Book reviews
http://euro-math-soc.eu/book-reviews
Book reviews published on the European Mathematical Society websiteenPtolemy's Philosophy
http://euro-math-soc.eu/review/ptolemys-philosophy
<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>
Claudius Ptolemy (c 100 - c 160 AD) is best known for his <em>Almagest</em>, a collection of 13 books that formed a treasure trove for astronomers and historians because of the historical collection of astronomical data they contain. He also wrote the <em>Tetralibris</em> consisting of four books about astrology, and in his <em>Planetary Hypotheses</em> he describes the universe as a set of spheres defining the movements of the celestial bodies either via an epicyclic or an eccentric model. Circular motion was essential as it represented mathematical and hence divine-like perfection. He was not only an mathematician/astronomer since he also left us his <em>Geographia</em> with geographic coordinates of the world as it was known in his days, and in the <em>Optics</em> he studies properties of light.</p>
<p>
All these texts are used as sources by Feke although her book is not about the astronomy or the mathematics, but about the philosophy that Ptolemy is conveying in his books. He lived in the midst of what is often considered to be a period of philosophical eclecticism where elements were freely imported from Aristotle, Plato, Epicurus, or Stoics. Elements of all these are indeed found in Ptolemy's vision, however with a twist of his own, as Feke clearly explains. In that respect the beginning of the <em>Almagest</em> and even more so his book <em>Harmonics</em>, in which he mainly discusses the application of mathematics to music, are crucial elements to understand the role of mathematics for his philosophical and ethical views. These are the ones that Feke cites most. The perfect ratios of the harmonic pitches, are reflected in the harmony of the stars in heaven and that divine commensurable organized constancy should also be a guide for the human soul. Mathematics is the only way to gain knowledge and mathematical ethics will furnish a good life since it will transform the soul into a godlike condition by mimicking the perfection of the stars. Let me quote the strong statement by which Feke summarizes Ptolemy's vision in the last sentences of her book:</p>
<blockquote><p>
[...] according to Ptolemy, the benefits mathematics provides are epistemological and ethical. Mathematics is the only field of enquiry through which human beings can acquire knowledge, and it is the only path to the good life. In Ptolemy's philosophy, the best way an individual can live his life is to live the mathematical way of life.</p></blockquote>
<p>
To come to such a conclusion, some explanation is in order.</p>
<p>
</p>
<p>
According to Ptolemy, theoretical philosophy has three parts: theology, physics, and mathematics. The object of theology is the immaterial "Prime Mover" who governs the motion of the stars but who is imperceptible. Physical objects are of course perceptible but mathematical objects are in between, since the latter can be observed with your senses or thought of as abstractions. Mathematics is however the only way to generate knowledge because the Prime Mover is too far out and cannot be perceived, while in physics, one has to measure and make observations, which are necessary to define the parameters of the mathematical model but observations are always inaccurate. Thus theology and physics can only lead to conjectures, while mathematics generates knowledge. Moreover, mathematics is useful since if can help to predict the outcome of astronomical and physical phenomena.</p>
<p>
Feke claims that the ethical aspects of Ptolemy's vision are the result of his solution for a contemporary debate over the relation between theoretical and practical philosophy. Ptolemy wrote a pamphlet about this, known as <em>On the Kritêrion and Hêgemonikon</em>. It gives a criterion for truth, explains how knowledge can be acquired and describes the structure of the soul, of which the <em>hêgemonikon</em> is the ruling part, located in the brain. His authorship is somewhat controversial since it is not completely in line with what he wrote elsewhere. So Feke assumes that it was written at an earlier stage before his <em>Harmonics</em> in which his mathematical methodology was fully developed. Ptolemy considers the soul to be something mortal and physical, with faculties in different parts of the body, only it consists of much finer particles than the body, so that it evaporates much faster after death. The same harmony of perfect ratios of the musical pitches, and the mathematical study of this harmony will generate the beauty in mathematical objects. This has ethical and practical applications because mathematical models will transform physical matter into a perfect form, and likewise, when applied to the soul, it will lead to an orderly, calm, commensurable good life that, as was quoted above, resembles the astronomical properties of the stars, and hence approximate a perfect divine status.</p>
<p>
What Ptolemy describes in his <em>Harmonics</em> are the rational and unavoidable rules that govern musical pitches, the heavenly bodies and the human soul alike. These harmonic relationships and the role of mathematics is what Feke explores in further detail in the subsequent chapters discussing Ptolemy's astronomy, the human soul, astrology, and cosmology. I will not report on all her fine dissections, but I will just mention how she sees Ptolemy's view on the role, the interplay, and the relative importance of the classical subjects of the quadrivium: arithmetic, geometry, music, and astronomy.<br />
According to Ptolemy, geometry and arithmetic are not mathematics but they are just methods. Geometry can be applied to study optics via observations, arithmetic can handle the harmonic ratios that are provided by mathematical theory. Astronomy relies on both geometry and arithmetic: it requires geometry for the optical observations, while arithmetic can deal with the harmonics that are intrinsically derived from the theoretical model.<br />
Cosmology (the study of the heavenly bodies) and astrology (the effect of the movement of the stars on sublunary bodies) are both physical sciences that Ptolemy bases on astronomy which he thus considers superior to cosmology and astrology. Therefore the results of the latter two are predictive but remain just like in physics only conjectural,</p>
<p>
In a book like this, the precise meaning and nuances of a word are important and for that Feke gives precise references and quotations from the work of Ptolemy. Often the translation is in English in the text, while the Greek text from the source is in a footnote. She also relates meticulously the interpretation that Ptolemy attached to some concept and she discusses the difference or the similarity to corresponding notions of Aristotle, Plato or other of the ancient philosophers. Occasionally she also refers to interpretations given by more modern authors who wrote on Ptolemy's work and ancient Greek philosophy. That does not happen very often because her book is mostly historical. Only in her concluding chapter she briefly discusses Ptolemy's influence in later centuries. Even the way she is harvesting an idea of Ptolemy from his different books is impressive. One can read sentences like "Ptolemy used this word at only one other place namely...". Ptolemy's philosophical ideas are indeed scattered all over his mathematical books and it requires a thorough knowledge of all the texts to crystallise them into clear English sentences.</p>
<p>
As I said before, Feke shows that mathematics is for Ptolemy the science that rules everything and to which humans should submit to lead a good life, but her book is far from dealing with mathematics as such. This is a book written by a philosopher mainly for fellow philosophers. Thus if the reader is a mathematician who is not so familiar with the philosophical jargon and writing style, it may require some pages to adapt, but Feke does a good job to make it a comfortably readable text also for an interested non-philosopher. I am not a philosopher myself, so I may not be aware of all the professional philosophical literature, but my search on the Web seems to confirm that this is indeed the first fully-fledged book bringing the philosophy of Ptolemy to the foreground (as it is announced in the blurb on the back cover). Neither have I the background to agree or disagree with the conclusions that Feke distils from the available sources, so I leave that to her peers. As a mathematician, I can only be warmed by a tingling ASMR experience while reading the conclusion that she arrived at in the quote formulated above. I am so pleased by Ptolemy's tap on my shoulder reassuring me that it has not been a waste of time to have devoted the larger part of my life to mathematics.</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>
By a careful analysis of the philosophical elements in Ptolemy's mathematical work, Feke shows that mathematics is the driving force for his philosophical and ethical views that can be summarized as follows. Only through mathematics one can acquire knowledge, and the mathematics that can analyse the rations of harmonic pitches in music is equally applicable to the stars in heaven, to predict the behaviour of physical objects, and to the human soul. Hence mathematics will lead humans to live a good life.</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/jacqueline-feke" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jacqueline Feke</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">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">978-0-69117958-2 (hbk); 978-0-69118403-6 (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">£ 30.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">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://press.princeton.edu/titles/13256.html" title="Link to web page">https://press.princeton.edu/titles/13256.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/00a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A30</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/01a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A20</a></li>
<li class="field-item odd"><a href="/msc-full/97e20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97E20</a></li>
</ul>
</span>
Thu, 08 Nov 2018 09:23:52 +0000adhemar48815 at http://euro-math-soc.euA short course in Differential Topology
http://euro-math-soc.eu/review/short-course-differential-topology
<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>There are several very good books on Differential Topology, the first to mention Milnor's gem that everybody knows and loves. However, it is difficult to use those books as a text for an introductory one term 60 hours course where you want to teach the basics form scratch (as far as possible, taking into account the nature of Maths degrees nowadays) and get in the end to some interesting applications. This being the situation one confronts often in the real classroom today, this text by Professor Dundas is a wonderful proposal to welcome. Indeed, the topics (summarized in the short description below) are presented in a very friendly and appealing way to motivate abstract notions; this includes many nice examples and assorted references to web links worth clicking. Quite useful too, some parts are explicitely marked when they can be omitted to have all the same a good program. There are in addition some pluses to stress: (1) the use of germs to discuss tangencies, (2) a presentation of vector bundles in general, (3) an introduction to Morse functions and (4) a very nice proof of Ehresmann theorem. Another remarkable feature is the care taken to provide a good bunch of meaningful exercises.</p>
<p>Summing up, the book is highly recommendable for all publics. An interested student can very well go through it quite by her/himself and learn a lot. A professor surely can discover useful aspects that may have skipped her/his attention before. But most important, Professor Dundas offers us a very enjoyable reading!</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">Jesus M Ruiz</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 text designed for a real course on Differential Topology for postgraduate students. The topics are as expected: smooth manifolds, tangent spaces and derivatives, regular values and transversality, vector bundles, tangent fields and flows and some global basic facts. There are many examples carefully chosen and described for motivation, as well as many exercises with hints to help the student. A very good book on the 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/bjorn-ian-dundas" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Bjorn Ian Dundas</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/cambridge-university-press-cambridge-textbooks" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Cambridge University Press: Cambridge Textbooks</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">978-1-108-42579-7</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">50 EUR</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">263</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.cambridge.org/core/books/short-course-in-differential-topology/F961891A6076D7B5EDC1A12A29D8F5B1" title="Link to web page">https://www.cambridge.org/core/books/short-course-in-differential-topology/F961891A6076D7B5EDC1A12A29D8F5B1</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li>
<li class="field-item odd"><a href="/imu/topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Topology</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/58-global-analysis-analysis-manifolds" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">58 Global analysis, analysis on manifolds</a></li>
</ul>
</span>
Mon, 05 Nov 2018 21:47:53 +0000JMRz48809 at http://euro-math-soc.euMillions, Billions, Zillions
http://euro-math-soc.eu/review/millions-billions-zillions
<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>
Journalists reporting on somebody else's results may easily be mistaken in citing the numbers that are not theirs, Perhaps authors change numbers on purpose to bias their arguments. In such cases unprepared and naive readers are easily deceived. In this book you can learn how to defend yourself from mistakes as an author and from deceptions as a reader.</p>
<p>
Kernighan is the guide on a tour where he shows all the number traps that people can easily fall into. There are of course the big numbers like millions and billions mentioned in the title. It is hard to get a mental idea of what they actually mean. As a consequence an interchange of millions and billions is a mistake easily made without being noticed. One should also be aware of what the really big numbers actually stand for when they are indicated by prefixes like mega, giga, tera, and peta. Even exa, zetta, and maybe yotta can come into the picture. In modern texts these terms regularly occur referring to the large amounts of digital data stored in for example the Deep Web. At the other end of the spectrum we should know something about micro, nano, pico, femto, atto etc. when reading about the tiny parts of the hardware on which these data are stored, or even further down the scale when reading about the Theory of Everything where the natural playground is at a subatomic level. Like in mathematics the super large and the super small often match to keep everything within finite boundaries.</p>
<p>
To avoid errors of course the units have to be right. Mistaking barrels for gallons, years for months, or hours for seconds may give quite unexpected and hard to believe interpretations of the numbers that are on display. Besides mistakes in scales, there is the extra complication that there are different systems of units like American miles, gallons, or degrees Fahrenheit that should somehow match with European kilometres, litres and degrees Celsius. Even a mile can have many different meanings in different contexts. All this requires carefully introducing the proper conversions. Just picking up a number from a website may easily lead to such mistakes if the proper conversion is not made. Another typical mistake is to confuse a square mile and a mile squared. This means that you should be aware that if you double the length of the sides of a square you get a surface four times as large. For a volume it is even more dramatic because that will give a volume or mass that is eight times as large.</p>
<p>
The previous examples are possible sources of mistakes. How should we detect them and how could we protect ourselves against malicious attempts to deceive us? We could compare different sources or different ways of computing. When the results are approximately the same we can probably trust the numbers. It is of course useful to check with some numbers from your own experience or numbers that you know like for example the population of your country. Approximate rounded values for these numbers can be used in crude and simple computations to get at least an idea about the magnitude of the number that should be expected. If the number presented to you deviates considerably, either it is fake or your computations are wrong. For example <em>Little's Law</em> is applicable for a simple check in many situations. Given the population of your country and an average human life span, Little's Law allows you to estimate for example the number of people that will turn 64 twenty years from now.</p>
<p>
Usually numbers appear in non-scientific texts with approximate rounded values. If one spots a specious number that is given with many digits, then it is probably the result of a computation or conversion and only the most significant digits or rounded values are actually meaningful. They are probably the result of some conversion, like a mountain over 4.000 meter high should not be referred to as over 13.123 feet. Statistics is another possible source of deception: the median is not the average, a correlation does not imply causation. It might also be interesting to know who did the statistics and the sampling or polling. Results may be consistently biassed towards the results of which some lobbyist or pressure group wants to convince you. Another well known trick is to fiddle with the scales used in the graphical representation of the numbers in pie or bar charts. Numbers representing a percent are again possible pitfalls that can put you on the wrong foot. A percent is definitely different from a percent point and you should also be aware that a percent increase is computed in terms of the lower number, while a percent decrease is referring to a percent of the larger number: a 50% decrease can only be compensated by a 100% increase.</p>
<p>
Kernighan gives many examples of all these issues, mostly from newspapers and websites. He also keeps his readers alert by continuously pushing them to do some mental calculation to estimate some results for themselves. As some kind of a test at the end of the book he gives many such problems that one should be able to approximately solve (he also gives his own estimates): How many miles did Google drive to get the pictures for Street View (for your country)? How long did that take? How much did it cost? Or, if you have a garden with some trees in it, how many leaves do you have to rake every autumn? And there are many of these so called <em>Fermi problems</em> throughout the book. Kernighan gives some tricks to solve them, hence the "test" at the end. However it certainly requires a lot of practising and training which the reader has to do for him or herself to acquire some routine in this,</p>
<p>
In this time of "fake news" and in a society that is more and more spammed by numbers, it seems like problems of numeracy among a general public is gaining interest and public awareness. More books devoted to different aspects of this issue seem to appear lately. Among the earlier examples are the books by John Allen Paulos <em>Innumeracy</em> (1988) and <em>A mathematician reads the newspaper</em> (1996). Kernighan mentions them in his survey of "books for further reading". However several more books appeared since 2010. Almost simultaneously with this one I received <em>Is This a Big Number?</em> (2018) by Andrew Elliott which is also reviewed <a href="/review/big-number" target="_blank">here</a>. Elliott has a more positive approach to the problem: how should I interpret the numbers presented to me (assuming they are correct) while Kernighan is more defensive: how to to stay away from mistakes or deceiving numbers.</p>
<p>
It should also be noted that Kernighan uses American units most of the time, and his examples are mostly related to the American situation, or from the American newspapers. His advise is of course generally applicable, but it might be an extra hurdle to take for European readers who are used to the metric system of metre-litre-gram. It is surprising that Kernighan does not discuss the difference between the short scale (a billion is $10^9$) and long scale (a billion is $10^{12}$) hence also missing the milliard ($10^9$) and billiard ($10^{15}$) and giving different meaning to trillions, and nomenclature higher up. These are also obvious ways to get the wrong numbers cited.</p>
<p>
This book does not need mathematics to read and it is actually not about mathematics at all. There is not a formula made explicit, even though the rule of 72 is explained (it takes 72/x units of time to double your capital when it has a compounding interest of x percent) and an idea is given about what it means to grow exponentially or by powers of 10 or powers of 2. The style of Kernighan is fluent and casual, but not particularly funny. The charm sits in his continuous teasing to make you think of these Fermi problems, and of course in his ample illustrations of how often authors are mistaken in citing numbers and how easily a reader can be deceived.</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 about numeracy learns you how to defend yourself against making mistakes with numbers or to recognize incorrectly cited numbers by explaining what the possible sources of these errors are. On the positive side, one learns how to solve Fermi problems, that is to make rough numerical estimates of certain quantities, using little available data and where the few computations can me made "on the back of an envelope".</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-w-kerninghan" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Brian W. Kerninghan</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">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">978-0-691-18277-3 (hbk), 978-0-691-19013-6 (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">£ 17.99 (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/14171.html" title="Link to web page">https://press.princeton.edu/titles/14171.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/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/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li>
<li class="field-item odd"><a href="/msc-full/01a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A80</a></li>
</ul>
</span>
Wed, 24 Oct 2018 12:08:48 +0000adhemar48773 at http://euro-math-soc.euIs That a Big Number?
http://euro-math-soc.eu/review/big-number
<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>
Mathematics is much more than numbers, but in an historical as well as in an educational context numbers are certainly essential for the development of mathematical skills. Counting is one of the primitive mathematical activities, but as societies became more complex, numbers became much more essential for the socio-economical fabric. Sometimes the balance tips the wrong way by reducing something or even someone to just a number expressing an amount of a particular something that can be associated with them. It has become more and more important to have an idea of what a number means or what it stands for. Whether it is small or big in the context it is used. Think of the kcal of the food we eat, the GNP or the debt per capita or the happiness index of your country, the meaning of Olympic records, or the area of forest destroyed by wildfire. This book is about the numeracy that people should gain to be able to understand the meaning of numbers in our daily life so that they can discuss about them knowing what they are talking about. This understanding of numbers has been promoted before by J.A. Paulos in his book <em>Innumeracy</em> (Hill and Wang, 2001) and his semi-autobiographical sequel <a href="/review/numerate-life-mathematician-explores-vagaries-life-his-own-and-probably-yours" target="_blank"> <em>A Numerate Life</em></a> (Prometheus, 2015).</p>
<p>
To get an idea of what is small and what is big, one should be able to get a mental picture of what a number means. So the book starts with enumerating five ways to attach a meaning to the adjectives "big" or "small".</p>
<ol><li>
<em>Landmark numbers</em> are numbers that one should memorise so that they are readily available for comparison</li>
<li>
<em>Visualisation</em> means to use your imagination to get a mental picture of the amount</li>
<li>
<em>Divide and conquer</em>: break a larger number up into smaller parts (e.g. x rows of y columns having stacks that are z high)</li>
<li>
<em>Rates and ratios</em> are usually smaller and often more relevant for what the numbers mean</li>
<li>
<em>Log scales</em> will bring numbers varying over a wide range to within reasonable bounds and separates them better in dense parts of the range</li>
</ol><p>
These ideas are illustrated abundantly with an incredible number of facts. Some are marked in frames containing possible landmark numbers, others are enumerated in lists of random alignments or number ladders. Random alignments are lists of (unrelated) numbers that happen to appear in almost perfect ratios (like for example the height of St. Paul's in London which is 100 times the height of R2D2 from Star Wars). Such lists are usually found at the end of the chapters. Number ladders are lists of increasing or decreasing numbers that run through a whole scale (for example the weight of animals ranging from an Indian flying fox (1 kg) to a blue whale (110 tons)). Each chapter also starts with a multiple choice question for the reader to test his/her numeracy (like which is largest in a list of four populations). The answers and references for the latter are given at the end of the book.</p>
<p>
</p>
<p>
The chapters are grouped into four parts that reflect the context in which the numbers appear. They are also somewhat arranged in historical order. There is in fact also a lot of history and etymology about the origin of names that have been invented to indicate units by which quantities are measured, or the origin of terms used in for example our monetary system. The metric system has simplified a lot, but before that, there were many different words for units to measure a quantity, and it may be a surprise to read how many are still in use that have escaped metric standardisation.</p>
<p>
Part 1 is about counting, which is our oldest use of numbers. When it gets to really big numbers (how many stars in the sky?) or in other instances (what percentage of alcohol in your beer?) it involves more than "just counting 1,2,3,...". The reader is instructed how to "visualise" numbers mainly by the first two methods of the list above. Methods 3 and 4 of the list are better illustrated in the second part.</p>
<p>
In that second part, we learn about measures. A spatial dimension was historically often measured using body parts. This still resounds in our names for units of length like inches, feet, fathoms, and other less known units. Time has for obvious astronomical reasons escaped the decimal subdivision that is now used in the naming of space dimensions. It is the reason why twelve or sixty (which happen to have many divisors) have played a basic role in early mathematical cultures, and we still use the terms "dozen" and "gross" today (strange that Elliott didn't mention these two terms). The hierarchical subdivision of (pre)history in time spans like ages < epochs < periods < eras < eons (where x < y means that y consists of several x's) is an illustration of the fact that a divide and conquer technique is a way to get an idea of our geological time scale. To discuss history, it is important to get at least a rough idea about the rise and decline of the different civilisations that directly or indirectly had an influence on the age we are currently living in. So both historical and prehistorical periods of time get much attention in the book. In this part it is also illustrated that to measure areas, size needs to be squared and for volumes (and hence also for mass and weight) size must be cubed. It explains why the legs of an elephant are relatively much thicker than the legs of a mosquito. The strength depends on the cross section (size squared) and the weight on the volume (size cubed). That is why a giant Godzilla cannot be real. Scales are used to quantify wind speed and hurricanes or earth quakes. These scales are actually logarithmic. So log-scales are first introduced in a (lightly) technical intermezzo (which also mentions Benford's law, the slider rule and mortality rates).</p>
<p>
Numbers get much bigger in the third part dealing with numbers in science. For example naming astronomical distances, measuring energy or capacity of a digital memory, or quantifying the information content of a text. Also measuring the complexity of solving combinatorial problems dives into the large numbers very fast. The fourth and final part is probably the most important for the general public since it treats numbers in a political and socio-economical context. This means among other things money (exchange rates) and economy (GDP), population (quantities and dynamics), quality of life (Millennium Development Goals and happiness index), etc.</p>
<p>
The book illustrates well that knowledge (being numerate) is power. So we should be able to unveil the true value of a number used in an argument and know whether it has to be considered alarmingly big or small, and hence defend ourselves against deceit, fake news, and false arguments which has become an essential skill in our current society. This book is an essential tool if you want to work on this skill for personal use.</p>
<p>
However, what the book does not discuss is that one should be careful with just ranking the numbers. Numbers do represent something, but it may not always be clear what that "something" really is. GNP is considered a measure for the wealth of a nation, but is it? Should military expenses, included in the GNP, be taken into account to measure wealth? Is an IQ really a measure of intelligence? A number will represent the result of a test or a poll, but what the test or poll is supposed to measure is not always clear because it may not correspond to what is actually measured. Moreover, these usually refer to just an sample while more general statements about a much wider population are concluded. In a world where everything is being managed, numbers are used to manipulate and define strategies, but often reality can not be caught in just a number. Believing that the number stands for some effect may be a generally accepted hoax. So the numeracy discussed in this book is just one aspect of being knowledgeable about a topic. Knowing whether a number is small or big is only one element to be taken into account for the interpretation of numbers. Even more important is to know what exactly these numbers measure and how they were obtained. This is not so important for the first three pars of this book since most of the numbers there concern quantities that can be objectively measured. Only in the fourth part when numbers are discussed in a social context, one should be careful with their interpretation and drawing conclusions.</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>
he book aims at promoting numeracy for everybody. When using numbers in arguments, one should have a good idea what these numbers mean and one should know when they are to be considered "small" or "big". An overwhelming set of numeric facts are provided together with some techniques that you can train to obtain a certain level of numeracy. Numbers and facts are situated in the context of simple applications such as counting and measuring and later in more complex situations that can be encountered in science and society.</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/andrew-elliott" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Andrew Elliott</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">978-0-198-82122-9 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£18.99 (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">352</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/is-that-a-big-number-9780198821229" title="Link to web page">https://global.oup.com/academic/product/is-that-a-big-number-9780198821229</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/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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
Wed, 03 Oct 2018 09:18:51 +0000adhemar48716 at http://euro-math-soc.euGreat 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>
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<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>
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<li class="field-item even"><a href="/msc-full/01a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A30</a></li>
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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>
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<ul class="field-items">
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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>
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</ul>
</span>
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<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.eu