European Mathematical Society - the mathematical association of america
https://euro-math-soc.eu/publisher/mathematical-association-america
enHalf a Century of Pythagoras Magazine
https://euro-math-soc.eu/review/half-century-pythagoras-magazine
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is the English translation of the book <em>De Pythagoras Code</em> that appeared in Dutch in 2011. It brings an anthology of puzzles, and papers that appeared in <em>Pythagoras</em>, a magazine appearing in The Netherlands since 1960 and that wants to bring recreational and popular mathematics to youngsters (and of course for anyone casually or professionally interested in mathematics).</p>
<p>
There are of course many books on the market today that bring a mixture of puzzles, history, and art all related to mathematics. Many of the items discussed in these books are circulated and iterated so that they became a part of mathematical folklore. This book is essentially different in that most of the items are original contributions that were harvested from the issues published in half a century of the magazine's existence. And there are some real gems among them. The offer is also very diverse, both in type of contribution, in complexity and in skill required to find the solution. The problems are often challenging and their solutions are surprising. Enough properties to attract any passionate puzzler.</p>
<p>
The book starts off with a hundred brainteasers. Short problems, formulated in only a few lines, and occasionally a plot. For example how to cut the ying-yang symbol into two congruent pieces or how to generate all numbers from 2 to 20 in the simplest possible way using only four 4's combined in brackets and algebraic operations.</p>
<p>
A second chapter has more complex problems, like card tricks, board and other games (and their winning strategies), cutting puzzles, knot puzzles, tangrams, logic puzzles, etc., they are all there.</p>
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Some text related to art and mathematics are collected in the next chapter. Of course, there is Escher and impossible objects (Bruno Ernst, Escher's biographer was one of the founders of the magazine). Artist Popke Bakker's raw material are wooden beams with a square cross section. He cuts them under certain angles and glues the pieces back together after rotation, which gives artistic skeleton structures. Many art forms are represented such as perspective drawing, architecture, but also the writings of Raymond Queneau and others.</p>
<p>
There is a rather extensive chapter four devoted to geometry, plane geometry as well as three-dimensional geometry. Here mainly classical problems and theorems are discussed. For example how one may compute the diameter of the Earth. The reader is amazed by the surprising height above the ground that is obtained when you enlarge a rope around the Earth by one meter and then lift it uniformly above from the ground. But also classical theorems in triangles, Platonic solids, Euler's formula for polyhedra, Penrose tiling, etc.</p>
<p>
Numbers is the title of chapter five. We meet series and sequences, of course pi and e, the abc conjecture (an as yet unproved conjecture that claims that if a and b are relative prime positive integers with sum equal to c, and r is the product of the prime factors of a,b, and c, then c is smaller than the square of r), etc.</p>
<p>
The last chapter collects 50 of the original puzzles that Dion Gijswijt provided for the problem section of the magazine. These are somewhat like the brainteasers of the first chapter, but more challenging. If ever, you do not succeed in finding the solution for yourself, (and I believe there is a reasonable chance that this will happen for some of these problems), then you may look up a solution for these (and for all the other problems) at the end of the book.</p>
<p>
Every puzzle lover will enjoy this book and find many original gems, unless he or she had a subscription to the magazine, but then the Dutch edition of this book would have been a previous occasion to acquire this collection. They may still be happy with this English translation because it is an excellent give away present if they have non-Dutch speaking friends. For anyone who does not read Dutch, this is a marvelous occasion to have the best of the best of the magazine available to them too.</p>
<p>
A last word about the marvelous cover of the book. Fritz Beukers plotted the interval [0,1] by covering every rational number p/q in the interval by a disk with of radius depending on q and using different colors. This does not only give a nice picture, but there is also a surprising theorem attached to it: if the radius of the disk covering p/q is c/q² then the interval is completely covered for c > (3-√5)/2 but for 1/3 < c < (3-√5)/2 there is a finite number of points left uncovered.</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 collects the best of the magazine <em>Pythagoras</em>. That is a Dutch magazine whose goal is to popularize mathematics among youngsters (and interested adults of course). The book has a selection of brainteasers, witty mathematical puzzles and some texts relating mathematics and art.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alex-van-den-brandhof" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alex van den Brandhof</a></li><li class="vocabulary-links field-item odd"><a href="/author/jan-guichelaar" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jan Guichelaar</a></li><li class="vocabulary-links field-item even"><a href="/author/arnout-jaspers" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Arnout Jaspers</a></li><li class="vocabulary-links field-item odd"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/mathematical-association-america" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">the mathematical association of america</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">2016</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">9780883855874 (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">£29.00 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">317</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.maa.org/press/books/half-a-century-of-pythagoras-magazine" title="Link to web page">http://www.maa.org/press/books/half-a-century-of-pythagoras-magazine</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span>Mon, 02 May 2016 10:40:35 +0000Adhemar Bultheel46912 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/half-century-pythagoras-magazine#commentsCalculus and Its Origins
https://euro-math-soc.eu/review/calculus-and-its-origins
<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 an historical approach to calculus. In ten chapters, a rough outline of calculus is given, approaching the topics discussed such as they were originally addressed by their "inventors", but using familiar modern concepts, terminology, and notation. To give an example: chapter 1 discusses infinite sums and how they also show up in the ideas of Archimedes on computing the area of a parabolic segments, filling up the area with smaller and smaller triangles. In a concluding section called "Furthermore" it is again Archimedes who uses successive polygons to approximate the area of a circle and hence the value of π. It is a further challenge/exercise to find a formula that gives the sum of the squares and cubes of the first n integers.</p>
<p>
The other chapters follow a similar pattern, each time ending with the "Furthermore" section giving additional information of historical mathematicians, and some (usually rather elementary) exercises. In this style we meet Ibn al-Haytham an Jyesthadeva again on infinite sums. Among others, the reader meets Fermat and Descartes and later Cavalieri and Roberval and their study of curves and the computation of areas and volumes, which bring the reader to the 16th and 17th century. Although the exposition still relies largely in graphics, there is gradually more algebra sneaking in. Still looking at areas under curves like the hyperbola leads to the concept of logarithms and the exponential function as developed by de Sarasa, Brouncker and Wallis. Enter Newton and Leibniz. They lay the foundations of differential calculus, the interpretation of differentials and introduce a notation that looks familiar to a reader of the 21st century. The final chapter is then about continuity as discussed by Bolzano, Weierstrass, Dirichlet, and others. They introduce more rigor and are gradually leaving intuition behind. The few names of mathematicians that I mentioned above are only some examples. Many more that have contributed are mentioned in due course.</p>
<p>
The topics are treated in a rather intuitive way, using extensively the many figures of the text. These are manually produced with ruler and pencil, which in time of computers is somewhat surprising, but it blends well with the general approach of the book. There are of course formulas, but no formal proofs are included. So it is not surprising that the book stops with the chapter where the mathematics take off to more abstraction and less intuition.</p>
<p>
The book is a collection of topics that more or less follows the evolution of calculus throughout the centuries. It is neither a full history of mathematics or even calculus, nor is it a collection of biographies of prominent mathematicians. It is a mixture of these, with emphasis on the mathematics itself. If used to teach calculus, then it is certainly an unusual approach. Many topics are not discussed and there is no classical set of drilling exercises for derivatives and integrals as is usual in a calculus course. It may be a good source of inspiration to formulate some assignments for homework if used as a textbook besides a more traditional course.</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">A. Bultheel</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">KU Leuven</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 10 chapters, some historical approaches to elements from calculus are explained. This involves summation of infinite series, computation of surfaces and volumes, limits, continuity and differentiability. Proofs are by figures and intuition. Each chapter ends with problems additional information and suggestions for further investigation.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-perkins" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">david perkins</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/mathematical-association-america" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">the mathematical association of america</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">2012</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">978-0883-8557-51 (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">£45.00 (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">180</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="http://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/calculus-and-its-origins" title="Link to web page">http://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/calculus-and-its-origins</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li></ul></span>Fri, 08 Feb 2013 17:10:28 +0000Adhemar Bultheel45487 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/calculus-and-its-origins#comments