often so sleep deprived by the time I arrive at my destination that I look like a badly drawn version of the person on my passport picture. At least this explains why we are no longer allowed to smile when they take our picture: we all look grumpy upon arrival anyway.</p></blockquote>
<p>
Or this one where he is explaining what a 287-dimensional hypercube means</p>
<blockquote><p>
I think that some people spend less time looking for a partner than for their keys (even when they know they're in their handbag). (...) [A room in 287 dimensions]... already has more corners than electrons in our observable universe. Then again, who needs keys anyway? It is relatively safe to leave the door unlocked in 287 dimensions, since your house has even more walls than corners, so it does take a stubborn burglar to find your door. And although it's quite unlikely that you will catch your children playing soccer indoors, for the simple reason that the size of their ball will shrink into nothingness, I do not recommend you put them in a naughty corner: it might take them more than a few decades to be back for supper.</p></blockquote>
<p>
This attempt to characterize the content should make clear that there is indeed mathematics, but there is also a lot of jokes. In my opinion it is more a fun book for mathematicians than it is a book for non-mathematicians to learn some serious mathematics from. There is indeed much informative mathematics presented, although not really fundamental, but neither the mathematician nor the layman can deny that it is a lot of fun.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is Monty Python's Flying Circus of mathematics. It reads like attending a stand-up comedy show based on popular mathematics stories, but also making fun of the life of mathematicians, students, and anything that comes to mind of the author in a three hundred page long stream with a baroque decoration of countlessly many puns. Prescription: to be administered in smaller portions!</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-eelbode" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Eelbode</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/academia-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Academia Press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9789401462617</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">€ 24.99</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">311</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.lannoo.be/nl/fundamental-mathematics" title="Link to web page">https://www.lannoo.be/nl/fundamental-mathematics</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Wed, 11 Dec 2019 08:33:57 +0000Adhemar Bultheel50058 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/fundamental-mathematics#commentsCurves for the Mathematically Curious
https://euro-math-soc.eu/review/curves-mathematically-curious
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Julian Havil has already produced several popular math books. Some of them have been reviewed here: <a href="/review/impossible-surprising-solutions-counterintuitive-conundrums" target="_blank">Impossible?</a> (2008), <a href="/review/irrationals-story-numbers-you-cant-count" target="_blank">The Irrationals</a> (2012) and <a href="/review/john-napier-life-logarithms-and-legacy" target="_blank">John Napier</a> (2014). In this book, containing an anthology of ten iconic curves, he takes another angle of approach to tell more stories about mathematics. Havil's popular math books are more of the recreational kind. I mean that while telling his story, he is not hiding away the mathematics. There can be many formulas and derivations that are however easy to follow with some background in basic calculus.</p>
<p>
The curves selected have names. That is because they are in some sense important. If it is the name of a mathematician, it is, like often in mathematics, not always the name of the one who first studied the object. This is again illustrated by Havil in this book when he explores the history underlying the origin of the curve. There is one chapter for every curve. Sometimes it is just one particular curve described by a unique formula like the catenary, but often these curves have parameters or it is just a whole collection of curves with a special property like space filling curves. "Why these ten?" is an obvious question to ask, and Havil has anticipated this because he opens every chapter with a section that explains why he has chosen this curve. Whatever reason he gives, what is important for the reader is that there is always a story or stories worth telling that can be connected to that curve and in some cases these also have a very long history. Let me illustrate the yeast of the book by a telegraphic survey of the ten chapters.</p>
<p>
1. <em>The Euler spiral</em>. Its parametrizations are analyzed and the connection with elastic curves and Fresnel integrals. It is also known under other names (e.g. Coru spiral and clothoid), and Havil also explains the history of how and why this has happened.</p>
<p>
2. <em>The Weierstrass curve</em>. This is defined as an infinite sum and it is probably the first fractal ever described: a continuous function that is nowhere differentiable. The proof of Weierstrass for these properties is included.</p>
<p>
3. <em>Bézier curves</em>. This is an introduction to the characterization of these curves and how they are constructed by the Casteljau algorithm. There are two fun stories connected to these curves. One is about a Bézier curve called Lump which is the name of a dachshund as it was sketched by Picasso caught in one smooth Bézier curve. Havil provides its control points. Another story on the side is about how these curves are used to design letter fonts.</p>
<p>
4. <em>The rectangular hyperbola</em>. This is an excellent occasion to tell the history of how logarithms were invented. This is of course described in much more detail in Havil's book about John Napier.</p>
<p>
5. <em>The quadratrix of Hippia</em>. The history of this curve is connected to the classic Greek problem of trisecting an angle using only compass and straight edge, but the story would not be complete if one did not recall also the other "impossible" problems of squaring the circle and doubling the cube. The quadratrix is formed by the intersection of two moving lines one translating and another rotating at constant speed. If one could construct that curve, then trisecting an angle and squaring the circle became possible as well as constructing segments whose length is a unit fraction or a square root. The latter are examples of how Havil manages to add some extra mathematics of his own to a well known story.</p>
<p>
6. <em>Two space-filling curves</em>. Cantor, Hilbert, and Peano, are three names connected with these curves. The construction of these curves is of course related to the study of cardinality. The Peano curve is a continuous map from a unit interval to a unit square but it is not surjective.</p>
<p>
7. <em>Curves of constant width</em>. These are curves like the Reuleaux triangle that looks like a triangle that is slightly inflated, and yet shares many properties with a circle. If it is used as a drill, it will produce square holes (with slightly rounded corners). But there are several generalizations to study. Again, the latter are typical examples of Havil's mathematical extras.</p>
<p>
8. <em>The normal curve</em>. This bell shaped curve is probably best known since it represents the normal probability distribution and it is related to the accumulation of rounding errors in long computations. No introduction to probability or statistics is possible without it. There are a few less known names of mathematicians that show up in the birth history of this curve.</p>
<p>
9. <em>The catenary</em>. This is the curve formed by a chain loosely hanging from its fixed extremes. It looks deceptively like a parabola, but it isn't and that has fooled some mathematicians of the past. It is of course a place to discuss also the other hyperbolic functions. This is one of the curves that has been used to shape bridges and arches. It is also the shape of the road on which one can smoothly drive with square wheels.</p>
<p>
10. <em>Elliptic curves</em>. These are the most complex curves of the book. They are related to Diophantine equations and they are most famous for their use in cryptography.</p>
<p>
It is clear that the variety of topics is very broad: form constructions with compass and straight edge to cryptography and from the foundations of mathematics to the design on fonts with Bézier curves and the Casteljau algorithm. There are also seven short appendices explaining some preliminaries or expanding on some topics. However the first appendix is a surprise. On one of the very first pages of the book (page ii, before the title page) there are two 13 × 41 blocks of decimal digits or a number <em>N</em> of over 500 digits spread over 13 lines. No reference, no explanation. The explanation comes in the first appendix. It shows a complicated formula whose main ingredient is a modulo 2 formula for an expression depending on <em>x</em> and <em>y</em>. It thus gives a 0 or 1 depending on <em>x</em> and <em>y</em> which are assumed within certain bounds. The bounds for <em>y</em> depend on a number <em>N</em>, It turns out that it describes the pixel values within a rectangle of a page that will reproduce a pixelated image of the formula on a 106 × 17 pixel grid. Thus the <em>N</em> is the decimal representation of the binary number with 106 × 17 = 1802 bits giving the bit pattern of the pixmap one wants to generate. The two blocks at the beginning of the book give the two <em>N</em> values needed to reproduce the title of this book in pixel-form. The idea is from a 2001 paper by computer scientist Jeff Tupper.<br />
A few pages further at the beginning of the book on page vi shortly after the title page, there is a mathematical doodle with nine wild curves symmetrically arranged in a 3 x 3 matrix, and a trigonometric formula. No further explanation, hence leaving it as a puzzle and a challenge to tease the reader.</p>
<p>
There is more serious mathematics to be found in some other relatively long excursions in the chapters. Many of them are following some historical evolution of the problem. For example in the chapter on the normal distribution there is a lot of formula manipulation to move from a binomial distribution, via summing binomial coefficients and Bernoulli numbers, to finally arrive at the exponential expression. The discussion that a bijective map from the unit interval to the unit square cannot be continuous is illustrated by following the steps of the proof of continuity and non-differentiability as given for the Peano curve. The move from an parametrization of the Euler spiral to a simple one, parametrized by arc length, is fully explained and variations in the parametrization can produce very frivolous curves. And there are more not-so-trivial derivations in other chapters that can set the reader on a DIY path for further exploration. The fun items on page ii and vi will certainly trigger the interest of the mathematical puzzlers to find explanations or variations. The conclusion of the book is that $x^2+(\frac{5}{4}y-\sqrt{|x|})^2=1$ is the most important curve of all and it is indeed a lovely one.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In this book Julian Havil selects ten iconic curves to tell entertaining stories about mathematics. The stories are written for a broad audience, but still there is also a lot of juggling with formulas. Some basic background in mathematical calculus should however suffice.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/julian-havil" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">julian havil</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691180052 (hbk), 9780691197784 (ebk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 24.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">200</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/algebraic-and-complex-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebraic and Complex Geometry</a></li><li class="vocabulary-links field-item odd"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/books/hardcover/9780691180052/curves-for-the-mathematically-curious" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691180052/curves-for-the-mathematically-curious</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/53a04" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53A04</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/14h50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">14H50</a></li></ul></span>Mon, 02 Dec 2019 07:18:48 +0000Adhemar Bultheel50003 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/curves-mathematically-curious#commentsOpt Art
https://euro-math-soc.eu/review/opt-art
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Robert Bosch is a mathematician who has produced computer generated art that can be found on his <a href="http://www.dominoartwork.com/" target="_blank">website</a> and that he also presented at the Bridges mathart conferences. The pleasing optical effect of his graphical art is generated by solving some constrained optimization problem. The objective function is often simple, but the challenge is to design and formulate appropriate constraints.</p>
<p>One possible example is the construction of mosaics to represent a given image in a recognizable way where the elementary tiles of the mosaic are used like dots in halftone images or in pointillist paintings. Given is a restricted set of (small) tiles to choose from and some (large) picture. Put a relatively fine square grid over the picture and replace each square of the grid (let's call them pixels) with a tile from your set whose greyscale approximates the original greyscale of that square. The first example in this book is using flexible Truchet tiles to build the picture. A Truchet tile is a square tile with a diagonal dividing the square into a black and a white triangle. By introducing a parameter to move the midpoint of the diagonal along the other diagonal it is possible to generate tiles whose average greyscale ranges from 75 percent black to 75 percent white. Finding the optimal parameter for each tile to match the greyscale of the corresponding square pixel of the image is a large constrained optimization problem. Note that a Truchet tile has four rotationally symmetric siblings. This can be taken into account by defining larger composed tiles in which the unit tiles have a prescribed orientation. Obviously what is done for squares can be extended to any other shape of tile with rotational symmetry that fills the plane. Representing coloured instead of greyscale images can be an extra complication.</p>
<p>As an example of a constrained optimization problem some elementary introduction to the simplex method is given and it is illustrated how such a problem can be solved. It is not necessary to understand all the mathematics since computations are done by optimization software. One only needs to know how to feed the problem to the software. Most applications in the book use the Gurobi optimization software, except for the travelling salesman problems, for which the Concorde TSP package is used. Once the software is available, the previous idea of Truchet tiles matching the greyscale of a picture can be applied by using any dictionary of tiles with different average degrees of darkness. It becomes more challenging when one uses domino tiles, which for this purpose are double nines, that means that the number of dots on half the domino is not between zero and six as usual, but they have between zero and nine dots. Thus there are 55 domino tiles in a complete set ranging from double blank to double nine without repetition. The extra complications with respect to the previous problems are that these tiles are rectangular, and one has to decide how they are oriented to cover two square pixels of the original image. Moreover one can restrict the available dominoes to be only a finite number of complete domino sets that should be used completely. Thus there is only a finite number of copies of each tile. Two methods are explained to solve this problem.</p>
<p>Another goldmine to dig from is the travelling salesman problem (TSP) and all its applications. First it is explained what the optimization problem is and how to solve it approximately and how to avoid disjunct subtours. One should first select random points that are closer together where the image is dark and sparser where the image is light. That is called stippling. Again there are algorithms to solve this stippling problem and they will generate a set of points. This is based on MacQueen's unsupervised learning algorithm to detect clusters. Once the dots are chosen, they need to be connected by one and only one TSP tour, hence producing a piecewise linear Jordan curve that connects all the points in one non-intersecting tour that is of minimal length, at least approximately. Plotting this path with a black line, will show a graph that from a distance will again give a greyscale representation of the original image. It you do not like the piecewise linear curve, it is of course possible to modify the path slightly to turn it into a smooth curve. On the other hand, one could plot a white ribbon on a black background, weaving into a knot like on some Celtic or Arabic graphics. After stippling and finding the TSP contour using extra constraints that prevent points on the ribbon and the contour to "cut" the ribbon where it is not allowed, we are left with the ribbon as blank areas. Depending on symmetry conditions and the imposed constraints it is quite remarkable to detect what is the inside and what is the outside of the Jordan curve that represents the route of the salesman. Some parts of the ribbon are inside while others are outside, which is counter intuitive since from a distance the ribbon looks like in one piece.</p>
<p>Other abstract designs can be obtained by visualizing the knight's tour on a chess board. Another challenging problem is to design a nontrivial maze in such a way that within the outer boundary of the maze, all the squares should be visited exactly once to reach the center. The fine-touch is to design it in such a way that it shows some pleasing pattern. Under the title "Mosaics with side constraints" we find several other variations on the previous techniques that obey some extra restrictions to make the problem more challenging. A very nice idea is based on Conway's game of life. Also this game of life is played on a square grid where every square in the grid represents a cell. The game is a discrete dynamical system in the sense that a cell will live (a dot is present) or die (no dot) depending on some simple rules like the number of its neighbours that are alive. One may collect a number of cells in a larger composite tile that remains stationary under these dynamics and, depending on the number of living cells (dots) it will represent a greyscale tile, which can be used in the previous way to form a mosaic. However, more challenging is to generate composite tiles with cells that alternate between two states, but that do not interact with neighbours. Then one can obtain a dynamic image with a blinking effect.</p>
<p>It is clear that there are many ways to apply the idea of using optimization problems with carefully designed constraints to generate some nice pictures like Robert Bosch illustrates here. I love the book. If you want to start designing yourself, you will find it is far more challenging and probably far more addicting than any game that has been designed for you.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>It is explained how (constrained) optimization can be used to generate pleasing visual mathematical art with optical effects like mosaics, or nicely designed mazes.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/robert-bosch" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robert Bosch</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691164069 (hbk), 9780691164069 (ebk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 24.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">200</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/control-theory-and-optimization" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Control Theory and Optimization</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematical-aspects-computer-science" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Aspects of Computer Science</a></li><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/books/hardcover/9780691164069/opt-art" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691164069/opt-art</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/90c90" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">90C90</a></li></ul></span>Mon, 25 Nov 2019 09:46:26 +0000Adhemar Bultheel49950 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/opt-art#commentsDo dice play god?
https://euro-math-soc.eu/review/do-dice-play-god
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Stewart considers six ages of uncertainty. Clearly people have always been fascinated with the future and have tried in many ways to remove its inherent uncertainty. So there was an age of belief in external powers such as gods, oracles, horoscopes and reading the future from the bowels of a slaughtered goat.</p>
<p>With the development of mathematical instruments like Newton's laws of motion it was possible to predict the trajectories of the planets and to describe the dynamical behaviour of objects on earth. This triggered the conviction that we were living in a deterministic mechanical world organized like a clockwork and that everything was predictable provided that we could measure all initial conditions and all the parameters involved. Uncertainty still existed but only as the consequence of our inability to measure everything with sufficient precision.</p>
<p>Gambling is also something of all ages, but since around the sixteenth century, patterns were observed and ideas of frequencies of random events and the probability of how often they can be expected became useful tools for gamblers and they were successfully applied. Cardano, Fermat, Pascal, Huygens, and Jakob Bernoulli developed the basics of the theory and coin tossing and rolling dice became common instruments to generate random sequences. Observation errors were also considered to be a random phenomenon. Their analysis showed the bell shape of the normal distribution when sufficiently many are accumulated. Linear regression and least squares fitting were born. Quetelet started to apply this kind of analysis, originally used by astronomers and physicists, to social and other data, and this became the origin of expectations and an abstract, non-existing, "average person" was distilled from the data. This is how gradually statistics came about. But there were problems since probabilities seemed to change depending on prior knowledge leading to fallacies and paradoxes contradicting common sense. Bayes eventually formalized all this with formulas. Ardent discussions about Bayesian versus frequentist interpretations were the result. Still today many counter-intuitive results can be the origin of a lot of Fake News.</p>
<p>The fourth age started at the beginning of the twentieth century when mathematicians thought to have uncertainty well under control. But then nature forced quantum mechanical mysteries upon the experimental physicists, and uncertainty became an inherent property of the world we live in.</p>
<p>But also in mathematics, uncertainty was reintroduced when mathematicians started to model nonlinear dynamical systems. These are deterministic, but they can be supersensitive to tiny perturbations, a phenomenon popularized as the butterfly effect. Thus in this fifth age even deterministic systems became unpredictable.</p>
<p>The sixth age is the age we are living in today. Since uncertainty is not going to go away, mathematicians and scientists are trying to manage uncertainty. Sometimes we can even use it to our advantage, but there are still many open problems to solve.</p>
<p>It is clear that there is no crisp boundary between these periods. There are for example even today still people believing they can read the future from tea leaves or they think they get messages from "the other side". Therefore also Stewart cannot separate and treat these six ages in a strict chronological order. The eighteen chapters are more thematic and some topics may require to go way back in time to trace the origins. However, as we read on, we see how insights into uncertainty is growing and how we can bring it somewhat under control.</p>
<p>Stewart has written many books already and knows better than anyone else how to bring a story about mathematics to a broad audience. So the mathematics are painlessly made crystal clear. What is most interesting here are the side tracks. Among these I count the fact that physically throwing a dice or tossing a coin is not as random as one would expect. There are also these seemingly impossible results like if a family has two children and you know one of them is a girl, what is the probability that they have two girls, which is very different from the problem where you know that the eldest is a girl. Stewart also gives some confronting examples of people found guilty in court based on wrong statistics.</p>
<p>On a more theoretical side there is a good discussion of difficult concepts like entropy, information and the arrow of time. Of course quantum physics is more difficult and requires a rather extensive discussion. Entropy as well as quantum theory is still today subject to different interpretations and Stewart adds his own vision to the discussion. He is also explicit about Bell's theorem (1964) which shows that the EPR (Einstein-Podolsky-Rosen) paradox is inconsistent with the theory. Stewart explains some loopholes in Bell's theorem that have been raised and adds some of his own.</p>
<p>That known uncertainties are most influential on our modern society is illustrated with other examples. Strange attractors and the dynamics of weather forecasting are explained, and how climate change cannot be denied, and what mechanism is playing in the consequential disasters, the so called extreme events. We are still suffering from the consequences of the 2008 crisis in the bank sector, and there are heated discussions by people objecting against vaccination. These are two other important topical issues of our society today. Finally it is shown that for simulations it is important to generate random sequences, but it is a complicated problem to generate one that is "as random as can be". And how should randomness be measured anyway?</p>
<p>So this is far from a dull introduction to probability theory and statistics. It is a lively story with historical roots but with many relevant references to how managing uncertainty is important for our everyday life, as well as for the big challenges that our society is facing today.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Ian Stewart deals in his characteristic way with the history of uncertainty. This starts with the belief in gods, ghosts or horoscopes to deal with an uncertain future. Then probability and statistics were developed to measure the amount of uncertainties about the future as it is computed in simulations, but eventually it turns out that we live in an inherently uncertain world of quantum physics and chaotic dynamical systems where we have to learn to manage uncertainty and even employ it to our advantage where possible.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/ian-stewart" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ian stewart</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/profile-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">profile books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781781259436 (hbk), 9781782834014 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£20 (hbk), £15.80 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">304</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links field-item even"><a href="/imu/probability-and-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Probability and Statistics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://profilebooks.com/do-dice-play-god.html" title="Link to web page">https://profilebooks.com/do-dice-play-god.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/60c05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">60C05</a></li></ul></span>Mon, 25 Nov 2019 08:52:18 +0000Adhemar Bultheel49945 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/do-dice-play-god#commentsPatently Mathematical
https://euro-math-soc.eu/review/patently-mathematical
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Jeff Suzuki is a mathematician teaching at Brooklyn College who has written two books about mathematics in an historical context, but in his previous one he shifted gear and he wrote about the mathematics as used in the US Constitution. That is quite revealing since it is well known that politicians and lawyers are usually not the most skilled mathematicians. He is an active blogger and vlogger with a <a target="_blank" href="https://www.youtube.com/user/jeffsuzuki1">YouTube channel</a> about mathematics. He has the knack for explaining mathematical concepts in a remarkably simple way. This skill is again one of the major characteristics of the present book which is exploring the mathematics that is underlying approved patents in the US.</p>
<p>Mathematics is characterized by abstraction and implementing a formula or a method in some specific application is often a straightforward thing to do. Applying the theory in the context of a diversity of applications is pretty obvious to anyone "skilled in the art". When the abstract result is obtained, the mathematician is satisfied and then often looses interest in the implementation or the application. Patents are approved to encourage innovation, but it should not prevent the exploration of a broad range of related research. So, mathematics or an algorithm are often considered as general abstract ideas that cannot be patented, and only when it is implemented on a device for a specific application, a patent can be approved for that particular case. Mathematicians and researchers in general have a culture of publishing and sharing their results with the idea of advancing science. Companies on the other hand want to hide away their innovative results from competing companies by claiming their ownership in patents and preventing others to build on the same idea. But what if that idea is basically just mathematics? Unfortunately patent agents are not mathematicians and patents have been approved whose core element is basically a simple implementation of a mathematical idea or formula. As we are living in a world that is becoming more and more digitized, mathematics has penetrated the smallest pores of society, and therefore these issues become more and more relevant. Can mathematical innovations be the subject of a patent, hopefully not, but where is the boundary and under what conditions can a patent essentially based on a mathematical idea be approved? Suzuki gives many examples of patents based on a mathematical idea and gives in his epilogue some concluding recommendations. First the mathematical core of any patent should be clearly defined and it should be proved that it does what it claims to do. This should prevent claims that are too broad and prevent any other development in the area. Secondly, since in the US patent agents have to prove their expertise, Suzuki suggests that developing mathematics coursework should be allowed as a proof of mathematical expertise. This is kind of a strange conclusion but it might refer to his own position. Finally, also mathematicians should be allowed as patent agents, which, in the US, is currently restricted to engineers or scientists.</p>
<p>The book describes in a very accessible way all the mathematics that are behind many patents. It starts with several indexing systems in the early days of the Internet. These indices or keywords should allow to detect similar or related documents. Then of course along came Google linking the queries to the appropriate pages with ranking. That was their reason for success putting the most probable sites sought for on top of the (long) list. This was based on Pagerank, which is basically just computing an eigenvector of a large network matrix. Patents were approved to competing search engines and for methods to prevent link farms, spamming or other fraudulent practices or techniques that abuse or disrupt the system. What is done for text documents can also be done for images, which poses additional issues of the way in which pictures are represented, compressed, and transformed. The same person or scene can be represented by images that correspond to possibly different views or the image has been edited and manipulated. Facial recognition is certainly a well developed area. Copyright issues for images that are spreading over the Internet is another issue to be resolved.</p>
<p>In the very different area of match making companies and dating websites, remarkably similar problems arise. How to characterize a candidate, how to characterize his expectations, and how to find possible matches? This is almost like matching websites to a search query. An additional problem may be that the requirements for a match put forward by a person may not be exactly what he or she is really desiring. Suzuki investigates even whether these patented algorithms really work. No hard proof is available so far. The problem of formulating the proper questions in order to evaluate what is really intended is a subtle problem that teachers are faced with when they have to evaluate their students. That is an important problem for all kinds of rating systems. That can be educational platforms but it is similarly important for e-commerce and advertisement. For example in e-learning, the evaluation by multiple choice exams can be deceptive, or the kind of question asked may not really test the skill of a student or her understanding of a topic that one intended to test. Oral interviews can mitigate that, but computers and automation through algorithms is so much faster and objective. But don't forget that these are also very stupid and just follow the prescribed rules, and these may not always be the rules that were intended. The math underlying all these companies may not do what is claimed in their patent applications.</p>
<p>From this point on, the applications described by Suzuki become a bit more technical, but the mathematics are still explained in the same easily understandable way. How can we measure the strength of a password, and how to defend against eavesdropping? Here cryptography is an important tool, but that may not completely solve the problem of authentication or the related problem of how to prevent giving away our identity. We can be identified by our way of touching the keyboard, or by our surfing behaviour traced with cookies, all highly desired data for advertising, spamming, or phishing. Other data are collected about how we are digitally connected. This can be used to propagate an idea or a product in a network just like a virus spreading in an epidemy. This requires an analysis of a network graph. Optimization problems with constraints in large networks raises combinatorial problems that can only be solved with heuristics like simulated annealing.</p>
<p>Compression techniques of images (jpeg, DCT, wavelets), encoding of bit sequences (Huffman coding), fractals (e.g. fractal antennas) and space filling curves, cellular automata are all explained with simple examples. But also RSA and other crypto systems are illustrated for simple cases. These require more advanced mathematical techniques like modulo calculus, prime number factorization, discrete logarithm, Chinese remainder theorem, elliptic curves,..., but traditional techniques are challenged by quantum computing. It will not be a surprise that all these essentially mathematical techniques have been encapsulated in some patent.</p>
<p>This book illustrates why Suzuki has mixed feelings towards patents. There are a lot of mathematical ideas that can potentially be turned into a commercial patent, but at the same time there is the fear that a patent may kill the development and use of mathematics in a mathematically similar, although seemingly a quite different application.This is an important issue to be considered in an increasingly automated society. This is an important message and basically a political problem. What impressed me most in this book is the painless simplicity used by Suzuki to explain all these mathematics. Some illustrations and very few simple formulas suffice to communicate the mathematical ideas to inexperienced readers. This simplicity is of course an essential requirement if he wants to bring his message across to the politicians.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Mathematical formulas or pure algorithms cannot be patented. They need to be implemented in the framework of some application. But there are often simple mathematical ideas that form the heart and soul of an application or implementation that has been patented and that patent was the start of some very successful billion dollar companies. This book is a very readable introduction to the mathematics that are implemented in many approved US patents.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/jeff-suzui" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jeff Suzui</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/john-hopkins-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John Hopkins University Press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781421427058 (hbk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 34.95 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">296</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematical-aspects-computer-science" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Aspects of Computer Science</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://jhupbooks.press.jhu.edu/title/patently-mathematical" title="Link to web page">https://jhupbooks.press.jhu.edu/title/patently-mathematical</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li></ul></span>Mon, 25 Nov 2019 08:46:14 +0000Adhemar Bultheel49944 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/patently-mathematical#commentsMath Art
https://euro-math-soc.eu/review/math-art
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Mathematicians consider some mathematics to be beautiful, and there has indeed been scientific research measuring that mathematicians showed increasing brain activity in the frontal cortex when seeing mathematical formulas. This brain activity is similar to what is observed when people see a beautiful painting or listen to music. So there must be some link between mathematics and art. Several mathematicians are known to be also great musicians of produce visual art and most mathematicians have a soft spot for a certain kind of visual works of art that have some mathematical flavour. The <a href="http://bridgesmathart.org/" target="_blank">Bridges Organization</a> promotes the interaction between mathematics and visual art, music, architecture, and it organizes an annual conference around these ideas.</p>
<p>
Several books are available in which mathematics is linked to art. Some are coffee table books, mainly consisting of pictures, others are philosophical essays with some illustrations. This beautifully illustrated book by Ornes is about the "mathematical art" of nineteen contemporary visual artists. Its format is a careful balance between a sketch of the artist, a short discussion of his or her work, and an easily accessible introduction to the "mathematics behind the art". Some of the artists were once represented at these Bridges conferences and some are at the MoMath museum. For obvious reasons Ornes is mainly interested in visually pleasing work, with a strong mathematical background. Most of the artists are still living and Ornes is quoting some of them, which shows that he has interviewed or at least spoken with these artists.</p>
<p>
Divided into four parts, Ornes discusses 19 artists and their work. A telegraphic survey of the contents and the list of artists is given at the end of this review. Some of the artists are professional mathematicians, but others are just inspired by mathematics. The selection of the art is quite diverse. It can be monumental sculptures, weaving, computer generated curves, quilts, 3D-printed objects, wood carving, or crocheting. The work presented is selected to serve the purpose of this book. The artists can have other work of a quite different nature, or it can be early work and they may have moved more recently to a different kind of work. The appendices about the mathematics that served as an inspiration is diverse as well. There is pi and phi (the golden ratio), the Fibonacci numbers and primes, the Pythagoras theorem, set theory and infinity, geometry with classical Platonic and Archimedian solids, fractals and non-Euclidean geometry, topology and the Moebius band, space filling curves and tilings, computer science with complexity theory, algebra with symmetries and groups, and more. An appendix is linked to one artist, but there are cross references to other artists as well. Clearly the selection of topics and artists is very diverse, but this is only a very small section from a vast domain showing a growing interest for this kind of interaction between mathematics and art.</p>
<p>
The size of the book is nearly square (9 x 9.5 inches) and it is printed on glossy paper. So it can serve as a coffee table book but it has more to read than it has to see. The cover is black with a white design by Bathsheba Grossman. I could not find the reference in the book for the cover picture (although all other pictures are properly credited). The picture is actually a dodecahedron based design for a <a href="https://www.materialise.com/en/mgx/collection/quin-mgx">lamp</a> that is 3D printed by Materialise. It is also an illustration on Grossman's <a href="https://en.wikipedia.org/wiki/Bathsheba_Grossman" target="_blank">Wikipedia page</a> (3 Sept 2019). Grossman has also a Klein bottle opener, i.e., an operational bottle opener in the shape of a Klein bottle.</p>
<p>
I like the book very much. Unlike some other popularizing math books, it literally illustrates the beauty of mathematics, and makes this beauty accessible to non-mathematicians. Hopefully they will be triggered by the beauty of the pictures, to also read the mathematical appendices, which are written at a level that can be read and understood by anyone.</p>
<p>
To conclude, a quick summary of the 19 cases that are collected in four parts.</p>
<ul>
<li>
Part 1: <em>Making sense of the universe</em>.
<ul>
<li>
The art of pi - <em>John Sims</em>, who among other work, produces quilts like coloured QR codes where colours are defined by the digits of pi.</li>
<li>
Geometry in motion - <em>John Edmark</em> designs objects that require dynamics, and here one should consult <a href="https://www.johnedmark.com" target="_blank">his website</a> to understand and appreciate his work. The mathematics here deals with the Fibonacci numbers and the golden section.</li>
<li>
The proof is in the painting - <em>Crockett Johnson</em> has paintings that are inspired by graphical proofs of the Pythagoras theorem.</li>
<li>
One to one to infinity - <em>Dorothea Rockburne</em> produces abstract art, sculptures and installations, that draw inspiration from set theory. The mathematical appendix discusses set theory and different orders of infinity and gives a proof that there are infinitely many primes.</li>
<li>
The many faces of geometry - <em>George Hart</em> makes sculptures by weaving several identical components together that shapes Platonic solids. The mathematics is about regular and classical polyhedra and their stellations.</li>
</ul>
</li>
<li>
Part 2: <em>Stranger shapes</em>
<ul>
<li>
Space and beyond - <em>Bathsheba Grossman</em> makes sculptures that are periodic minimal surfaces or projections of the 120-cell in 3D space.</li>
<li>
The consequences of never choosing - <em>Helaman Ferguson</em> has monumental sculptures like an umbilic torus decorated with a Peano space filling curve. This and other space filling curves are discussed in the appendix.</li>
<li>
The tangled, torturous universe of fractals - <em>Robert Fathauer</em> produces fractal organic sculptures. Fractals are introduced in the appendix but is continued in the next case.</li>
<li>
The mystical and the mathematical - <em>Melina Green</em> focusses on the Mandelbrot set and generates an image of the set that suggests the shape of a Buddha.</li>
<li>
The equations of nature - <em>David Bachman</em> is a topologist and his art was originally the result of describing nature by equations and then generate artificial objects that look very natural. More recently his work visualizes more abstract ideas. The appendix is discussing topology.</li>
</ul>
</li>
<li>
Part 3: <em>Journeys</em>
<ul>
<li>
The wandering mathematician - <em>Robert Bosch</em> produces a piecewise linear Jordan curve that is denser at some places which, from a distance, gives the impression of a grey-scale reproduction of for example the Mona Lisa of whatever other image one cares to choose. The construction of the curve is based on a traveling salesman algorithm which is discussed in the appendix together with the P versus NP problem.</li>
<li>
The curves in the machine - <em>Anita Chowdry</em> is inspired by the Lissajous curves and produces some steampunk instrument to generate such complex curves.</li>
<li>
The algorithms of art - <em>Roman Verostko</em> (born in 1929) has embraced the first computers and designed algorithms to produce graphical art. The appendix discusses some elements from complexity theory and quantum computing.</li>
<li>
Projections - <em>Henry Segerman</em> has work inspired by stereographic projection, producing a Riemann sphere that is the projection on the sphere of for example a regular grid in the plane.</li>
</ul>
</li>
<li>
Part 4: <em>(near) Impossibilities</em>
<ul>
<li>
Following yarn beyond Euclid - <em>Daina Taimina</em> is known for her crochet work representing hyperbolic geometry. The appendix explains and illustrates rather well hyperbolic geometry with the Poincaré disk or half plane models.</li>
<li>
Bounding infinities - <em>Frank Farris</em> produces symmetric images and transitions in wall paper groups using deformed photographs as a stamp. Some of his work is discussed in <a href="/review/creating-symmetry-artful-mathematics-wallpaper-patterns" target="_blank"> <em>Creating symmetry</em></a>.</li>
<li>
Connections - <em>Carlo Séquin</em> is a computer scientist who produces complex large mathematically inspired sculptures. The appendix discusses symmetry and group theory.</li>
<li>
Math and the woodcarver's magic - <em>Bjarne Jespersen</em> is a wood carver who produces a wooden object where a sphere is capture inside a polyhedral structure that, unable to take it out or put it in. The appendix is about tessellations that cover the plane.</li>
<li>
The possibilities - <em>Eva Knoll</em> uses many different media to express herself among which weaving where some relative prime repetition of patterns creates some extra pattern on top of the underlying one. The appendix gives a discussion of algebra and all its different meanings in mathematics.</li>
</ul>
</li>
</ul>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a richly illustrated book discussing the relation between mathematics and art by describing the work of 19 contemporary visual artists and explaining the mathematics that is behind their artwork.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/stephen-ornes" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Stephen Ornes</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/sterling-publishing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Sterling Publishing</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1454930440 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">$24.95 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">208</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.sterlingpublishing.com/9781454930440/" title="Link to web page">http://www.sterlingpublishing.com/9781454930440/</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Mon, 09 Sep 2019 14:07:48 +0000Adhemar Bultheel49707 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/math-art#comments