Book reviews
https://euro-math-soc.eu/book-reviews
Book reviews published on the European Mathematical Society websiteenDark Data
https://euro-math-soc.eu/review/dark-data
<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>Dark matter and dark energy in cosmology is the matter and energy that cannot be directly observed with the techniques currently available, but we know that it must be there since it is the only explanation for the data that we observe and that cannot be explained by all the measurable matter and energy. This may be our first insight into the existence of dark data. Currently we live in a period of big data thanks to computers and the World Wide Web (part of if is called dark as well) and there are many different ways to collect and process these data. However there are also many different ways in which our analysis may lead to the wrong conclusions because part of the data are missing or wrong, or "dark" as Hand calls it. These dark data can exist for many different reasons. This book is taking a closer look at the phenomenon. Countless examples are described in the book (mostly for UK data). What type of obfuscations can darken our data? Why are some datasets dark? What are the consequences? What could possibly be done to remedy the situation?</p>
<p>Hand starts with a kind of taxonomy of dark data. In what kind of situations are we dealing with dark data? He describes, with many examples, fifteen different phenomena that can lead to dark data. Some are quite obvious like missing data that we know are missing (known unknowns), but there might also be data missing that we are not aware of (unknown unknowns). Some data are intentionally wrong (falsification) or unintentionally (over-simplification or rounding). Conclusions may be obtained for a whole population or over a larger period of time based on data that were only collected for part of the population or were only valid at a certain moment in time (extrapolation), etc. Quite often, the data are wrong or misused for more than one reason.</p>
<p>There is not a formal definition in this book, but nevertheless, using his examples, Hand explains what the different types of dark data are and how they come about, and identifies some of the concepts that he uses throughout the book. For example dark data caused by "self-selection" refers to the fact that data are corrupted because some participants, invited for and online poll, decide not to participate, or prefer not to answer some of the questions. There are problems of designing the sample (even a sample can be big data, in any case the data collected should represent the whole population for which the conclusion is supposed to hold), one has to be careful not to miss what really matters (like causality between data used and the conclusion derived), data can be corrupted by human errors, by summarising or simplifying or rounding the data. People can manipulate data in a creative way (like tax evasion) or corrupt data by deliberately feeding false data (criminal activity, insurance fraud).</p>
<p>Hand also has a chapter on science and dark data, not only were scientists in the course of history tricked in their conclusions by dark data, some also contributed by falsifying published data intentionally, or they may have been biassed by a general belief or intuition. John Ioannidis threw the cat among the pigeons with his 2005 paper <a target="_blank" href="https://doi.org/10.1371/journal.pmed.0020124"><em>Why Most Published Research Findings Are False</em></a>. Reproducibility was recognised as a major problem and research institutions are demanding data management plans in funding applications. Data management has grown into big business interfering with problems of privacy and GDPR. Distinguishing truth from reality, has become increasingly difficult in our digital world. Encryption, verification, identification, authentication, etc. can hardly keep up chasing creative fictionalisation. Artificial Intelligence algorithms based on machine-learning try to analyse the data that are too massive for humans to deal with, but even these machines can be led astray by dark data.</p>
<p>Thus it has become a major problem to recognise dark data and to know how to deal with it and avoid wrong conclusions. This is what Hand is discussing in the first two chapters of part II of the book (part II has a third chapter that is also the last chapter of the book which is summarising the taxonomy that was described in the early chapters). First we need identify why some data are missing. Here Hand considers three different types: it can be a random phenomenon but related to the missing data (UDD = Unseen Data Dependent) like some may be reluctant to give their BMI when it is high. Or missing data may depends on the data previously observed (SDD = Seen Data Dependent) like a BMI not given because it has increased since the last registered observation. Finally, data are missing but that does not depend in any way on the data observed (NDD = Not Data Dependent). Recognising the mechanism behind the missing data is important because it defines how one should deal with the data, for example on whether and how to complete the missing data or not. Effects of NDD and SDD can be be cured, but UDD is more difficult to deal with.</p>
<p>Dark data can also be beneficial if it is detected and if that leads to a reformulation of the question that we want to answer, or it may lead to strategic elimination of some data that would bias the result. To avoid dark data, it helps to randomise the sample and even hide data from the researcher (like not revealing who did and who did not get the placebo). An obvious way to fill up missing data is to use averages, but a somewhat strange advise is to fill up these data by simulation. It is a valid way to generate data in case of a simple model with known probability distribution, but when it involves a complicated model, then these models are simplifications of reality built upon observations that may involve dark data. Similarly machine-learning techniques involve massive analysis of data that may be corrupted. In these cases it can only be hoped that over the repeated simulations or over the whole learning process the wrong data are not systematic and are averaged out over the iteration. Or one may apply techniques such as boosting and bootstrapping to reduce bias. Bayesian statistics helps to test hypotheses and thus confirm or refute intuitive assumptions. Cryptography can help to make data anonymous so that people are more willing to provide correct data or to prevent the introduction of false data by fake persons. It may even help to make some data deliberately dark, not making them available to users but still using them in computations.</p>
<p>As mentioned above, the description is an exploration of a major problem in data analysis with an attempt of classification, analysing causes, mechanisms, and to some extent also suggest mitigations. However most of the book consists of examples and particular cases to clearly explain the ideas. Nowhere, however is there a concrete or general well defined statistical, mathematical, or algorithmic solution given. This is mainly a wake-up call that clearly points at a major problem that any scientist has to be aware of and that he or she should think about how to deal with. Certainly statisticians, applied mathematicians, computer scientists, but in fact anyone dealing with data (big or not) should be well aware of the "darkness" of their data.</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 a description of how important it can be that in our treatment of data, some of them are missing, or fake. Conclusions derived from these corrupted data can be biassed or wrong. How do we recognize the dark data? How can we deal with the phenomenon? These are answers that Hand deals with in this book. The approach is mainly descriptive with an abundance of examples, mainly from data related to the UK situation. Suggestions are given, but no concrete precise or detailed mathematical or statistical analysis or algorithms is discussed in detail.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/david-j-hand" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David J. Hand</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">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">9780691182377 (hbk), 9780691198859 (ebk), 9780691199184 (abk) </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">£ 26.00 (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">344</div></div></div><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/9780691182377/dark-data" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691182377/dark-data</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/62-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62 Statistics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/62d99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">62D99</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/68p99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68P99</a></li>
<li class="field-item odd"><a href="/msc-full/68t09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68T09</a></li>
</ul>
</span>
Mon, 03 Feb 2020 12:29:51 +0000adhemar50375 at https://euro-math-soc.euEuler's Gem
https://euro-math-soc.eu/review/eulers-gem-0
<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>
The fact that the book is reprinted in its original version as a volume of the <em>Princeton Science Library</em> is a quality label as such. For completeness, I should here also mention the subtitle: "The polyhedron formula and the birth of topology". This rules out that by the "gem" is not meant the other famous Euler formula $e^{i\pi}+1=0$, but that it concerns the polyhedral formula <em>V</em>−<em>E</em>+<em>F</em>=2 (the letters stand for Vertices, Edges and Faces).</p>
<p>
A short introduction serves as a teaser for the reader and explains the kind of problems that will be discussed in the sequel. A reasonable way to start the full story is to give a biography of Euler. He is recognized as one of the greatest mathematical minds of all times. Next, the reader is surprised by the fact that it is not so straightforward to define a polyhedron, or at least to identify the ones that one wants to focus on. So the reader is warped back to the Greek origin when the five Platonic solids (discussed in Book XIII of Euclid's Elements) and the thirteen Archimedean solids were studied. We then have to take a leap to the Renaissance of the fifteenth century before the polyhedra and the Greek knowledge was rediscovered. Kepler later built a whole world view and a solar system on polyhedral shapes. And then came Euler, who detected his above mentioned "gem". It is so simple an observation that it comes as a real surprise that, as far as we know, it was missed by everyone so far. Some explanation may be that previously one concentrated on the vertices and the faces (or the solid angles), but Euler also considered the edges as essential components of a polyhedron. Richeson reproduces Euler's proof (1750-51), but the formula does not hold for all (regular) polyhedra. When does it hold and when does it not? The reader is expecting to read the answer in the next chapter, but then Richeson surprises again by revealing that Descartes may have been the first one to have discovered the formula because a similar relation was found posthumously in his notes (notes that were miraculously saved from oblivion). However, a complete rigorous proof was only given by Legendre (in 1794), a proof that Richeson also explains later in the book.</p>
<p>
From the story told so far, there are already many historical mathematicians involved and Richeson gives every time some short biographical sketch to situate him (so far only men) as a person who existed and lived a life of his own. It is not just some abstract name used to identify a result. Note also the way in which Richeson builds up his story. He takes the reader along to think about what polyhedra are, for what polyhedra does the formula hold, and how could it be adopted to hold in more general case? Euler and his proof are some kind of a climax, but then Descartes shows up as an unexpected twist, and Legendre's proof is not based on properties of planar faces but (another surprise) requires geometry on a sphere with geodesics. This shows how well the book is written and how Richeson manages to fascinate the reader, and make her curious about what is coming up next.</p>
<p>
And next chapter is again some kind of a surprise because it introduces the problem of the Bridges of Köningsburg and how Euler solved the problem which is considered as the origin of graph theory. Not so surprising though if you know that a graph consists of vertices and edges. Cauchy uses this idea by projecting a polyhedron on a plane, giving a plane graph that can be analysed to prove Euler's formula. Now Richeson's story takes off into graph theory and applications: recreational mathematics (the game of sprouts and Brussels sprouts invented by Conway), the four colour problem for planar maps (and other graph colouring problems). Graph based proofs for the polyhedra formula and generalisations concludes the graph theory subject.</p>
<p>
Next Richeson embarks on proper topology as the rubber sheet version of the usual geometry. This requires some new concepts and a classification of all surfaces. Therefore one needs to know when a surface is or is not homeomorphic to another and thus are topologically the same. For example, a torus is a sphere with a handle, a Möbius band is the same as a cross cap, and the projective plane is a sphere with a cross cap. Classification is connected to the definition of the Euler characteristic (or Euler number as Richeson calls it). Make a finite partition of the surface and count the "rubber versions" of vertices, edges and faces, then the Euler formula gives the characteristic <em>χ</em></p>
<p>
which is an invariant for the surface (2−2<em>g</em> for a sphere with <em>g</em> handles, and 2−<em>c</em> for a sphere with <em>c</em></p>
<p>
cross caps). This characteristic and the orientability of the surface allows some classification as started by Riemann but only completed in 1907 by Dehn and Heegaard.</p>
<p>
I have to say that, although now Richeson is still explaining things at an introductory level of topology, (and continues to do so), it will take a more persistent and motivated topological layman to follow in pace and read on. We arrived now at about two thirds of the main text of the book and the mathematical level is not decreasing for the last part. It continues with knots and Seifert surfaces (whose boundary is a knot or link), the hairy ball theorem for vector fields on a sphere and more generally the Poincaré-Hopf theorem on surfaces with boundary, Brouwer fixed point theorem, the angle excess theorem for a surface, the Gauss-Bonnet theorems about the total curvature of an orientable surface, Betti numbers, and Richeson ends with an epilogue about the Poincaré conjecture. All of these are nicely presented in a smooth and logical succession by Richeson, but they are too technical to be discussed at the level of this review. However, for example an undergraduate mathematics student should not have a serious problem to read on.</p>
<p>
Everything in the book is very well illustrated with insightful graphics that, together with the text, make results almost like being obvious. Richeson also adds in an appendix building patterns that can be used to make paper models of polyhedra, of a (square and edgy, yet topologically a perfect) torus and even (the paper realization that looks like) a Klein bottle, or a projective plane. In the text he also gives advise on how to prepare the liquid to make soap-bubble models. These are aids to help visualising the surfaces if the graphics of the text do not suffice. There is a long list of papers referred to in the text, but also an appendix with an annotated survey of recommended literature.</p>
<p>
Except for an additional preface by the author, the book is the unaltered reprint of the original version of 2009. Thus for example the facts of Martin Gardner passing away in 2010 and Perelman refusing the Millennium Prize for proving the Poincaré conjecture were still unknown in 2009. Although the latter was to be expected since he had already declined the Fields Medal in 2006 and an EMS Prize.</p>
<p>
The first half of the book can be considered as a popular science book on a mathematical subject written for everyone. Depending on the motivation or knowledge of the reader this might or might not include the part on graph theory. Once Richeson dives deeper into topology, it becomes more a popular science book for the mathematics student of at least an amateur mathematician. People who are interested in this book may also be interested in a more recent book by Richeson that has also been reviewed here <a href="/review/tales-impossibility" target="_blank"><em>Tales of Impossibility</em></a>.</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 the book of 2009, that is now reprinted in the <em>Princeton Science Library</em>. Richeson gives an account of 2500 year of mathematical history that runs from the Greek's approach to regular polyhedra to the modern problems of topology, all centred around Euler's polyhedral formula <em>V</em>−<em>E</em>+<em>F</em>=2 and its generalisations.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/david-s-richeson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David S. Richeson</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">9780691191379 (pbk), 9780691191997 (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">£ 14.99 (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">336</div></div></div><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/paperback/9780691191379/eulers-gem" title="Link to web page">https://press.princeton.edu/books/paperback/9780691191379/eulers-gem</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li>
<li class="field-item odd"><a href="/imu/topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Topology</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/55-algebraic-topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">55 Algebraic topology</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/55-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">55-03</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/52-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52-03</a></li>
<li class="field-item odd"><a href="/msc-full/54-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">54-03</a></li>
<li class="field-item even"><a href="/msc-full/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li>
<li class="field-item odd"><a href="/msc-full/51m20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51m20</a></li>
</ul>
</span>
Fri, 31 Jan 2020 10:49:42 +0000adhemar50363 at https://euro-math-soc.euThe 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="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/mircea-pitici" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mircea Pitici</a></li>
<li class="field-item odd"><a href="/author/ed-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(ed.)</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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><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="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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 +0000adhemar50361 at https://euro-math-soc.euCalculus Reordered. A History of the Big Ideas
https://euro-math-soc.eu/review/calculus-reordered-history-big-ideas
<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"></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">Ángeles Prieto</div></div></div><div class="field field-name-field-review-appendix field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://euro-math-soc.eu/sites/default/files/book-review/BressoudCalculusReordered.pdf" type="application/pdf; length=66685">BressoudCalculusReordered.pdf</a></span></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 very-well written book can be strongly recommended as a resource for instructors. Also students will benefit greatly from Bressoud’s journey through centuries to explain how calculus evolved into its current structure.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/david-m-bressoud" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David M. Bressoud</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">978-0-691-18131-8</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">$29.95</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">242</div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li>
<li class="field-item odd"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History 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/9780691181318/calculus-reordered" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691181318/calculus-reordered</a></div></div></div>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/26-real-functions" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26 Real functions</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/26a-functions-one-variable" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26A Functions of one variable</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li>
</ul>
</span>
Mon, 27 Jan 2020 17:04:58 +0000Ángeles Prieto50335 at https://euro-math-soc.euA 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="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="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="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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><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="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A30</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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 +0000adhemar50115 at https://euro-math-soc.euPerspective and Projective Geometry
https://euro-math-soc.eu/review/perspective-and-projective-geometry
<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 a workbook to learn the techniques of perspective drawing and the theory of perspective and projective geometry. The exercises range from very practical to proving theorems, but it is essentially based on experiments and discovering the theory by practicing on the examples.</p>
<p>
The book starts with an example of a very practical experiment. One person (the director) has to stand immobile with one eye closed in front of a big window at a few meters distance. He or she is looking at the landscape or the buildings outside. The view should have preferably many straight lines. The window is used as a canvas, on which the outside world has to be projected and the director has to instruct other students (the artists) to fix waste tape on the window where the director sees the projected straight lines of the outside world. This is a way to detect how a 3D world is represented on a 2D window-canvas. The first module has several questions that can be answered in the blank space that is left open for that purpose or one has to check true/false answers or choose from multiple choice possibilities. All the pages of the book are perforated so that they can be torn out and handed in for correction or feedback. There is also a section for homework with more questions to answer (marked with circled E: Ⓔ) and art assignments like make drawings or take pictures (marked with a triangulated A) or more theoretical exercises related to theorems and proofs (marked with a squared P). In an appendix to this first module some explanation is given that should lead to a definition of sketches in n-point perspective, which is what the subsequent modules will work to on a more theoretical basis.</p>
<p>
This first module described above is an example of how all the 13 modules are organized, although some are more theoretical, and none of the others have an appendix. Some of the questions are incomplete and the student has to guess what the question is. That has of course been prepared in previous questions, but still it may be a problem to correctly complete the sentence, which makes it impossible to continue with the next questions. Therefore, I believe this is not a workbook for self-study, a teacher should be guiding the process but it remains a challenge for the student whose responsibility is to discover the proper way to go or to detect the concepts and the theorems that support the constructions.</p>
<p>
To illustrate how one moves from the window taping experiment to the theory, we note that in section 2, one has to analyse and complete the graphic representation of a 3-dimensional construction of two thick tiles on top of each other in the form of the letter T, drawn in (a 1-point) perspective. And then module three is stuffing up the theory with definitions and properties of points, lines, segments, planes, and module four is introducing geometry in $\mathbb{R}^2$ and $\mathbb{R}^3$ with the announcement of Ceva's and Menelaus's theorems. Module 5 extends the Euclidean space in (2 and 3 dimensions) by introducing ideal points, lines and planes, that are essentially the points at infinity. This gives the extended spaces $\mathbb{E}^2$ and $\mathbb{E}^3$. This seems to complicate things, but it actually simplifies life since no exception has to be made for these ideal objects which is the whole idea of projective geometry. To formalize the perspective drawings, meshes and maps are defined in $\mathbb{E}^3$</p>
<p>
. The latter allow to project a 3D scene onto a 2D plane and it can be used for example to correctly project equispaced segments (say vertical poles of a fence or a square tiling of a floor) onto non-equispaced distances on the canvas or how to correctly draw a poster on a wall that is represented on a canvas in perspective.</p>
<p>
The theory continues as above, but always in connection with practical problems related to the plotting of 3D scenes on a 2D canvas. The next issue is Desargues theorem (two triangles are in line perspective if and only if they are in point perspective). It is formulated in terms of meshes in $\mathbb{E}^3$ and a proof is to be derived. On the other hand, this module also introduces exercises in GeoGebra (a free interactive software package for geometry, algebra, and other computations). An elementary introduction to using GeoGebra is added in an appendix. Now the student should know enough about projective geometry to move objects around with concepts like (perspective) collineations, homologies and how these connect with harmonic sets. One may now experiment by moving points in a GeoGebra plot.</p>
<p>
Herewith the modules move somewhat towards numerics. For example the position of the designer in the taped window example could be found by someone who moves herself into a position where the scene outside aligns with the tapes on the window. But now, at this stage of the book, it is possible to compute the distance of the viewer's eye to the canvas from the perspective drawing. Or it should be possible to derive from the 1-point perspective projection of a box whether it is a cube or not. Module 10 allows to draw boxes in 2-point perspective, respecting the actual distances and module 11 catches the cross ratio of four points as a numerical invariant allowing to draw lines in perspective that are equidistant in reality, a problem that was also previously considered in module 6. Another invariant is a more complex $h$-expression relating distances between 8 points in a rectangular configuration. This is named after and proved by Howard Eves (1911-2004 — the authors give wrong dates 1913-2000). Also the Casey angle of 4 collinear points is another invariant as proved by M. Frantz and named after John Casey (1820-1891) (the theorem here is not to be confused with what is generally known as Casey's theorem).</p>
<p>
The last two modules introduce the Cartesian coordinate system, projective coordinates and linear algebra and even some topological concepts like the Möbius band and the possible shape of $\mathbb{E}^2$ (the four colour theorem does not hold for this space since six colours are necessary in general).</p>
<p>
Every module starts with a one-page graphic, which is to be worked on as one progresses in the module, and thus it is somehow the target and motivation for the module that is coming up. For reference, the main definitions and theorems are summarized in an appendix. A last appendix deals with writing mathematical prose, and that includes style, punctuation, use of formulas and words, etc. This is of course important for any student who has to learn to write mathematics, but even more so it will be important for students who follow the course with an artistic background, and who have not been exposed very often to mathematical texts. I assume however that they may find the abstract mathematics of this text a bit hard to do.</p>
<p>
The book is a nice mixture of mathematics (with rather abstract concepts and proofs), but with practical introductions of these concepts and interesting applications in art as well as in practical situations. Moreover, it gives an introduction to the computer system GeoGebra, and a hint towards linear algebra, analytic geometry and topology. There are excursions into historical aspects, music, photography, and many other tracks. But foremost it is a workbook with an extensive list of many different assignments ("cool" problems to use the language of the authors). More material related to these modules is available at the <a href="http://www.fumikofutamura.com/mathart" target="_blank">Futamura website</a>.</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 a workbook to instruct the techniques and the mathematics of perspective and projective geometry. It has many blank spaces where the student can write the answers to the questions and of course there are many graphics, partially set up, that he or she has to complete. The pages are perforated and can be torn out to be handed in for correction and feedback.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/annalisa-crannell" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Annalisa Crannell</a></li>
<li class="field-item odd"><a href="/author/marc-frantz" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Marc Frantz</a></li>
<li class="field-item even"><a href="/author/and-fumiko-futamura" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">and Fumiko Futamura</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">9780691196558 (hbk), 9780691196565 (pbk), 9780691197388 (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">£ 97.00 (hbk), £ 40.00 (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">296</div></div></div><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/9780691196558/perspective-and-projective-geometry" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691196558/perspective-and-projective-geometry</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/51-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51 Geometry</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/51-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-01</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/51n15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51N15</a></li>
<li class="field-item odd"><a href="/msc-full/51a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51A05</a></li>
<li class="field-item even"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li>
</ul>
</span>
Fri, 20 Dec 2019 14:54:13 +0000adhemar50114 at https://euro-math-soc.euRepublic of Numbers
https://euro-math-soc.eu/review/republic-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>
In twenty short biographical chapters it is sketched how the role of mathematics in the American society and its educational system has evolved from the early 19th till the late 20th century. That is one chapter per decade, but the life span of the individual mathematicians is of course wider: From Nathaniel Bowditch (1773-1838) to John Nash (1928-2015). In the 19th century, the US were expanding and fighting for independence. Importing slaves was gradually abolished which entailed a civil war between the Northern and Southern states. In the 20th century they participated in global conflicts and survived a cold war. Around 1800, there were only nine colonial colleges, (for white men only), and they mainly trained lawyers, physicians, and clergy. In rural regions teaching to read and write was for the lucky ones and it was forbidden to teach slaves. In the 1990's, there are numerous renowned universities and a regular educational system was established, with mathematics taking an important place at all levels of education. How did this come about? That is what Roberts is illustrating with this selection of 23 biographical sketches (some chapters treat two persons simultaneously). He did not take the leading mathematicians to illustrate the evolution (only few were famous) but there is a diversity of characters and people who were in some sense related to mathematics, and often they were involved in educational issues.</p>
<p>
Here are some names from the first of these two centuries. Simple calculations were sufficient for every day life in 1800, except for navigation which required some knowledge of celestial mechanics. Nathaniel Bowditch taught himself mathematics which he needed as a sailor and wrote a book on navigation and later translated work of Laplace. Sylvanus Taylor had some education when entering the military. Later he became the director of West Point, the US military academy that he modeled after the Ecole Polytechnique in Paris and whose alumni played an important role in professionalizing mathematics in other places. Abraham Lincoln did not become a mathematician, but in his youth, he maintained a scrap book with elementary mathematical problems. Only some of its pages have been recovered. Catherine Beecher and Joseph Ray were authors of popular math text books, and Daniel Hill was a popular educator at West Point. J.W. Gibbs became famous as a mathematical physicist with his work on thermodynamics. Charles Davis was a naval officer who supervised the computation of the <em>Nautical Almanac</em>. and was later superintendent of the Naval Observatory. After the civil war (1861-1865), the educational system became more tolerant for women, Christine Ladd was one of the first women to become a researcher at John Hopkins University. She fulfilled all the requirements for a PhD but it was only awarded 44 years later in 1927 when she turned 80. Kelly Miller is an example of an African American who attended the "black" Howard University, and wrote a math textbook and essays on popular mathematics. H. Hollorith, known from the punch cards named after him, was also founder of the Tabulating Machine Company, which later grew into IBM and E.H, Moore is a mathematician known for several things like the Moore-Penrose inverse. He had some students that became famous mathematicians: G. Birkhoff, L. Dickson, and O. Veblen. Those names bring us to the end of the 19th century, with data processing on the horizon and mathematics and mathematicians being imported from Europe on a larger scale raising mathematics to a higher level.</p>
<p>
The list of names from the 20th century is started with E.T. Bell, a popularizer of mathematics whose <em>Men of mathematics</em> became a classic. By this time, education had been formalized. Classes were split according to the age of the pupils, lessons were separated by a bell signal, and schools had a non-teaching management. The <em>Mathematical Association of America</em> (MAA) was established in 1915 as an offspring of the <em>American Mathematical Society</em> (AMS). The <em>National Council of Teachers of Mathematics</em> (NCTM) with the first president Charles Austin was founded in 1920 as a follow up for the <em>Men's Mathematics Club</em> of the greater Chicago area. Edwin B. Wilson was a student of Gibbs and became mainly involved with statistics. The couple Liliane and Hugh Lieber are known for their series of booklets popularizing math and science with text in free verse format for easy reading by Liliane (maiden name Rosanoff, an emigrate from Ukraine) and drawings by Hugh. Their best known title is <em>The education of T.C. MITS: what modern mathematics means to you</em>. T.C MITS stands for The Celebrated Man In The Street. With WW II, computers came into vision and Grace Hopper designed a computer language that was a precursor of what later became COBOL. Izaak Wirszup studied mathematics under Zygmund in Poland, and survived a Nazi concentration camp. Zygmund, who had escaped the Nazis, invited him to the US where Wirszup became mainly involved in math education. The 1960's was the period where African Americans were fighting racial segregation and Edgar L. Edwards, Jr., was one of the first black teachers at the University of Virginia. Also Joaquin Diaz, although an American citizen from Puerto Rico, was subject of racial discrimination because he was considered Hispanic, and non-American. As an applied mathematician working on fluid dynamics, he was involved in NACA (precursor of NASA). The <em>math wars</em> of the 1980's was the fight over traditional versus "new" mathematics that was abruptly introduced in the US, a reform supported by the NCTM. Frank Allen, who was a believer in the original ideas of <em>New Math</em>, and who had been involved in NCTM became an active polemicist in the debate. The last man in the row is John Nash whose life is well known because of the biography <em>A beautiful mind</em> by Sylvia Nasar and the eponymous film.</p>
<p>
This enumeration of names shows that Roberts is not focussing on mathematical research at university level, but rather at the historical evolution of mathematical education at a lower level, which is of course not independent of what happens at the universities. Why these names? I guess any list of names can be criticized, but I think Roberts chose a good mixture of sex and race, that somehow represents how political and social circumstances have influenced the mathematical education. In the beginning, navigation and the military interest were stimulations for doing mathematics. The military definitely remained to have an important influence and WW II has given a boost to the development of math and science in the US because of the many scientists that fled Europe for the Nazis, which made a <em>Space Race</em> possible during the <em>Cold War</em> period. The latter events are however more important at a research level, and that is not so present in this book. Nevertheless the USSR having Sputnik first is related to the forcing initiative to introduce the <em>New Math</em>.</p>
<p>
Roberts has assigned one particular year to every chapter. Each chapter starts with an epigraph and the description of a particular event that happened in that year to the person that is going to be discussed. That introduction takes only one to three pages and should serve as an appetizer for the longer biography that is following. There is some discussion of the mathematics but it is nowhere technical (no formulas), and there is a photo of each of the mathematicians discussed (except for Abraham Lincoln who is not a mathematician anyway). There are notes and references for what is mentioned in the text but no extensive list for further reading. The book is a very readable survey that will be of interest to any mathematician and non-mathematician alike, but maybe more so for those who are particularly interested in the history of math education.</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>
With 20 short biographical chapters, this book illustrates how the US evolved from the early 19th century with schools where children learned to read and write while mathematics was mainly of interest to navigators and astronomers to the end of the 20th century where mathematics had become a main ingredient at all levels of education.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/david-lindsay-roberts" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Lindsay Roberts</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">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">9781421433080 (hbk), 9781421433097 (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 29.95 (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">252</div></div></div><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/republic-numbers" title="Link to web page">https://jhupbooks.press.jhu.edu/title/republic-numbers</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li>
<li class="field-item odd"><a href="/msc-full/01a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a60</a></li>
<li class="field-item even"><a href="/msc-full/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li>
<li class="field-item odd"><a href="/msc-full/01a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A80</a></li>
</ul>
</span>
Fri, 20 Dec 2019 14:36:53 +0000adhemar50113 at https://euro-math-soc.euFUNdamental 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="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="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="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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><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="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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 +0000adhemar50058 at https://euro-math-soc.euCurves 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="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="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="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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><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="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field-item odd"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
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<li class="field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li>
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<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
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<li class="field-item even"><a href="/msc-full/53a04" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53A04</a></li>
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Mon, 02 Dec 2019 07:18:48 +0000adhemar50003 at https://euro-math-soc.euOpt 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="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="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="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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><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="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="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="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>
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Mon, 25 Nov 2019 09:46:26 +0000adhemar49950 at https://euro-math-soc.eu