Book reviews
http://euro-math-soc.eu/book-reviews
Book reviews published on the European Mathematical Society websiteenGreat Circle of Mysteries
http://euro-math-soc.eu/review/great-circle-mysteries
<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 role of mathematics in nature and why in sciences it is so effective in catching the laws that govern the phenomena that we observe, has been the subject of speculations and conjectures throughout human history. As more recently we were able to demystify some of the elementary building blocks that make up life and we also got some insight into the processes by which our brain functions, some circle is being closed. Indeed, it is with the help of mathematics that scientists were able to unravel the mysteries of life, and to model our brain while on the other hand mathematics is an abstract construction of the brain and the brain/mind defines the identity of a living organism. This vague circularity starts shimmering faintly through but it is still largely mystical and far from being understood.</p>
<p>
In 2011 the <em>Fondation Cartier pour l'art contemporain</em> organized an exhibition <em>A Beautiful Elsewhere</em> in Paris. Part of the exposition was a <em>Library of Mysteries</em>. This work was realized by David Lynch in collaboration with rock icon Patti Smith and the geometer Misha Gromov. Using quotes from great scientists, Lynch visualized the mysteries of time, space, matter, life, knowledge, and mathematics, presented subsequently as</p>
<p>
</p>
<ul><li>
the mystery of physical laws</li>
<li>
the mystery of life</li>
<li>
the mystery of the mind</li>
<li>
the mysterey of mathematics</li>
</ul><p>
</p>
<p>
An excellent report on this exhibition by Michael Harris was published in the <a href="http://www.ams.org/notices/201206/rtx120600822p.pdf" target="_blank">Notices of the AMS 2012 vol 59, no.6, p.822-826</a>.</p>
<p>
</p>
<p>
Gromov was asked to prepare this book as a "naive mathematician's version" of what Lynch had done in Paris, and this is how this book came about. If you know some of the films by David Lynch in which he creates his unearthly mysteries, then you know that they can take some shocking bends that lift the spectator out of reality wondering what had just happened. A sense of humour is not completely absent. It seems like an impossible task to translate this atmosphere into a book format. I was not able to attend the 2011 exhibition in Paris, but I know some of the work of David Lynch and with Harris' detailed report, I can imagine what it must have been like, and I assume that Gromov succeeds well in keeping the original ideas. After all he knows them very well because of his involvement in the 2011 project. This being said, you may not expect to read a linearly structured straightforward book. It uses different colours for (parts of) sentences and different fonts to give a precise meaning to words. Some would classify it as a painting with words and ideas or even poetry. It is full of quotes and ideas and it is seasoned with this this particular slight humorous undertone. Every section, and almost every sentence is an invitation to contemplate about its deeper meaning and the consequences. Some explanation is required, and a lot of these extras are included, not only in the text but also with several footnotes that can be found on each and every page. There are many illustrations, most are borrowed from the web, but their role is more to "give some air" to the text and make it lighter to work your way through. The hope is that the reader will play with the ideas and opinions and whatever is in between those two, so that he or she will experience the <em>Beautiful Elsewhere</em> called Mathematics.</p>
<p>
It is difficult to summarize the content because it is multi-branched like a fractal tree that floats on a stream of conciousness. A first part of some seventy pages is mainly a discussion of quotes that reflect ideas of scientists of all times who gave an opinion on science in general, on numbers, laws (physics and others), truth, life, evolution, the brain, and the mind. All these are elements in the chain that eventually will form the circle. The conclusion is that mysteries remain. We do not know much about how space/time/matter/energy is transformed into life/brain, how the latter brings about what we can classify as mind/thought. It seems that we cannot conceive these relations but by using mathematics. This closes a circle because we therefore need to understand how mathematics can be the result of this brain/mind/thought complex and mathematics is precisely the instrument that tells us something about space/time/matter/energy. Only in this last relation we know something as shown by the results of physicists. The mathematics(?) to describe the other connections/transformations/mechanisms are still unknown. This book wants to be a first attempt to throw a hook at these missing mathematics.</p>
<p>
The second part is called <em>Memorandum ergo</em> that one wants to complete spontaneously with the missing <em>sum</em>. The ideas/opinions/conjectures in the remaining 130 pages are analysing the finer structure of the components in the volatile circle of mysteries sketched above. For example, there is definitely a difference between brain and mind. The term "ergo" appears for the first time in the conjecture that there is something like an ergo-brain that is not accessible by introspection and that is responsible for unconscious thoughts. It contains structural patterns that we can recognize for example in natural language. The idea is that this ergo-brain is an instance of a wider ergo-system that hopefully can be analysed using mathematics and that eventually will also shed light on mathematics itself. This is the ergo-project and it requires to investigate all the components of this very complex system. The ergo-brain is alert to the unexpected and is bored by the ordinary, repetitive impulses. It is alert to the signals that are "interesting", not the ones that are "obvious" or "logical". It is responsible for a child learning a language or to read and write, or even walk. How do we learn things? It is not sufficient to understand the electrochemical system at the level of brain cells to explain how we attach a meaning to signals that we receive when seeing or hearing something. A strong ergo-brain is probably also responsible for children being gifted for mathematics or music or chess. It is responsible for goal-free learning, meaning that it is <em>not</em> the result of evolution which defines behaviour as maximizing the chance of survival. Evolution has a big hand in forming our ego-mind. The ego-mind is rational and intelligent. It plays by the rules and has common sense, while the ergo-brain wants to be free and wanders around always looking for the new and interesting. A cave-man with a super-ergo-brain would probably not survive, but it made Ramanujan fill up his notebooks with remarkable formulas. The ergo-system is responsible for our agility with our language, for finding pleasure in playful and "useless" activity like solving sudokus, for getting bright ideas, for progress in science and mathematics. The problem is that the ego-mind has no access to the ergo-system. Direct observation is impossible. Moreover it is not logical in the usual sense but requires some ergo-logic to deal with it.</p>
<p>
All the elements that play a role in the whole process are analysed in the book: how external signals arrive in the brain, how language has to be analysed, how do we recognize structure, etc. This allows to formulate some principles (16 rules of the ergo-learner) of how we learn through the ergo-system. The trailing part of the book is more technical. It describes in terms of category theory how the ergo-system can analyse language and give meaning to words and sentences. More generally it has to recognize structure and units, classify them through partitioning and clustering and identify connections and relations between units. This is only a first attempt to formalise how an ergo-system can make sense of a text being read or being heard. Although formal in a framework of categories and functors, the description is still more qualitative than quantitative,</p>
<p>
The fact that the text is more or less written as a freewheeling stream of consciousness to, in the end, arrive at some result that is still somewhat fuzzy, is a perfect illustration of how our ergo-brain works. This is how it gives meaning to observations and thus how it is feeding the knowledge of our ego-mind. Reading the book is a strange experience that will certainly keep your ergo-brain on alert since what you read is "interesting" and certainly not boring like a standard text is. The book unfolds its ideas following an ergo-logic and therefore should be read by an ergo-brain.</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 is a spur of the exhibition <em>A Beautiful Elsewhere</em> organized in Paris 2011 where the author collaborated with the filmmaker David Lynch and rock icon Patti Smith to evoke the mysteries of life (including the brain, mind, and language), all existing in a physical world (with its time, space, matter, and energy), and how mathematics (a product of our brain) can be useful to, not only explain the physics, but also to explain how life, brain and mind can originate in this physical world. Category theory is used to make a first attempt to catch the mechanisms used by our ergo-system (an assumed autonomous system, well separated from our conscious ego-mind) can make sense of language.<br />
</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/misha-gromov" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Misha Gromov</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/springer-nature-birkh%C3%A4user-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Nature / Birkhäuser</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-53048-2 (hbk), 978-3-319-53049-9 (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">90.09 € (hbk); 71.39 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">208</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.springer.com/gp/book/9783319530482" title="Link to web page">https://www.springer.com/gp/book/9783319530482</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>
<li class="field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</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/18-category-theory-homological-algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18 Category theory, homological algebra</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/18-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18-02</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/18d35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18D35</a></li>
<li class="field-item odd"><a href="/msc-full/18-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18-03</a></li>
</ul>
</span>
Mon, 17 Sep 2018 05:25:32 +0000adhemar48686 at http://euro-math-soc.euThe Quantum Astrologer's Handbook
http://euro-math-soc.eu/review/quantum-astrologers-handbook
<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>
Gerolamo Cardano (1501-1576) was an Italian polymath. He studied medicine and earned reputation and wealth as a physician, but he was also gifted mathematically and he used negative numbers and imaginary numbers as square roots of negative numbers well before they were more generally accepted. As a gambler, he also laid foundations of probability theory a century before Fermat and Pascal worked it out and 200 years before Laplace finished the job.</p>
<p>
Michael Brooks has a PhD in quantum physics from the University of Sussex, but he switched career and became a journalist and most of all a science writer. In this book (his seventh) we learn how his science is entangled with what Cardano (or Jerome as Brooks calls him) did. It has been remarked before by the authors of the papers collected in <a href="/review/art-science" target="_blank"> <em>The Art of Science. From Perspective Drawing to Quantum Randomness</em> </a> (Springer 2014) that Cardano's findings (complex numbers and probability) laid the foundations for the elements that are so essential for the development of quantum theory. Brooks is exploring this trace by writing a novel-like biography of Cardano, and at the same time explaining his own field by discussing the history and the subtleties of quantum physics and the many questions that it has raised and that still remain unsolved even today.</p>
<p>
Collecting the data for a biography of Cardano is not a major problem since he wrote an autobiography near the end of his life. Born as an illegitimate son of Frazio Cardano, a jurist and mathematician, his birth is almost a miracle because his mother tried abortion, but he got born anyway and survived frequent illness and the plague to which his three siblings succumbed. He decided to study medicine against his father's desire. He applied several times to be accepted as a physician in Milan, but it was repeatedly refused which caused him to live in poverty. By the mediation of some influential friends he got eventually a professorship in mathematics in Milan and got his medical licence. He was now a respected scientist and the most popular physician of Milan. He got offers from kings of Denmark, France, and the Queen of Scotland, that he all refused. He did travel to Scotland though where he also treated the archbishop John Hamilton, whom would later save Cardano from the inquisition. With his wife, Lucia Brandini, who was the love of his life, he had three children but his oldest son got executed for poisoning his own wife, and he disinherited his youngest who was a gambler stealing from his father. His outspoken confrontational ideas (his book <em>On the Bad Practice of Medicine in Common Use</em> was a success but not gracefully accepted by colleagues) and influential jealousy of his success brought his reputation down and allegations about his behaviour made the inquisition decide to imprison him at the age of 69 for the obscure reason of having cast an horoscope of Jesus Christ. So he had to spend several months in jail. By an intervention of John Hamilton he got out but he lost his professorship and was forbidden to publish his work. Not feeling accepted in Milan anymore, he moved to Rome where he wrote his biography. He predicted his own death on September 21, 1576 presumably by committing suicide.</p>
<p>
Brooks has interwoven this biography with the evolution of quantum physics by using a fictional component in which he is visiting Cardano while he is imprisoned waiting for his release or conviction. Cardano writes in his biography that he was visited by a guardian angel, and Brooks is taking up this role and they have a conversation of which this book is a reflection. We learn about the ups and downs in Cardano's life, the love of his life, the misery he has with his children, and the well known dispute with Tartagli and del Ferro about revealing the formula to solve a cubic equation. At the same time Brooks explains to Cardano (and thus also to the reader) the principles of quantum physics. He writes:</p>
<blockquote><p>
Jerome's views on astrology mirror our own on quantum physics. In quantum experiments we see things appear in two different places at once, or an instantaneous influence over something that is half a world away. We cannot make sense of it, but we don't dismiss it as ridiculous. We have the evidence of our experiments, after all, just as the astrologers have the 'evidence' of experience. (p.22-23).</p></blockquote>
<p>
Quantum physics is real as Brooks describes his history and evolution, but we still do not understand why the experiments give the results they do. He goes through all the possible interpretations from the Copenhagen interpretation to the multiverse theory and the superdeterministic interpretation, the pilot wave theory, the Penrose interpretation, etc. Cardano (and the reader) learns all about the main protagonists, the double slit experiment, Schrödinger's equation, the EPR thought experiment and its verification, and even some particle physics and string theory.</p>
<p>
</p>
<p>
Because in his conversation with Cardano, Brooks, coming from the future, knows things that did not happen yet. However, using the mysterious possibilities that quantum physics provides, Brooks can convince the reader to accept these anomalies. So the following twist comes as a surprise, and I think it is an amusing find. Brooks suggests to Cardano in prison that all this misfortune is the result of Tartaglia's doing. But Cardano answers that he doubts that because Tartaglia is dead for more than a decade. Then Brooks realizes that he read that in a book by Alan Wykes <em>Doctor Cardano</em> (1969) but Wykes may have used this historical flaw for the sake of his story. So it leaves Brooks blushing with shame in front of Cardano. At this moment Brooks is simultaneously a character in his own book and the biographer of Cardano who is correcting another biographer about historical facts. Later a similar trick is used when Brooks suggests that Cardano should write to Hamilton for help. It is then Cardano who doubts that Hamilton is still alive. But Brooks insists since he knows who has helped Cardano to get out of prison.</p>
<p>
The parallels between Cardano and Brooks, and the similarities between Cardano's science, inventions, and philosophy and the modern quest to explain quantum physics is very inspiring. Many of the findings that Cardano pioneered were way ahead of his time. For example his idea of the <em>aevium</em> even hints at a higher dimensional universe. Whatever the eventual faith or the proper interpretation of quantum physics may be, currently we are still in the dark. Perhaps we shall look in a century upon our present guesses and beliefs like we now look upon Cardano's astrology and his horoscopes, that were fully rational to him, as much as they are unscientific to us. So it may be symbolic when near the end of the book, Cardano steps out of his prison cell, leaving Brooks behind sitting on the bed.</p>
<p>
As far as I know, there is no biography-novel-popular-science-or whatever-you-call-it book produced that mixes all these ingredients in a marvelous plot. There are of course very good historical novels that sketch a biography of some scientist or another historical figure (and some exist some for Cardano already), but none has mixed this with popularizing science in such an harmonic and entertaining way as Brooks has achieved here. A novel, a biography and a popular science book, none of these in a strict classical sense, and yet all of them at the same time. Its format is certainly original. A recommended read.</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 biography of Cardano and a popular science book on quantum physics, brought in the form and with the quality of a good witty novel. Brooks takes on the double role of a character in his book, playing the role of Cardano's guardian angel who has a dialogue with Cardano while he was imprisoned by the inquisition having cast an horoscope of Christ, and at the same time he is telling the reader the ups and downs of Cardano's life and explaining his own work as a quantum physicist to both Cardano and the reader, showing some remarkable parallels between ideas of Cardano and quantum theory.</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/michael-brooks" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Michael Brooks</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/scribe-uk" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Scribe UK</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">2017</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">978-1-911-34440-7 (pbk)</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">£ 16.99 (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">256</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://scribepublications.co.uk/books-authors/books/the-quantum-astrologers-handbook" title="Link to web page">https://scribepublications.co.uk/books-authors/books/the-quantum-astrologers-handbook</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/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</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/01a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A09</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/00a15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A15</a></li>
<li class="field-item odd"><a href="/msc-full/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</a></li>
<li class="field-item even"><a href="/msc-full/81-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-01</a></li>
<li class="field-item odd"><a href="/msc-full/81-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-03</a></li>
</ul>
</span>
Tue, 07 Aug 2018 12:01:28 +0000adhemar48636 at http://euro-math-soc.euApplied Mathematics: A Very Short Introduction
http://euro-math-soc.eu/review/applied-mathematics-very-short-introduction
<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>
If you are a mathematician, try to define what exactly is applied mathematics and in what sense is it different form (pure) mathematics, and you will realize that it is not that easy. Existing definitions are not univocal. Although in most cases you recognize it when you see it. To that conclusion comes also Alain Goriely, who is an applied mathematician himself, in the introduction of this booklet. Yet he isolates three key elements characterizing the topic. First there is the modelling: some phenomenon is (approximately) described by choosing variables and parameters brought together in equations. Then there is of course a whole mathematical machinery to support and analyse the model theoretically, and finally there are the theoretical as well as the algorithmic and computational methods that solve the equations. The digital computers that emerged after WW II have certainly contributed to the development of applied mathematics bifurcating from pure mathematics. These three elements (model, theory, methods) form the framework for the rest of this (very short) introduction to applied mathematics which is intended for a mathematically interested outsider. Like the other booklets in this series it is a compact (17 x 11 x 0.6 cm) pocket book that is entertaining to read, even on a commuting train or during some idle moments.</p>
<p>
The data that an applied mathematician has to deal with are numbers, but these numbers have a certain dimension (length, weight, time,...) and they need to be expressed in proper units (like mks) and at a proper scale. Only when all this has been taken care of in a proper way, one can start building a model to, for example, predict the cooking time of poultry as a function of their weight or try to solve the inverse problem: how fast mammals can loose heat. With the answer to the latter problem, one may deduce something about their metabolism as a function of their volume. Keeping track of the proper dimensions throughout the modelling and the computations is called <em>dimensional analysis</em>.</p>
<p>
Choosing a model is a matter of deciding which are the most influential variables. The finer the model, the more computing time it will require while its predictive power or insight will not increase correspondingly. A simple mechanism to arrive at a model is illustrated with the model describing our solar system. First there was the geocentric system, but anomalies in the observations made Copernicus propose his heliocentric alternative. The more precise observations provided when telescopes were being used (a lot of data were provided by Tycho Brahe), allowed Kepler to derive his laws which fit the data, but it was only Newton's gravitational theory that gave the eventual explanation, not only for Kepler's laws, but for gravitation in general. Nowadays, models are constructed in a similar although a more interactive and more complex way. Observations lead to simple models, that are checked by experiments, which require subsequently refinements of the simple model, which is then checked against new observations, etc.</p>
<p>
Once the model is shaped in the form of equations, it requires theory to analyse under what conditions there exist solutions and what properties these solutions will have. For example one may analyse when they have an explicit solution (in terms of simple functions). If not, the equations can be considered as defining equations for new (less elementary) functions. The celestial gravitational problem of two mutually attracting bodies was generalized to the three-body problem, which was only solved partially by Henri Poincaré who, by doing so, created chaos theory. A deterministic world view had to be left behind and a qualitative analysis of (nonlinear) differential equations was born. The Lotka-Volterra equations describes a prey-predator model has periodic solutions, but with three species involved they will have chaotic solutions. Also the Lorenz equations, a set of three simple differential equations, originally describing an atmospheric convection problem is a famous model generating chaotic solutions.</p>
<p>
When it comes to periodicity, then the wave equation is the example that pops to mind. However when non-linearities are involves, like with seismic P-waves that travel trough the earth mantle, or phenomena like rogue waves, then solitons are involved, which are bump-like shapes that travel along without changing shape. They have a particle-like behaviour, and thus they have potential as carriers of digital information in optical communication, which is an exciting recent research field.</p>
<p>
The applications mentioned in the remaining chapters are computer tomography, the discovery of DNA, and the use of wavelets in JPEG2000 for image compression. Other examples are illustrating that what originally were theoretical developments, eventually turned out to be of the highest importance for applications like complex numbers, quaternions, and octonions (this line of complication was eventually replaced by the concept of a vector space), and knot theory (which found application in DNA modification). Finally large networks and big data are fairly recent topics that are used for describing global phenomena. Even with the complexity and magnitude of these networks, they are still inferior to what a human brain is capable of. Accurate modelling of our brain is momentarily still a (distant) target exceeding our current computational capacity but we are closing the gap.</p>
<p>
The previous enumeration is just a selection of some of the topics discussed that should illustrate what applied mathematics is about. Of course this limited booklet cannot be exhaustive. The approach is partially historical and still manages to refer to topics of current research. While examples are rather elementary in the beginning, towards the end, the topics tend to be more advanced. But even when discussing these more advanced subjects, Goriely tries to convince the reader that even if math is not always simple, still it is fun to do. The many quotations from the Marx brothers (most of them from Groucho) sprinkled throughout the text are funny of course. Goriely even provides a play-list of pop music that you could play in the background while reading (at least some of these he used while writing). This makes it clear that he has enjoyed writing the book and some of this joy radiates from the text when you read it.</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 short survey, Goriely gives examples (rather than a precise definition) of how applied mathematics relates to and interacts with pure mathematics. Applied mathematics fills the gap between the abstraction of pure mathematics and the world we live in. He describes historical models as well as more recent applications and even reaches out to future targets.</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/alain-goriely" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alain Goriely</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/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-1987-5404-6 (pbk), 978-0-1910-6888-1 (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">9.99 € (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">168</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://global.oup.com/academic/product/applied-mathematics-a-very-short-introduction-9780198754046" title="Link to web page">https://global.oup.com/academic/product/applied-mathematics-a-very-short-introduction-9780198754046</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>
<li class="field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</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/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</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/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li>
<li class="field-item odd"><a href="/msc-full/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li>
<li class="field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li>
</ul>
</span>
Mon, 02 Jul 2018 09:27:22 +0000adhemar48570 at http://euro-math-soc.euIslamic Design: a Mathematical Approach
http://euro-math-soc.eu/review/islamic-design-mathematical-approach
<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 book is similar to Jay Bonner's <a href="/review/islamic-geometric-patterns" target="_blank">Islamic Geometric Patterns</a> that was recently reviewed here, although this one does not have the aplomb of a coffee table book. It is accompanying the database <a href="http://www.tilingsearch.org" target="_blank">www.tilingsearch.org</a> of the first author. Brian Wichmann also authored <em>The World of Patterns</em> (2001) which is a catalogue of 4000 plates of different tiling patterns, accompanied with a searchable cdrom. He has a mathematics degree from Oxford but he worked as a software engineer at NPL. The tilingsearch website is Wichmann's retirement project that conveniently replaces the cdrom. David Wade has written several books on Islamic art and he also has his own website <a href="https://patterninislamicart.com" target="_blank">patterninislamicart.com</a> describing patterns illustrated with some 4000 related photographs. Of course many of the photographs and drawings in the book are also fount on these two websites. Several other related websites can be found by a simple web-search.</p>
<p>
The books, just like Bonner's book, has two parts: the first is about the historical and cultural context, and the second about the mathematical analysis of the patterns. The first part has some mathematical interest as well because it sketches where this fascination of symmetry and patterns comes form. Its origin is the geometry of Pythagonas and Plato's philosophy. The monotheism of Islam created a sense of unity and this resulted in a successful <em>jihad</em> with a quick expansion of the young <em>Khalifat</em> in the period 650-700 CE. By conquering the existing ruling powers like Byzantium and Persia the Arabs assimilated also all their knowledge and cultural heritage. Hence neoplatonism came naturally into Islamic culture via encyclopedic efforts to translate the Greek philosophers.<br />
There is also a religious component of course. While the <em>Quran</em> is like the Christian Bible, the <em>Hadith</em> (revelations) is like the Jewish Torah, a book of law where it is written that human and animal representation is not allowed (a rule that has a Jewish origin). Moreover the iconoclasm was also a way of submitting the wealthier cultures they had conquered. On the other hand, the abstraction of a design gave a sense of transcendence, and the repetitive tiling character can symbolize eternity.<br />
Soon the Islamic territory became too large to be ruled by one central government and fights among different subregions based on religious, political, and cultural differences divided the superpower. Different styles were grafted on the basic ideas.<br />
All of these ideas are clearly worked out in separate chapters of this book. The timeline at the end of the book is thereby very helpful. It spans the period from the fall of Rome in 410 CE and the consecutive rise and disintegration of the Islamic superpower till the invasion of the Europeans into the Arab lands in 1500, Also the glossary of Arabic words and of special terms used to describe the designs are useful.</p>
<p>
The mathematical part has diverse topics to discuss. In the introductory chapter the key concept of a rosette is defined. A rosette has rotational symmetry with at its centre an <em>n</em>-pointed star. Most common are rosettes with <em>n</em> equal to 5, 6, 8, 10, 12, 16, 24, or, 32. This star is surrounded by an alternating sequence of kites and petals. Kites are rhombi with two long and two short sides having d2 (mirror) symmetry. The short sides fit into the inward pointing wedges of the star. The (usually larger) petals are 6-sided polygons, also with d2 symmetry and that in standard design have 2 radially oriented parallel sides. Think of a rectangle whose short edges are replaced by an obtuse and a sharp outward pointing arrowhead respectively. The petals fit in the wedges created by the kites and their sharp points touch the points of the star. (To properly understand this, just check one of the websited mentioned above.)</p>
<p>
The symmetry of the design is denoted both in Orbifold and Hermann–Mauguin notation (a survey is included as an appendix of the book). Once the symmetry is fixed by the central pointed star, it requires a detailed analysis to fix the lengths of the edges and the angles to produce all the tiles that are needed to generate the overall design that will have several identical or different rosettes. Once the <em>n</em> is chosen and the length of the smallest edge, little freedom is left to fill up the whole figure. A first example is worked out for a 16-pointed star, surrounded by eight 6-pointed stars. The central star has a vertex angle of 45°, and all angles involved will be simple fractions of 45°. If the central star edge is chosen as unity, then the length of all other edges can be computed to capture the whole design with mathematical precision.</p>
<p>
While in the mathematical design, the tiles fit tightly together, in a practical realization, the (mathematical) boundaries of the tiles can be replaced by lines with a certain width. Sometimes these lines are wide enough so that they can also be realized by tiles. These lines are like treads that run over the pattern. They can be painted in white or have different colours. If one follows one of these lines, then it will intersect with itself or with other lines. At intersections they can be strictly interlacing, meaning that they will interrupt the intersecting line or will be interrupted by it in a strictly alternating pattern. This give the visual impression that the line goes over or under the lines it crosses. If these lines are wider bands then they can have a quadrilateral tile at their intersection. A separate chapter is devoted to all these issues.</p>
<p>
Like the analysis of the example of the 16-pointed star, other configurations are discussed in subsequent chapters.<br />
The <em>kathem</em> is an 8-pointed star with vertex angles of 90° that is very common and which allows for a lot of variation. Only 17 different tiles are used with some variation in edge length to form all octagonal designs. Some variations are possible that have a central star that has 16, 24 or 32 points.<br />
Similarly decagonal patterns are analysed that leave the ratio of two edges as a degree of freedom to bring variations into the pattern. Here the central 10-pointed star can be surrounded by ten 5-pointed stars or the central star can be 20-pointed surrounded by 10-pointed ones.<br />
Designs with six-fold symmetry are called 6-fold delights in the book. Clearly here angles are multiples or simple fractions of 60°. The 6- or 12-pointed stars can be surrounded by stars with 5,6, or sometimes 9 points.<br />
Sometimes there is a symmetry to be discovered at different scales. There is the micro symmetry, as described above, but if the size of the pattern allows to look at it on a macro scale, some other patterns may occur.</p>
<p>
The above construction based on trigonometry does not always work for some patterns as is illustrated in the penultimate chapter. A different construction is then required which is based on the use of ruler and compass. Starting from one rosette and the centres and radii or neighbouring ones, all the rosettes and the intermediate pattern can be constructed using only these two instruments. This is explained in detail for an example of a design with 18 and 12 pointed stars with two (irregular) heptagons in between (a design from a door of a mosque in Cairo). A goniometric analysis is made for the pattern on a pulpit of another mosque in Cairo. Not all the authors agree on the mathematics that were historically used to make the designs. Because some boundary lines are wide, precise measurement of the pattern is sometimes impossible, or the design has been damaged or it has to be analysed using an unclear or noisy photograph. All this can leave some room for interpretation. The mathematical techniques (mainly trigonometry) used here should have been known at the time the patterns were made.</p>
<p>
For all these patterns, existing examples are given showing that some patterns are typical for certain regions. There is for example a typical Moroccan style and a Byzantine style and designs typical for India etc. With few exceptions, the mathematical analysis of these patterns is not in depth and it would have been nice if there had been more details about the software used to generate all these patterns. Often photographs of existing artwork are used as the starting point for the mathematically generated pattern. Because measures can be imprecise, sometimes it is not clear that there are small flaws in the design that will only come to the foreground when implemented on a computer. So there is a chapter that illustrates some of these small errors either in the design or in some published analyses of designs. There are also some small typographical errors in this book. For example, on page 94 the authors refer twice to triangle X, but these are different triangles and there is only one X on figure 10.10; page 129 refers to an interlace discontinuity in Figure 10.2, while I think the idea is to refer to the kink in Fig.10.5; page 143 refers to angle Y in Fig. 14.9, but there is no Y in that figure.</p>
<p>
The nice thing about this book is that it does explain many of the constructions, but it also shows that not all existing artwork is perfect and that different methods may have been used to generate the patterns. All these examples being generated over many centuries and in geographically very different regions explain the richness and diversity, and yet the underlying uniformity of these geometrical patterns. Note that just like Bonner's book, the analysis of this book is also considering only strictly geometric patterns.</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>
The book describes the richness of the Islamic decorative geometric designs exposing rosette-like symmetry. In a first part the historical and cultural background is explained. The second part describes how to compute, using goniometric formulas, all the angles and edge lengths of the tiles used to form the different patterns. The book accompanies the websites maintained by each of the authors, that provide more online information and illustrations.</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/brian-wichmann" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Brian Wichmann</a></li>
<li class="field-item odd"><a href="/author/david-wade" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Wade</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/springer-nature-birkh%C3%A4user-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Nature / Birkhäuser</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-69976-9 (hbk), 978-3-319-69977-6 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">103.99 € (hbk); 83.29 € (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">237</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.springer.com/gp/book/9783319699769" title="Link to web page">https://www.springer.com/gp/book/9783319699769</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/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</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/01a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A30</a></li>
<li class="field-item odd"><a href="/msc-full/51-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-03</a></li>
<li class="field-item even"><a href="/msc-full/05-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05-03</a></li>
</ul>
</span>
Mon, 02 Jul 2018 08:59:48 +0000adhemar48569 at http://euro-math-soc.euDo Colors Exist?
http://euro-math-soc.eu/review/do-colors-exist
<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>
Many a mathematics or physics student will know the popular site <a href="http://www.askamathematician.com/" target="_blank">Ask a Mathematician / Ask a Physicist</a>. The main contributor of that blog is Seth Cottrell, "the physicist", who has however a mathematics degree in quantum information from NYU. In 2008 at the <em>Burning Man</em> festival (an annual experimental festival in the Nevada desert) he, together with a friend Spencer Greenberg "the mathematician", set up a little tent with a sign "Ask a Mathematician / Ask a Physicist", an experiment that was later repeated in public parks around New York City. The idea is that strangers can just ask any question about the physics of our universe, which the physicist and/or the mathematician try to answer as well as possible. Later (2009) this took the more convenient form of the previously mentioned blog on the Internet where "the physicist" is definitely more active than "the mathematician", or perhaps the physics questions are more popular. This book is a collection of some of the Q&A from that blog. Thus also here most of them are more physics-related than directly mathematics-related. It is however interesting to note that on the FAQ of the blog it is written:</p>
<blockquote><p>
<em>It cannot be overemphasized how important math is. If you’re bad at math, then doing more math is the only way to get better. If you can’t get past something (looking at you, fractions), then admit it to your teachers (or anyone else who can help), ask lots of questions, and then: math, math, math. Math.</em></p></blockquote>
<p>
Cottrell admits that he started mathematics studies because of his interest in the physics and he needed mathematics to understand the physics. This book is a selection of the more extensive blog entries (there are now hundreds Q&A in the searchable blog archive).</p>
<p>
The reader is warned by the author that some of the questions (and their answers) are controversial and may be subject to critique and neither "the mathematician" nor "the physicist" are infallible. The questions are however most interesting, and I can safely assume that most of you sooner or later in life have asked some of these and answering them is sometimes surprisingly nontrivial. Since inquirers are often students or certainly not specialists, the answer tries to balance between a proper (but deep and technical) answer and a superficial (with some hand-waving) reply that remains readable (at least to some extent) for the person who asked the question. As popularizing science texts usually are, the style is colloquial, entertaining, and even funny. A special warning is given when things become more technical. This more technical or more advanced material is placed at the end and gets a section-title "gravy".</p>
<p>
The book has four parts called "Big Things" (about cosmology and the universe), "Small Things" (about atoms, particles and quantum physics), "In-Between-Things" (mostly about classical physics), and "Not Things" (about mathematical topics). The title of the book "Do Colors Exist?" is for example a question discussed in the "In-Between" part. Although we can define color by wavelength and we can take pictures beyond the human visual boundaries, what our eyes register are basically only three components from which our brain makes up a color. Some other questions here discuss why wet stones look different from dry stones, but also carbon dating, entropy, energy, plasma, etc, The cosmological questions are related to the obligatory big bang, relativity theory, dark energy, and expansion of the universe, but also: 'What if the Earth were a cube?' and 'What if we drill a tunnel though the Earth and jump in it?'. The description of what we would experience just before the Earth were hit by another celestial object of a similar size is mind-bogglingly frightening. The "Small Things" section answers questions about true randomness, or whether an atom is besides a few particles mostly empty space, furthermore quantum decryption, anti-matter, particle-spin, etc.</p>
<p>
These is of course some mathematics involved already in answering some of the previous questions but the more "purely" mathematical section contains 11 questions, which form a curious collection. Some of them are classical topics in popularizing math books like why 0.999... = 1, and the problem of 1/0: stumble stones in undergrad mathematics. Others involve modern cryptography and the Enigma machine, transfinite numbers, the number pi, prime numbers, and chaos theory. Somewhat less obvious are a discussion of Fourier analysis, fractional derivatives, and a topological problem of knots in higher dimensions, and what the "Theory of Everything" (ToE) stands for.</p>
<p>
All in all, an entertaining collection with some interesting physics questions. A skilled mathematician, may not be thrilled by the mathematical subjects, but I can imagine that many people are pleased with the mathematics answers as much as they are by the physics explanations. The whole book has some nice illustrations (sometimes more intended to be fun or just to be `illustrating' than they are explaining). Of course the "Ask a Mathematician/Ask a Physicist" site is not the only one of its kind. There are many similar initiatives, which is a blessing of the World Wide Web, but entails also the danger of innocent students being spammed by fake and incorrect information. Science in general and mathematics in particular is certainly happy with people such as Cottrell who take such initiatives to their heart and serve the interested and the curious only driven by their enthusiasm, with little or no financial support.</p>
<p>
It is true that Cottrell is not really avoiding formulas, since there are quite a lot, perhaps more than what some people are prepared to swallow. On the other hand, if the readers had a phobia for formulas, they would probably not be asking the question. Most people will be more than satisfied with the answers provided. But be warned that to <em>really</em> understand the physics (or the mathematics), it will require a handbook to look up de details, although I must admit that for some explanations the answer will not directly be found there, and it will require to work up your way to a well founded answer starting from first principles. In that case Cottrell is your guide, pointing the way to follow.</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 collection of Q&A from the popular blog <em>Ask a Mathematician / Ask a Physicist</em>. The majority of the items discussed is physics-related but is has also a part that is more directly mathematics. Since questions are usually asked by non-specialists or students, the answers are as accurate as possible, but remain sometimes a bit on the surface to be understandable. The style of the answers is friendly, colloquial, sometimes funny, like popularizing texts usually are.</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/seth-cottrell" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Seth Cottrell</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/birkh%C3%A4user-basel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser basel</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-64360-1 (pbk); 978-3-319-64361-8 (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">42,39 € (pbk); 32,12 € (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">291</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.springer.com/gp/book/9783319643601" title="Link to web page">https://www.springer.com/gp/book/9783319643601</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>
<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/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</a></li>
<li class="field-item odd"><a href="/msc-full/70-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70-01</a></li>
</ul>
</span>
Mon, 02 Jul 2018 08:50:24 +0000adhemar48568 at http://euro-math-soc.euMusic by the Numbers From Pythagoras to Schoenberg
http://euro-math-soc.eu/review/music-numbers-pythagoras-schoenberg
<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>
Music and mathematics have a long joint history. Music theory was part of the Greek quadrivium, and it has been designed and revised by mathematicians including Pythagoras, Simon Stevin, Kepler, etc. Many well known mathematicians were also skilled practitioners of some instrument (Einstein loved his violin, Feynman enjoyed playing the bongos, and Smullyan gave piano recitals,...). Of course several books were written on the subject already. For example D.J. Benson: <em>Music, A mathematical offering</em> (2007) or the monumental two-volumes historical survey by T.M. Tonietti <a href="/review/and-yet-it-heard-musical-multilingual-and-polycultural-history-mathematics-2-vols" target="_blank"><em>And yet it is heard</em></a> (2014). But also G.E. Roberts <em>Music and Mathematics</em> (2016); G. Loy <em>Musimathics: The Mathematical Foundations of Music</em> (2011); D. Wright <em>Mathematics and Music</em> (2009); N. Harkleroad <em>The Math Behind the Music</em> (2006). And the collection of papers J. Fauvel, R. Flood, R. Wilson (eds.) <a href="/review/music-and-mathematics-pythagoras-fractals-0" target="_blank"><em>Music and Mathematics</em></a> (2006), G. Assayag, H.G. Feichtinger (eds.) <em>Mathematics and Music</em> (2002). This is to name just a few. A simple internet search will give many more results.</p>
<p>
Maor is a writer of several popular mathematics books, and, although not a practitioner, he is a lover of music. In this relatively short booklet he draws a parallel between the history of mathematics and the history of music theory. It is again a book on popular mathematics for which no extra mathematics outside secondary school education is needed. However some familiarity with terms from music theory is advised, even though most of these concepts are explained. Maor selects some topics of (historical) interest and sketches evolutions both of mathematical history and of the historical approaches to music theory. Besides the obvious and obligatory topics, and a personal selection of the historical periods, there are also a number of side tracks added as curious anecdotes.</p>
<p>
Maor describes some pillars of the historical bridge that is spanning the wide gap of the eventful evolution of music and math since Pythagoras till our times. The opening chapter is describing the pillar on which that bridge is resting on our side of history. The early 20th century is the scenery where Hilbert challenges the mathematicians with his his list of problems. Solving some of them eventually leads to a crisis in the foundations of mathematics. Physics moves forward to a new era leaving Newtonian mechanics and entering an age of relativity theory. The rigid world of Laplace, acting as a clockwork, becomes a quantum world governed by probabilities. Likewise music changed its face. The fixed tonality, the reference frame, that had been the standard for ages was left and Mahler and Berlioz made this all relative, culminating in Schoenberg's twelve-tone system. This introduction sets the scene where the book will eventually lead to in some grand finale. But first we need to wade through the historical evolution to appreciate the meaning of these revolutionary ideas.</p>
<p>
Maor's guided tour starts at the other pillar of the history bridge at 500 BCE with a (physical) string theory by Pythagoras, defining a scale by introducing an octave, a fifth, and a fourth, which are logarithmic scales long before John Napier conceived logarithms. The Greek vision of a physical world dominated by integers was accepted during many centuries to follow and Galileo and Kepler were still Pythagoreans in this respect adhering to the music of the spheres.</p>
<p>
The Enlightenment was a first breach with the past. Galileo's father Vincenzo Galilei discovered that the pitch of the vibrating string was proportional to the square root of the tension of the string. Galileo in his <em>Dialogues</em> on the `New Sciences' was the first to have the word `frequency' in his book and Mersenne was the first to measure it. Although better known for his prime numbers, he was the first to write a book on vibrating strings: his <em>Harmonie Universelle</em> (1636). Even less known is Joseph Sauveur (1653-1716) who coined the term `acoustics' and who discretized the differential equation describing the vibrating string by considering it as an oscillating string of beads. Of course a true differential equations needs calculus that was being invented by Newton and Leibniz in those days and they have quickly conquered science in many aspects through the work of the Bernoullis (Jacob, Johann, Daniel), Euler, D'Alembert, and Lagrange. The differential equations of a vibrating string was related to music theory and harmonics, but it was only Fourier who finally discovered that almost any periodic function can be written as a sum of sine functions of different frequencies and this defines the acoustic spectrum and generalizes the idea of standing waves or the natural harmonics or overtones of instruments. These were further explored in the acoustic theory in books written by Helmholz in Germany and Rayleigh in Britain.</p>
<p>
The physics being established, Maor returns to music theory. The history of how to subdivide the octave has caused much confusion and disagreement, and has not only defined musical temperament but also heated the temperaments of the protagonists. As a transition to a discussion on rhythm, meter and metric, Maor introduces the tuning fork and the metronome as musical gadgets. When composers started using variable meters, a parallel is drawn with the local metric on Riemannian manifolds, just like Einstein used a local reference system for his relativistic observations. This idea is extended to other disciplines using reference systems such as cartography and the relativistic use of perspective in visual arts as explored in the work of Escher's and Dali.</p>
<p>
That brings Maor back to the nearby pillar of his narrative tension in a chapter where Schoenberg, a contemporary of Einstein, develops his relativistic music in the form of a strict twelve-tone system. However, while Einstein's theory has practical applications still used today, Schoenberg's experiment was less successful and he didn't have many followers. Maor closes the circle completely with some remarks on string theory in current theoretical physics, which of course links up with the strings studied by Pythagoras.</p>
<p>
Most interesting are also some of Maor's excursions on the side (there are five) about the musical nomenclature, the slinky (a periodic mechanical gadget in the form of a spiral that can `walk' down the stairs), some musical items worth an entry in the Guinness Book of Records, the poorly understood intrinsic rules that govern the change of the tonic to different keys, and the <em>Bernoulli</em> (an instrument invented by Mike Stirling with 12 radial strings equally tempered as like on a Bernoulli spiral and that actually looks like a spiral harp).</p>
<p>
Maor is an experienced story teller. His mixture of musical, mathematical, and physical history, enriched with personal experiences and some unexpected links and bridges are nice reading for anybody with a slight interest in music and science. No mathematical training required. Leisure reading. Do not expect deep analysis or high brow theoretical expositions. Just enjoy and let yourself be surprised.</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>
Maor gives a selection of historical parallels that can be drawn between the evolution of mathematics and music theory. From the strings of Pythagoras to the string theory of theoretical physics. His main message is that at some point mathematics and physics have abandoned an overall reference system and accepted local reference frames (think of relativity theory and geometry). At about the same time something similar happened in music theory when keys were no longer maintained over a long time but they became local which has resulted in atonality and Schoenberg's twelve-tone theory.</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/eli-maor" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eli Maor</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">2014</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">978-0-691-17690-1 (hbk); 978-1-400-88989-1 (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">24.95 USD (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">176</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/titles/11250.html" title="Link to web page">https://press.princeton.edu/titles/11250.html</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/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/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</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/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li>
<li class="field-item odd"><a href="/msc-full/97a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A30</a></li>
</ul>
</span>
Tue, 29 May 2018 06:34:03 +0000adhemar48509 at http://euro-math-soc.euThe History of the Priority Dispute between Newton and Leibniz
http://euro-math-soc.eu/review/history-priority-dispute-between-newton-and-leibniz
<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>
Priority disputes among mathematicians are from all times, but the one between Newton and Leibniz about the discovery of calculus is notorious. Many authors, and historians have written about it. Even during the lifetime of the protagonists, the Royal Society had a commission to investigate the matter. Their conclusion was that Newton was the first, but since at that time Newton was the president of the Royal Society, this conclusion may have been a bit biased.</p>
<p>
The supporters of Leibniz whose home base was Hanover were mainly from the continent, while most of the British defended their national hero. In those days mechanics, mathematics, optics, chemistry, alchemy, astronomy, and history were all part of the job of a prominent scientist. In Newton's case certainly also theology, history, and monetary politics. While Leibniz started as a lawyer and published on palaeontology. So the whole scientific (and political) community was involved.</p>
<p>
Newton introduced his fluxions inspired by physics. A fluxion is the instantaneous change in a fluent. We now say that it is the time derivative of a function of time (the fluent). The problem was that the notion of limit was still unknown, so his peers had problems with computations that used infinitesimal small (but nonzero) quantities, that seemed to vanish when appropriate and remained nonzero at other instances. This was directly connected to the construction of a tangent and what was called a quadrature, which is the computation of the area under a curve, thus what we now call an integral. Newton's great insights happened mainly during the period of the Great Plague in 1995-1667 when he retreated to Woolthorpe Manor to live with his mother. In that time he also developed his theory of gravitation, laid the foundation of classical mechanics, and explained the planetary motion. None of this was however published until much later. The mechanics were published for the first time in his <em>Philosophiæ Naturalis Principia Mathematica</em> in 1687 and two other editions in 1713 and 1726. His book <em>The method of fluxions</em> was only written in 1671 and published in 1736.</p>
<p>
Leibniz was educated as a lawyer ans got only interested in mathematics later in 1672 when he visited Paris and meets Huygens. He was mainly concerned with quadrature. The approximate length of a curve $ds$ could be considered as the hypotenuse of a rectangular triangle with sides $dx$ and $dy$. Using geometrical arguments and similarities of triangles he obtained a method to compute the quadrature of an arbitrary curve. This was around 1674, but it was not published before 1684. He used the notation $dy/dx$ for the derivative, which was conceptually much easier to work with than Newton's fluxion notation which used the dot atop the fluent variable. This of course becomes problematic for higher order derivatives. Leibniz also introduced the integral sign ∫ as a elongated 'S' for sum, that we are still using today and which is included in the title of this book by writing "Dispute" as "Di∫pute". It is clear, and generally agreed by now, that Leibniz and Newton developed their theory independently by following different methods. However in the heat of the controversy Leibniz was accused of blatant plagiarism. Strangely enough, it were not Newton and Leibniz that stood in the barricades most of the time. In fact they exchanged polite and friendly letters. It were their followers, friends, and believers who did all the fighting on the front line, although they were of course backed up and sometimes directed by the protagonists. Newton remained more on the background, but when accusations became too direct, Leibniz had no choice but to protest against an open insult by a warrior from the opposite camp.</p>
<p>
Among the historical defenders of Leibniz were Jacob and John Bernoulli. Among Newton's warriors were John Collins, John Wallis, and Nicolas Fatio de Duillier, which is called Newton's monkey by Sonar. This Fatio has put the fuse that lit the powder keg by openly accusing Leibniz of plagiarism. At a later stage John Keill became the `army commander' of the group defending Newton. Some of the problems arose because the first correspondence was not directly between Newton and Leibniz but passed via others like Henry Oldenburg, the secretary of the Royal Society, who was not a mathematician. Oldenburg was advised on matters of mathematics by Collins, an outspoken nationalist, who was naturally opposing anything that came from the continent. There were misunderstandings, half spoken truths, and hesitation to disclose results that oxygenated the fire. The war went on, even beyond the grave. Clearly the new calculus found applications, and because Leibniz's formalism was easier, his calculus was the eventual winner. In fact it caused a drop back of the English mathematical scenery. While they were at a comparable level with the mathematics on the continent when the controversy started, they were not able to keep up with the development of calculus and analysis for a while in the eighteenth to nineteenth century post-Newton era.</p>
<p>
This fight may be well known, but disputes in those days were very common among others as well. Newton and Hooke became personal enemies over a priority dispute in optics (Newton did not want to publish his <em>Opticks</em> until after Hooke died), Huygens rejected Newton's corpuscular theory of light. He also fought with Heuraet over the rectification of curves, and he quarrelled with Hooke over a clock mechanism. Newton and Flamsteed, the Astronomer Royal, were fighting over the trajectory of the Great Comet of 1680, which Newton explained with gravity. And there were other such disputes that are also described by Sonar in this book.</p>
<p>
Thomas Sonar is from Hanover and before he engaged in the study of this history, he was rather convinced that it was a good-hearted Leibniz that was the one who was maltreated and unjustly accused by a quarrelsome and short-tempered Newton and his disciples. Sonar may have started his research with the idea of defending Leibniz, when he finished the original German version of this book in 2016, his conclusion was much more mollified. Leibniz also had not always told the truth and he wasn't the saint attacked by the devil Newton. He also had his pawns in the war and used them. This conclusion becomes clear only after meticulously investigating all the original correspondence of the seventeenth century and of all the books and papers that were published about the matter. This is the most thorough discussion of the matter that has been published so far and that still remains very readable with a minimum of mathematical knowledge, hence available for a general readership. In fact Sonar starts with an elementary introduction like a modern introductory calculus book would, so that the reader should know what calculus is about, or at least grab the meaning of derivative and integral. Then he introduces the `giants on whose shoulders Newton claimed to stand': John Wallis, Isaac Barrow, Blaise Pascal, Christiaan Huygens. So we find a biography of these people, and what they did for mathematics. In retrospect it is clear that calculus was on the doorstep, and that it only took some great minds to bring it in the open.</p>
<p>
But Sonar also gives a detailed description of the political situation and events of those days in England, France, Spain, and the Netherlands. Of course these are not really essential for the mathematics, but it sketches the framework in which scientists were working. It were usually political leaders that employed the top scientists and they made the start of academies financially possible. This part of the book has several very useful timelines, and there are many beautiful pictures throughout the book. Just reading this political prequel to the main dish is already a wonderful experience. Then of course we meet both Newton and Leibniz, how they grew up and studied and how they arrived at the discovery of their new calculus. At first there is some friction (Sonar calls it a cold war) between the two, then there is a period of relaxation, but when things get published the smoldering fire becomes a real war. Sonar includes many quotes from the letters that go back and forth about the matter with precise dates of when they were written, whether it was as an impulsive reaction to a previous message or it was written only after a long time of postponing it, possibly who was the messenger, and, not unimportant, when the letters arrived. With every new player we are given his or her background and some biography.</p>
<p>
Fortunately the excellent and smoothly reading English translation comes so shortly after the German original and was done by Sonar himself with the help of Keith Morton, his Oxford thesis advisor and later by his advisor's wife Patricia Morton. I can highly recommend this book if you have just a slight interest in history and/or mathematics. Perhaps the professional mathematical historians may not find much new or innovative material, since this 'cold case' has long been settled and solved, I believe they will still enjoy reading this book.</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 English translation of the German original that appeared in 2016. It is a detailed analysis of this famous controversy that is brought in an easily accessible format for a general readership. It starts with a brief introduction to differentiation and integration (that can be skipped if you don't need it), then sketches the political situation in England, France, Spain and the Netherlands of the 17th century, en finally elaborates on the rise and decline of the controversy backed up by many quotations from the letters that were mailed back and forth on this matter.</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/thomas-sonar" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Thomas Sonar</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/springer-international-publishing-birkh%C3%A4user-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer International Publishing / Birkhäuser Verlag</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-72561-1 (hbk); 978-3-319-72563-5 (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">137.79 € (hbk); 107.09 € (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">576</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.springer.com/gp/book/9783319725611" title="Link to web page">https://www.springer.com/gp/book/9783319725611</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>
</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/01-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-02</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/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li>
<li class="field-item odd"><a href="/msc-full/01a45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A45</a></li>
<li class="field-item even"><a href="/msc-full/26-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-03</a></li>
</ul>
</span>
Tue, 29 May 2018 06:18:34 +0000adhemar48508 at http://euro-math-soc.euErnest Irving Freese's Geometric Transformations
http://euro-math-soc.eu/review/ernest-irving-freeses-geometric-transformations
<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 its simplest form, a geometric dissection refers to subdividing a polygon into a finite number of pieces and to reassemble them to form another polygon. It is proved that this is always possible, but the challenge is then to obtain it with a minimal number of pieces. Proving the minimality for an orthogonal polygon is shown to be NP-hard. If every piece can pivot around a vertex that stays connected to a neighbouring piece during the transformation, it is called a hinged dissection. When unfolded, it forms a string of connected pieces. Piano-hinged means that the connection is not at the vertices, but that edges are connected so that this corresponds to folding a paper version, but these are not considered in the present book.</p>
<p>Frederickson starts this book with a short history of dissections. They were first studied in the 19th century by some mathematicians a.o. Farkas Bolyai, but they became popular in the 20th century when they featured in newspaper puzzle columns by Sam Lloyd and by H. Dudeney, (who collected them also in their collected puzzle books) and later M. Gardner. Harry Lindgren provided a way to derive dissections by overlapping tessellations of the plane around the middle of the 1900s. That is also the time that Freese was preparing his work on the topic.</p>
<p>In his book Dissections: Plane & Fancy (1997) Frederickson mentions some 'over 200 plates' prepared by Freese of which a few loose copies were circulation, but the whole manuscript was not located. In his second book Hinged Dissections: Swinging & Twisting (2002), Freese has disappeared from the reference list, but in Piano-hinged Dissections: Time to Fold! (2006), Freese's work is very prominently present in the form of appendices to the chapters written by Fredericson. Some of the plates are reproduced and Frederickson provides comments. What has happened? Frederickson gives an explanation, which is actually a very remarkable story. That story is retold in this book. Freese, who obviously knew about these dissection problems from the puzzlers columns and from puzzle books that he got as a present from his wife. He had collected his ideas on dissections in the form of 200 plates that he finished a couple of months before he died. He wrote in 1957 to a friend that he had been intensively busy preparing them, that a blueprint could be obtained for $28.00, but that probably nobody would be interested in publishing his drawings. After his death several people tried to obtain the manuscript from his wife Winifred but she had written a letter to Ginsburg (a friend of Freese and editor of a journal) asking him to take care of the manuscript. However Ginsburg had died three weeks before her husband, so he never answered and the manuscript was forgotten. Frederickson obtained Winifred's address only after she passed away, but her son Bill was living in the house now. Frederickson wrote him a letter but the manuscript was not found. After Bill died, a cousin found the letter and the manuscript so that Frederickson finally could make a copy of Freese's plates in 2003. That explains why Freese features in the appendices of his 2006 book.</p>
<p>Because Frederickson also got hold of some other material, he can add a biography of Ernest Irving Freese (1886-1957) as the second chapter of this book. A very wild adventurous life this Los Angeles architect has lived. Mostly self-taught, he first worked for an architectural firm, later as an independent architect, but he also travels the country as a tramp, and later goes on a bicycle world tour, working when he needs money or as a crew member on a ship to pay for his fare. He published articles in cycling magazines and in architectural and construction journals. After an earthquake in 1933 he started a campaign to construct safe schools (he was by then father of three). He was an assertive man with strong opinions.</p>
<p>The main purpose of this book is to finally publish the manuscript by Freese. It was originally not conceived as a commercial product, so it is a notebook that consists of loose hand-made geometric constructions with little text in Freese's elegant slanted handwriting. Frederickson has kept the order of the numbering of the places, subdivided them into chapters and provided an introductory text and explaining notes, references, and new results per chapter (Freese has no references) and this text is followed by the relevant plates. So, while in his 2006 book Frederickson used Freese's results as an appendix to his own work, here it is the other way around, it is Freese's work with Frederickson commenting. The plates are beautifully reproduced after being digitally processed to remove stains. Their original size is 8.5 x 11 inch (21.6 x 27.9 cm) which is also the somewhat unusual format of this book. Moreover to keep the originals intact and in the state that Freese has created them, these pages get no headers or page numbers, they only have the original encircled plate numbers.</p>
<p>Freese had divided the plates into sections corresponding to the geometry of the objects. The subdivision into smaller chapters is a decision of Frederickson. The idea is that chapters will group transformations that are somewhat similar so that a common introduction is possible, although Frederickson is also commenting on all the separate plates within each chapter. The chapters are then about transforming isoscele or equilateral triangles, followed by squares, crosses, rectangles, and n-polygons and n-polygrams up to n = 12 and they conclude with some unclassified miscellaneous figures. All the dissections are two-dimensional, so no 3D generalizations. It is made clear with references that some of the dissections were found later by others, when Freese's work was unavailable. Everyone interested in geometric dissections, and this kind of puzzles, either mathematically or recreationally will embrace this publication. But also the readers interested in the history and certainly those who became curious about this mystery man and his manuscript, after reading Frederickson's 2006 book, will be fully satisfied with this respectful reproduction eventually made available for a general public.</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>The main purpose of this book is to publish 200 plates illustrating geometric dissections that were produced by E.I. Freese, a Los Angeles architect, shortly before his death in 1957. Due to circumstances, the plates got lost and was only recovered by G. Frederickson in 2003. This book contains a short history of geometric dissections, and a biography of Freese, followed by the reproduction of the plates subdivided into chapters and introduced and commented by Frederickson.</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/greg-n-frederickson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Greg N. Frederickson</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/world-scientific-publishing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific publishing</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-3220-46-1 (hbk), 978-981-3220-47-8 (pbk), 978-981-3220-49-2 (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">GBP81.00 (hbk), GBP32.00 (pbk), GBP26.00 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">432</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.worldscientific.com/worldscibooks/10.1142/10460" title="Link to web page">https://www.worldscientific.com/worldscibooks/10.1142/10460</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/52-convex-and-discrete-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52 Convex and discrete 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/52b45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52B45</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/52-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52-01</a></li>
<li class="field-item odd"><a href="/msc-full/05b45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05B45</a></li>
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
Thu, 10 May 2018 06:34:22 +0000adhemar48456 at http://euro-math-soc.euThe Paper Puzzle Book
http://euro-math-soc.eu/review/paper-puzzle-book
<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 subtitle of the book : All you need is paper (and scissors and sometimes adhesive tape if you want to be picky), might be tricking you into an imaginary situation of a kindergarten with children producing some artwork for mom, dad, or one of their grand parents. This is a completely different kind of book. You need a well trained set of brains and a strong puzzler's attitude to solve the puzzles that are collected by some of the best.</p>
<p>Ilan Garibi is an Israeli origami specialist, David Goodman is a designer of (mechanical) puzzles, and Yossi Elran is a mathematician, head of the Davidson Institute Science Education Accelerator of the Weizmann Institute in Rehovot, and a big puzzle fan. When they met at a meeting of recreational mathematics and games, the idea for this book was born.</p>
<p>In the best of Martin Gardner's tradition 99 puzzles are collected. Some are classics, some are found in the literature, and others are new. The authors are kind enough to give the origin of the puzzles when appropriate. The number of 99 is just a rough indication because there may be 99 problems formulated, but their solutions, which are given at the end of the chapters, sometimes propose variations or end with an extra challenge left open for the reader.</p>
<p>It may seem not very easy to represent with a static image (or images) in a book, all the necessary operations of folding an cutting that have to be performed in 3D and that sometimes even result in a 3D object. However the different steps are represented using some pictoral vocabulary that is explained in the beginning and that is remarkably clear and easy to read.</p>
<p>The puzzles are grouped according to techniques and topics in ten chapters. Sometimes puzzles are sequential, i.e., you first need to solve puzzle x before you solve puzzle x+1 because solving x is a subproblem of x+1. The puzzles are also rated with one up to four stars. Sometimes the shape of the paper is important for the technique to work: it need to be square or A4, but in other cases it can be just rectangular, or it has to be a long strip. Here is a list of the chapters with some simple illustrative example:<br />
1. Just folding. For example fold a square paper into an equilateral triangle with a follow-up problem to fold the largest possible equilateral triangle that is contained in the square.<br />
2. Origami puzzles. These need so called Kami paper whose sides have different colours, for example black and white. A first exercise is to fold the paper such that the visible areas of black and white are equal. This chapter is rather extensive.<br />
3. 3D folding puzzles. Given a strip of size 1 by 7, fold it into a cube with side 1.<br />
4. Sequence folding. Here one is given for example a square paper with a 2x2 grid defining 4 squares that are marked with the numbers 1 to 4 in lexicographical order. The problem is to fold the paper until it has size 1x1, but such that the squares on the folded stack have the natural order 1,2,3,4. Many variations are possible, starting from different configurations, or allowing a few cuts, etc.<br />
5. Strips of paper. Here of course the Möbius band plays a prominent role, but there are other puzzles to formulate with strips.<br />
6. Flexagons. This is an invention of Artur Stone of 1939 and popularized by Martin Gardner and later picked up by several others. Paper is folded into a polygonal form in such a way that that it has a front and a back side, but it allows for an simple flipping operation such that it is so to speak turned inside-out, showing different faces. One could define it as a flat folded configuration that has more than two faces. As a simple example one could start from a particular configuration of 6 connected squares (neighbouring squares have exactly one edge in common). Both sides have two squares marked 1, two marked 2 and two marked 3. Counting both sides, there are thus four 1's, four 2's and four 3's. This has to be folded into a 2x2 square and the 'first' and 'last' square are taped together so that one gets a sort of Möbius ring object that will allow only a limited number of hinged flips. The 2x2 square has to show the four 1's on the front and the four 2's on the back. By 'flipping' it, one gets all 3's on one side and all 2's on the other. There are three faces that can be shown in turn by flipping.<br />
7. Fold and cut. For example, you have to fold a piece of paper in a certain way and cut it with one straight cut to obtain a prescribed shape like a cross or a star.<br />
8. Just cutting. A classic is to cut a hole in an A4 size paper, such that a person can step through the hole without tearing the paper.<br />
9. Overlapping paper puzzles. It is clear that, given three paper squares, one may arrange them in a partially overlapping way such that all three are only partially visible. This is impossible with four squares. Problems based on this principle can be formulated putting restrictions of the number or size of papers you start with, or restrictions on the shape of the outer boundary of the stacked papers.<br />
10. More fun with paper. This is the miscellaneous section with many diverse fun constructs like putting together a rotator or an helicopter, performing magic tricks, solve (seemingly) impossible bets, etc.</p>
<p>The examples I gave above are just to illustrate the idea of what kind of puzzles are possible. They are usually the first kick start puzzles for the chapter rated with one or two stars. Sometimes these innocent looking problems can be be surprisingly difficult to solve even if they get the lowest difficulty rating. Although the solution methods for the puzzles are reminiscent to geometry, no mathematics is required. It reminds me of the ancient Greek idea of constructions using only compass and straightedge, but this is definitely different and even more basic: there is no compass, and there is no ruler. It is for example difficult to divide an edge of a square in three (or in n if n is odd) equal parts. That is only possible using an iterative pinching procedure. Such basic techniques are explained in an appendix. There is also a (limited) list of books, papers, and websites for further reading.</p>
<p>This is a marvellous book. The diversity of possible puzzles that can be given with these very limited resources, which are basically some paper and scissors, is overwhelming, and the challenges are sometimes very tough. Even the two-star problems may be hard for an untrained puzzler. This is medicine against boredom on long rainy days, but be careful not to get addicted or it may suck up your less empty and sunny days as well.</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 marvellous set of about a hundred puzzles that have to be solved by only folding and/or cutting paper. They were collected by three experts: an origami specialist, a puzzle designer, and a mathematician. Many of these innocent looking problems are really hard to solve, and others seem to be impossible at first sight. It requires geometrical thinking, but no mathematical knowledge is needed. As with many of these mathematical puzzles you need to be able to think outside the box, and sometimes to visualize things in 3D.</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/ilan-garibi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ilan Garibi</a></li>
<li class="field-item odd"><a href="/author/david-goodman" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Goodman</a></li>
<li class="field-item even"><a href="/author/yossi-elran" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Yossi Elran</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/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li>
</ul>
</span>
<div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-3202-40-5 (hbk), 978-981-3202-41-2 (pbk), 978-981-3202-43-6 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP42.00 (hbk), GBP25.00 (pbk), GBP20.00 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">264</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.worldscientific.com/worldscibooks/10.1142/10324" title="Link to web page">https://www.worldscientific.com/worldscibooks/10.1142/10324</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/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
Thu, 10 May 2018 06:28:44 +0000adhemar48455 at http://euro-math-soc.euMandelbrot the Magnificent
http://euro-math-soc.eu/review/mandelbrot-magnificent
<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>Benoit Mandelbrot finished his autobiography shortly before he died in 2010. After some editing, it was published as a book entitled <a target="_blank" href="/review/fractalist-memoir-scientific-maverick"><em>The Fractalist, Memoir of a Scientific Maverick</em></a> in 2012 with a foreword by his wife Aliette. In that book he describes in three parts (1) his youth in prewar Warsaw and Paris and later in Tulle (France) during WW2 (2) his education and scientific life in the period 1944-1858 and (3) his life after recognition, i.e. the period 1958-2004.</p>
<p>Liz Ziemska is a Polish born literary agent who came to the US when she was seven. She earned a Bachelor of Science in Biology and a Master of Fine Arts in creative writing and she has written several stories that appeared in fantasy and science fiction collections. What do you think will come out as a story if she first read Mandelbrot's <em>The Fractalist</em> and then, impersonating a juvenile Mandelbrot, she would re-tell the first five chapters using the facts provided by Mandelbrot and mix these with her own fantasy? You do not have to guess because that story has been written and it is called <em>Mandelbrot the Magnificent</em>.</p>
<p>The facts: Benoit Mandelbrot is born in Warsaw (1924) in a family with a long Russian-Jewish tradition. His mother was a dentist and his father was a business man who became also a tailor, forced by the circumstances. His father's younger brother, uncle Szolem, was mathematically gifted and introduced Benoit to mathematics at a young age. When Szolem got a position in France in 1936, Benoit together with his younger brother Léon and his parents, moved first to Paris and three years later, when Szolem was appointed at the university of Clermont-Ferrand, they moved to the nearby village of Tulle. After the Germans invaded France, Tulle fell under the "Free" France of the Vichy Régime headed by Marshal Pétain. Szolem escaped the war because he got an appointment in Princeton and migrated with his wife to the US. When also the Vichy France was invaded by the Germans, life became pretty dangerous for Jews and Benoit and his brother narrowly escaped from being arrested. Tulle is infamously remembered for the Tulle massacre in 1944, three days after D-Day, when civilians were executed and many taken captive by an SS tank division as revenge for a successful action of the French Résistance. That is a summary of the first five chapters of <em>The Fractalist</em> and these are also the facts that Ziemska uses in her story.</p>
<p>If she were only repeating these facts, then there would be no point in writing her novella since also the original text is well told, and it is first hand. So, Ziemska adds some fractal imaginative detail-adornments and some more large-scale fantasies. Examples of the latter are certainly the Mandelbrot family hiding from the Germans in a fractal structure invented by Benoit. She also introduces the <em>sefirot</em> as an essential element in Benoit's life. It is a densely but well structured esoteric graph from the Kabbalah with ten nodes that represent all manifestations of an infinite God (or of "G-d, the Mathematician" as Ziemska writes). In Ziemska's view it gave Mandelbrot the insight of iterated function systems, an essential tool for the generation of fractals. To increase the narrative tension, she also added the character of Emile Vallat, a student in Benoit's class in the Tulle period who is another bright student, competing with Benoit for the best grades in mathematics, but Emile's family (his mother is the local librarian where Benoit finds his Book of Monsters, a ficticious book on mathematical objects) is sympathizing with the Germans and so, he is constantly teasing and humiliating Benoit and his brother who try hard to be discrete and hide their Jewish background.</p>
<p>Of course fractals and more generally mathematics are well represented announcing Mandelbrot's future career as the inventor of fractals and their omnipresence in nature. So several names of mathematicians are mentioned: Kepler, Poincaré, Gaston Julia,...; and mathematical terms, not really mathematics, just some name dropping without explanation: Zeno's paradox, Fibonacci numbers, Hausdorff dimension, the volume of a sphere as a multiple integral, the golden section, and of course the Mandelbrot set; and there are the illustrations from The Book of Monsters: the Sierpinski triangle, Koch's snowflake curve, Peano's curve,.... These are all very curious elements to embed in an imaginative novella, making it literally "extraordinary". The beginning and the end of this story is Mandelbrot finishing his memoirs while his wife Aliette is serving him cauliflower for his eightieth birthday, his favourite dish in which he admires the fractal structure.</p>
<p>This is a well told story, with many Jewish elements like for example the role of the <em>sefirot</em> and the dreadful situation of Jews during the war, and the atrocities that in fact any war does to a society. The latter is unfortunately very real, but, although based on facts, it is not a biography of Benoit Mandelbrot's youth. For that component it should be taken for what it is intended to be: a mixture of facts and fiction. Somewhat disappointing from a mathematician's point of view is that geometry is illustrated by "monsters" (and not by "gems") but Ziemska blames Poincaré for that. She uses one of his well known quotes where he claims, referring to an example produced by Weierstrass of an everywhere continuous function that is nowhere differentiable (a typical property of fractals) that "logic can sometimes make monsters that would have to be set grappling with this teratologic museum". But it is certainly Ziemska's fantasy that makes mathematics seem to be some Kabbalistic pseudo-science placing it in the same category as a magician's magic, with the magician anxiously hiding the secrets of his tricks from his public, so that he can perform disappearing tricks in some Hausdorff dimension, impenetrable for ordinary people. But of course this works out very nicely when used in a fantastic story.</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 novella in which Ziemska impersonates the adolescent Benoit Mandelbrot as the narrator telling the story of his youth first in Warsaw and mainly in Tulle in France during the second World War. Ziemska has added several of her own imaginative components to the story.</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/liz-ziemska" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Liz Ziemska</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/torcom" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Tor.com</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">2017</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">978-0-7653-9805-5 (pbk)</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">10.04 USD (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">128</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.tor.com/2017/10/24/excerpts-liz-ziemska-mandelbrot-the-magnificent/" title="Link to web page">https://www.tor.com/2017/10/24/excerpts-liz-ziemska-mandelbrot-the-magnificent/</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>
<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/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li>
</ul>
</span>
Tue, 10 Apr 2018 09:17:45 +0000adhemar48379 at http://euro-math-soc.eu