European Mathematical Society - 00a65
https://euro-math-soc.eu/msc-full/00a65
enMusic by the Numbers From Pythagoras to Schoenberg
https://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="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/eli-maor" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eli Maor</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/titles/11250.html" title="Link to web page">https://press.princeton.edu/titles/11250.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</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/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li><li class="vocabulary-links 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 +0000Adhemar Bultheel48509 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/music-numbers-pythagoras-schoenberg#commentsThe Turing Guide
https://euro-math-soc.eu/review/turing-guide
<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>
Jack Copeland is a professor at the University of Canterbury, NZ, director of the <a href="http://www.alanturing.net/">Turing Archive for the History of Computing</a>, co-director of the <a href="http://www.turing.ethz.ch/" target="_blank">Turing Center of the ETH Zürich</a>, and he has written or edited several books about Turing and his work. So he seems to be also the driving force behind this new collection of papers devoted to the life and the legacy of Alan Turing. Only four authors are explicitly mentioned on the cover of this book, but the collection contains 42 papers authored by 33 persons with very diverse backgrounds. Fifteen of the 42 papers were (co)authored by Copeland. Four of the papers by older authors (three of them have known or collaborated with Turing) are published posthumously.</p>
<p>
Alan Turing (1912-1954) hardly needs any introduction. Most people will know him as a codebreaker of the German Enigma at Bletchley Park during the second World War. They probably also have heard of his tragic death covered by a veil of uncertainty: was it an accident or suicide. He was convicted in 1952 to chemical castration for having a gay relationship. Only in 2013 he was rehabilitated by a royal pardon. Some may also have an idea of what a Turing Test is. A mathematician or a computer scientist will almost certainly also know that he proved independently but almost simultaneously with Alonso Church that Hilbert's <em>Entscheidungsproblem</em> was unsolvable. Turing proved it by reducing it to a halting problem which is undecidable on a universal Turing Machine. Many books and even films tell the story of Turing and of all the activities at Bletchley Park. The Turing Centenary Year 2012 which triggered the publication of many more and the recent (loosely biographical) film <em>The Imitation Game</em> (2014) have spread the knowledge about Turing in a broader audience. Bletchley Park may now be a major tourist attraction park, but the confidentiality that was kept by the British authorities about what was developed there during the war concerning cryptanalysis and the early digital computers has delayed the historical disclosure of the role played by Turing and other scientists in that period. Somewhat less known, but very familiar to biologists is Turing's work on morphogenesis which he developed during a later stage in his life. The book has eight parts that cluster papers about eight different aspects of Turing's life and legacy.</p>
<p>
Thus Turing was much more than just a codebreaker. His universal machine was an essential theoretical model in proving results about the foundations of mathematics, logic, and computer science. Because of his work at Bletchley Park while the first digital computing machines were being assembled during and just after the war, he was intensively involved in writing original software, a user's manual, and he has even contributed to the design of circuits and hardware. The introduction of machines that could be instructed to perform less trivial tasks raised concern about the future of Artificial Intelligence and Turing contributed with several variants of his Turing test in an attempt to define what intelligence really meant. He called his ultimate version of 1950 the 'imitation game'.</p>
<p>
It should not be forgotten, that, even though his scientific interest and contributions are broad, Turing was fundamentally a mathematician. It is less known that his Kings College Fellow Dissertation (1935) involved a proof of the Central Limit Theorem. It was little known that this was proved already in 1920 by Jarl Lindeberg and so Turing's result was never published. He also worked on group theory, in particular the word problem, on number theory (the Riemann hypothesis and normal numbers) and of course the code breaking involved statistical analysis and hypothesis testing. Turing exploited these statistics in his algorithms Banburismus and later Turingery. After the war he was also doing numerical analysis (LU decomposition, error analysis,...). His work on morphogenesis was also mathematical and involved diffusion equations that model the random behaviour of the morphogenes.</p>
<p>
This collection of papers is produced for an interested but general audience. Formulas are kept to a minimum and technical discussion is maintained at an accessible level. It may not be the best choice to read as a first introduction to Turing and his work. Better introductions that are less chopped up in different papers are available. On the other hand, if you have read already several books about Turing and his work, I am sure you will find here some anecdotes and historical facts that you did not know yet in each of the eight parts of the book.</p>
<p>
A first part is biographical. The timeline by Copeland is useful to place everything in a proper historical sequence. There is a testimony of Sir John Dermot Turing, Alan's nephew, and another by the late Peter Hilton an Oxford professor who worked with Turing at Bletchley Park.<br />
Part two is more history in which Copeland explains about the Universal Turing Machine conceived by Turing to solve the Entscheidungsproblem. It has also a noteworthy contribution by Stephen Wolfram, the creator Mathematica and Wolfram-alpha, who praises Turing for initiating computer science.<br />
The third part is the most extensive one and puts the codebreaking and Bletchley Park in the spotlight. Some of the texts are by people who worked there and who give an account of how everyday life was during the war, other papers are explaining how the Enigma machine worked and how it could be broken.<br />
In part four the first computers as they developed after the war are in the focus. The Colossus machines were computers that were used since 1943 for codebreaking, These facts were only declassified in 2000 so that one got the impression that the original ideas and prototypes came from von Neumann at Princeton who developed the ENIAC and the EDVAC. However, the University of Manchester had a small scale computer <em>Baby</em> (1948) that was running a few months before the ENIAC and Turing at the National Physical Laboratory developed the Automatic Computing Engine (ACE) that was operational in 1950. Turing even wrote a manual on how to program the machine to play musical notes.<br />
The fifth part is about computers and the mind: chess computers, neural computing, and the working of the human brain. It also has a remarkable text by novelist David Leavitt about Turing and the paranormal.<br />
The next two parts are about Turing's biological (morphogenesis) and mathematical (cf. supra) contributions. The final part has two papers contemplating the Turing thesis (1936) which claims that a Turing machine can do any task a human computer can do. Similar claims were made by Zuse and Church, but whether the whole universe can be seen as a computer, obviously depends on what you call a computer.<br />
In the last chapter about Turing's legacy in different disciplines we find many references to books and other media that can be consulted for further information.</p>
<p>
The remaining pages offer a short biography of the contributors, references to some books about Turing, and a list of published papers by Turing. The many references and notes from the contributions are also gathered at the end. The book ends with a very detailed index, which is of course very welcome and obviously non-trivial with that many different authors.</p>
<p>
In summary, this is a welcome addition to the existing generally accessible literature that gives additional testimony of the brilliant mind of Alan Turing. There is historical as well as technical material that will be appreciated also by specialists whatever their discipline: history, mathematics, biology, computer science, or philosophy.</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 papers about Alan Turing, his life and legacy. It has biographical and historical details and explains the influence of Turing on codebreaking, artificial intelligence, computer science, mathematics, biology, and philosophy.</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/jack-copeland" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jack Copeland</a></li><li class="vocabulary-links field-item odd"><a href="/author/jonathan-bowen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jonathan Bowen</a></li><li class="vocabulary-links field-item even"><a href="/author/mark-sprevak" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mark Sprevak</a></li><li class="vocabulary-links field-item odd"><a href="/author/robin-wilson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robin Wilson</a></li><li class="vocabulary-links field-item even"><a href="/author/et-al" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">et. al</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/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">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-1987-4782-6 (hbk), 978-0-1987-4783-3 (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">£ 75.00 (hbk), £ 19.99 (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">576</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/the-turing-guide-9780198747833" title="Link to web page">https://global.oup.com/academic/product/the-turing-guide-9780198747833</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/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</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/00a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a99</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a60</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/03d10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03D10</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/03b07" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03B07</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/68-06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68-06</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/68q05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68Q05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/92c15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92C15</a></li></ul></span>Tue, 13 Mar 2018 07:33:47 +0000Adhemar Bultheel48322 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/turing-guide#commentsAll sides to an oval
https://euro-math-soc.eu/review/all-sides-oval
<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>
It is not difficult to define mathematically what an ellipse is. Its Cartesian equation is well known. It is however less clear what an oval is. Most people will come up with the condition that it looks like an ellipse. It is a smooth convex closed curve of the plane with two orthogonal symmetry axes. But how to be more precise? Since antiquity, ovals have been used in architecture. So what was the construction used by the architects?</p>
<p>
There are the Cartesian and the Cassini ovals, that have a simple Cartesian equation, but they do not always have the symmetry (in th first case) or are not convex (second case). Historically however the ones that have been used most in arts, especially in architecture, are the polycentric ovals that consist of circular arcs that are stitched together in a smooth way. This is the only kind of ovals that is considered in this book.</p>
<p>
The easiest and most popular one that has been studied thoroughly is the so-called four center oval. It consists of four circular arcs that fit together smoothly. Two arcs with smaller radius are at the tops of the long symmetry axis and the ones at the end points of the shorter symmetry axis have lesser curvature because they belong to a circle with a longer radius. The crux is of course to choose these four centers of the circles in such a way that the arcs fit smoothly together at the four connection points. How does one have to select these centers and how large should the sector angle be that supports the arcs so that one does indeed get this smooth transition? Because of the symmetry, two centers are located symmetrically on the long axis and two on the shorter one. So it suffices to consider only a quarter of the oval and find two of the centers to define the arcs and the connection point where the arcs meet in that quadrant. Once the length of the axes are given, an ellipse is completely defined. For an oval, one needs at least one more parameter, like the distance from one of the centers of the circles to the center of the oval or the distance of the connection point to one of the symmetry axes.</p>
<p>
Once these arguments have been formulated, it needs some analysis of the geometry of the problem. And that is where this book gets started. The author has, besides other interests, a knack for polycentric curves like eggs or ovals. This book is restricted to ovals, and the first chapter analyses the properties that will enable us to relate the different parameters. Once this is cleared out, the construction with ruler and compass of an oval (actually a quarter of an oval, because the rest follows by symmetry) is given step by step depending on which parameters are prescribed. So one might choose three of the six possible parameters in many different combinations and that gives rise to twenty different ways to define and construct an oval satisfying the data. Some are more complex and some have more restrictive conditions than others. The solution may not always be unique. Everything is clearly explained and the many illustrations produced with geogebra are crystal clear. It might however be interesting to have a look at the associated website <a href="http://www.mazzottiangelo.eu/en/pcc.asp" target="_blank">www.mazzottiangelo.eu/en/pcc.asp</a> where you find links to YouTube videos showing animated geogebra constructions. The link goes both ways: you may consider this book as a manual for the online site, or the online site as an illustration for the book.</p>
<p>
Besides the parameters described above, one might also choose for one of the radii of the arcs or the ratio of the axes or the angle formed by a symmetry axis and the line joining the circular centers of the arcs. With all ten parameters, there are a total of 116 possibilities to construct the ovals, many of which, but not all, reduce to the twenty constructions mentioned before. Some of the constructions are historical and often pretty old, but others are surprisingly recent. For particular choices of the parameters, the construction may simplify considerably or the oval may have especially pleasing esthetic properties, which are discussed in a separate chapter.</p>
<p>
Towards more practical applications of stadium design, one may consider ovals circumscribing or inscribed in a rectangle. If the symmetry axes are the middle-lines of a rectangle and the diagonals of a rhombus, then all previous constructions circumscribe the rhombus and are inscribed in the rectangle. For a stadium one should find an oval circumscribing the inner rectangular field (for example a soccer field) and surround it by ovals like running tracks, all inside an outer rectangle defining the limitations of the stadium. Modern constructs however have straight parts for the running tracks along the long sides.</p>
<p>
While the constructions are mostly obvious, it takes more algebra and more formulas to express some parameters as a function of others. This is a short chapter, but essential to find ovals that are optimal in some sense. For example finding the "roundest" oval with given axes. They are also needed in geogebra animations when slider rules are provided allowing to see the effect of changing a parameter.</p>
<p>
The last two chapters discuss ovals in two famous architectures in Rome: the dome of the church <em>San Carlo alle Quattro Fontane</em> by the architect Borromini and the ground plan of the <em>Colosseum</em>. A careful study is made of the ovals of the base of the dome in the church, the rings of coffers, and of the lantern. It turns out that there are small defects making them deviate from perfect mathematical ovals. This has long been a mystery. It is suggested that the starting point was a mathematically perfect oval, but that practical restrictions entailed heuristic corrections. The solution that Mazzotti proposes here corresponds remarkably well with Borromini's original drawings.<br />
For the Colosseum, we have to leave the simple ovals with four centers and go to quarter ovals consisting of more than two arcs. Because of symmetry there have to be always $4n$ centers. Again constructions of such ovals are considered. In the case of the Colosseum, $n=2$, i.e., ovals consisting of eight circular arcs seem to match the ground plan perfectly well.</p>
<p>
This is a very nice geometric application that requires only simple algebra and that can be easily experimented with. You do not need to be a mathematician to enjoy it. It that sense, it might be interesting to have the geogebra source available somewhere, which is unfortunately not the case. Also historians might be interested in the last two chapters about historical buildings. For the mathematician, it is invaluable because it brings together so much information that was either not known or never writen down or if it was, then at least it was scattered in diverse publications. The graphics are very readable since they use colors (except for the pictures in the last two chapters, only red, green, and blue suffice for the mathematical constructions). As a LaTeX purist, I cannot resist mentioning my irritatin when seeing variables mentioned in roman font when in a sentence, while they are in a different font when used in a formula. Also, I do not understand why the ratio of the half symmetry axes is denoted at least twice as $\frac{p=\overline{OB}}{\overline{OA}}$ (p.20 and 148) and when at the end of a line $p=\frac{\sqrt{2}}{2}$ is split into $p$, which is left dangling at the end, and $=\frac{\sqrt{2}}{2}$ at the beginning of the next line (p.102). These are however minor flaws in an otherwise nice text, and as I am sure, these will disappear in a next edition. Do not let this prevent you from reading this most enjoyable book and you should certainly try out some of the constructions for yourself, either with ruler and compass or with geogebra.</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 restricts to polycentric ovals, which means that they concist of circular arcs that fit smoothly together. Some properties are derived to allow for many different ruler and compass constructions. The major part of the book is about the case of simple ovals, i.e., ovals concisting of four arcs. They can be constructed when 3 parameters are given (like location of the four centers, the length of the symmetry axes or the location of the points where the arcs meet). The book ends with the discussion of ovals in two historic buildings in Rome: the dome of the <em>San Carlo alle Quattro Fontane</em> church by Borromini and the ground plan of the Colosseum.</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/angelo-alessandro-mazzotti" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Angelo Alessandro Mazzotti</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/springer-international-publishing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer International Publishing</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-3-319-39374-2 (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">31,79 € </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">170</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/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li><li class="vocabulary-links field-item odd"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li></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.springer.com/gp/book/9783319393742" title="Link to web page">http://www.springer.com/gp/book/9783319393742</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/51-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51 Geometry</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/51-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-01</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/51m04" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51M04</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</a></li></ul></span>Wed, 22 Mar 2017 08:49:13 +0000Adhemar Bultheel47565 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/all-sides-oval#commentsThe Fibonacci Resonance and other new Golden Ratio discoveries
https://euro-math-soc.eu/review/fibonacci-resonance-and-other-new-golden-ratio-discoveries
<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 book deals with the Golden Ratio $\phi$, Fibonacci numbers $F_n$ and friends. And there are a lot of friends. The Fibonacci numbers satisfy $F_n=F_{n-1}+F_{n-2}$ with $F_0=0$ and $F_1=1$ and $\phi=(1+\sqrt{5})/2$ which is the limit of $F_{n+1}/F_n$. For this book, the Binet relation $F_n=(\alpha^n-\beta^n)/\sqrt{5}$ with $\alpha=\phi$ and $\beta=1/\phi=(1−\sqrt{5})/2$ is even more important. Throughout history in human artwork, the Golden Ratio 1:$\phi$ has repeatedly appeared, but also 1:$\phi^s$ with $s$ an integer (positive or negative) or one half. Another important player is the logarithmic spiral (the Golden Spiral) with polar equation $r=\phi^{b\theta}$ with $b=2/\pi$. This means that with every quarter circle, the value of the radius is an integer power of $\phi$ anchored at $\phi_0=1$ for $\theta=0$. Lucas numbers $L_n$ satisfy the same recurrence $L_n=L_{n-1}+L_{n-2}$ but they start with $L_0=2$, and $L_1=1$. Their Binet formula is $L_n=\alpha^n+\beta^n$. It is supposed that $F_{-n}=(-1)^{n+1}F_n$ and $L_{-n}=(-1)^nL_n$ for all integer $n$.</p>
<p>
All this is pretty well known, and it can of course be found in this book, but there is much more. It attempts to give a roundup of all that is known about Fibonacci numbers and friends, and adds to this a new insight from the author: the Fibonacci resonance. As the author states in his preface, this volume is bundling three books in one. The first one (here represented by Part I) is conform to what most popular science books mention about these issues. It is an historical survey of how $\phi$ and the Fibonacci numbers appeared in science, in nature, and in artwork in the past. The `third book' (appearing in the form of Part V) is a continuation of the first one, but these elements are less easily found in the literature, at least not in this accessible form, since it discusses several applications of $\phi$ and friends in more recent developments of science. The middle piece (Parts II, III, IV) is more mathematical and develops some original ideas of the author about what he calls the Fibonacci resonance which is based on the elements that were introduced in my first paragraph of this review. But let me start with the historical background.</p>
<p>
All the usual suspects appear in the historical survey. Of course the pyramids from Egypt, but also, and these may be less familiar to an average public, the megalithic Sun and the Moon gates in Bolivia, which are extensively discussed. Also the meter of classical Sanskrit involves mathematical patterns. Obviously on the list are also the mathematics of the ancient Greek, re-introduced in the West by the Arabs, and the scientific evolution since the 15th century with the naming of the Fibonacci numbers and the introduction of the symbol $\phi$. The spirals are introduced and their classical appearance in nature (nautilus, pineapple, pine cone, sunflowers, Roman broccoli, etc.). Also in music such patterns can be detected. Bartók, Debussy, and Xenakis are taken as examples. Paris became 'the capital of $\phi$' in the course of the 19th and the early 20th century when artists picked up the Golden Section credo that was propagated by scientists and theoreticians who strongly influenced the Parisian art scene. Among them Charles Henry, friend of the mathematician Édouard Lucas who studied the Fibonacci sequence, Joséphin Péladan who promoted $\phi$ on mystical grounds, Maurice Princet, the mathematician of cubism etc. Extensive discussions are devoted to paintings of Seurat, Toulouse-Lautrec, and Mondrian, the purist `par excellence'. Architecture is represented by Le Corbusier and his Modulor, a system of proportions based on an anthropomorphic scaling.</p>
<p>
The `third book' is called '$\phi$ science' and is also 'traditional' in the sense that it lies in the line of expectations. Here we meet the recent applications of the Fibonacci sequence. There we obviously find phylotaxis, not only in plants, but also in DNA, superconductors, and sunlight harvesting. A link not mentioned so far is sphere packing and tiling. Here we meet 3D crystal structures, Penrose tilings, and quasicrystals (extensively discussed), Islamic patterns, superlattices and composites (metamaterials) with the protagonists (Victor Veselago, John Pendry, Dan Shechtman) and applications (cloaking, plasmonics,...). Of course all these applications are practically important, but some topics such as Fibonacci word and Penrose tiling are interesting structures that invite to be studied at an abstract theoretical level.</p>
<p>
This leaves us with the most original, most surprising, and most mathematical part of this book. The first idea is to divide the goniometric circle into 32 equal parts. This results in an Ori32 geometry referring to 32 possible orientations. Menhinick got his inspiration from the fact that many applications depend on angles and orientations, rather than the classical point-line-plane approach to geometry. With these 32 parts, the smallest part of the disk is thus a wedge with angle $\pi/16$. This is used as a kind of unit and is called a MIK. It is denoted as $\fbox{1}$ and one can consider multiples of this unit. A quarter disk corresponds for example to $\fbox{8}$. The disk can be divided into wedges progressing in Fibonacci-like manner. Putting together $\fbox{1}$, $\fbox{2}$, $\fbox{3}$, $\fbox{5}$, $\fbox{8}$, and $\fbox{13}$, one get the full disk because $(1+2+3+5+8+13)\pi/16=2\pi$. Because of the periodicity, calculus with these MIK is modulo 32. If the radius of the circle is 1, then the arc length of $\fbox{5}=5\pi/16\approx1$. Similarly the arc of $\fbox{8}=\pi/2\approx\phi$ and for $\fbox{3}$ we get $3\pi/16\approx1/\phi$ etc. All these approximations are too large, so we may forget $\fbox{1}$ (which contributes $\phi^{-3}$) and get $2\pi\approx\phi^{-2}+\phi^{-1}+1+\phi+\phi^2=2\phi+3=4+\sqrt{5}$. Because the arc lengths are only approximately correct, Menhinick points to the analogy of the Pythagorean comma in music theory. His search for the necessary correction resulted in a generalization of the Binet formula: $F_n=F_s\alpha^{n−s}+F_{n-s}\beta^s$ for all integers $n$ and $s$. To visualize this, a golden spiral with equation $r=(F_s/\phi^s)\phi^{b\theta}$ is constructed for each $s$. These are in fact simple transforms of a (standard) golden spiral. The first term $F_s\alpha^{n−s}$ defines points on the spiral at its intersections with the coordinate axes. The second term is a quantized deviation to get $F_n$ from the previous spiral points namely $F_{n-s}$ times a quantum $\beta^s$. Similarly for the Lucas numbers, one has $L_n=\sqrt{5}F_s\alpha^{n-s}+L_{n-s}\beta^s$. The $\sqrt{5}$ rotates the axes over about 150 degrees and the intersections of these rotated axes with the spirals gives again approximations $\sqrt{5}F_s\alpha^{n-s}$ for $L_n$ with a quantized deviation $L_{n-s}\beta^s$. Note that each spiral has its own characteristic quantum $\phi^{-s}$. Then Menhinick considers the finest quantum to be half a wavelength. Since quanta for different $s$ always appear in integer multiples, this can be considered as standing waves of different frequencies that 'vibrate' in resonance: the Fibonacci resonance. It is also investigated whether there is fractal behavior but that issue seems not to be cleared out completely. This is followed by an extra part in which these ideas are applied to generalized Lucas sequences $U_n=PU_{n−1}+QU_{n-2}$, starting with initializations 0 and 1. The Pell and Pell-Lucas numbers are a special case for $(P,Q)=(2,−1)$. The analog of $\phi$ is here the Silver Ratio $\delta=1+\sqrt{2}$. They were recently (2014) studied in the book <a href="/review/pell-and-pell–lucas-numbers-applications" target="_blank"><em>Pell and Pell-Lucas Numbers with Applications</em></a> by T. Koshy, which is not in the (otherwise quite extensive) list of references of this volume.</p>
<p>
All the material that I discussed so far takes about 400 of the approximately 600 pages. The remaining one third of the book consists of appendices with technical and mathematical details, glossaries of terms and symbols used, a collection of formulas, and most of all an overwhelmingly extensive list of references (1004 items!). Also the index is well stuffed and useful for an encyclopedic work such as this book.</p>
<p>
There is no doubt that what is described as the first and the third book are useful additions to what is already available in the literature. There are certainly original contributions also there. About the second book, introducing Fibonacci resonance, I am not so sure where all this is leading to, and what to think of all this spiraling number magic. There are obviously interesting, and as far as my knowledge is concerned, new relations derived in this part. What I mean to say is that I can certainly appreciate the formulas underlying the concept, but the resonance interpretation hints to numerological significance that I believe unnecessary. A remark such as the fact that the right-hand side of $\sum_{n=−2}^7\phi^n=\frac{11}{2}(7+3\sqrt{5})$ (formula (12.3)) combines the first 5 primes: 2,3,5,7,11, is of course true, but it is in my opinion pure coincidence and has no further meaning. And there are other examples, including the resonance interpretation, which is amusing and imaginative, but otherwise with little mathematical significance.</p>
<p>
The illustrations are plentiful and helpful, except perhaps the 3D model of the different $s$-spirals which is for me only more confusing than what is already in the previous chapter. All the facts and persons of the book are extremely well researched and referenced. Also the pointers forward and backward are detailed and make it so much easier for the reader, and the typesetting in LaTeX is practically flawless (some italic instead of roman <em>log</em>'s and <em>arctan</em>'s here and there are minor exceptions).</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 an encyclopedic work about the history, theory, and applications of the Golden Ratio and the Fibonacci numbers (and their companions, the Lucas numbers). A remarkable addition to this is the middle part of the book in which Menhinick develops a theory of Fibonacci resonance that is based on recursion formulas for Fibonacci and Lucas numbers expressing them as deviations form certain points on logarithmic spirals.</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/clive-n-menhinick" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Clive N. Menhinick</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/onperson-international-ltd" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">OnPerson International Ltd.</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">2015</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">978-0993216602 (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">US$ 89.00</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">638</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li><li class="vocabulary-links field-item even"><a href="/imu/number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Number Theory</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.amazon.com/Fibonacci-Resonance-other-Golden-discoveries/dp/0993216609" title="Link to web page">http://www.amazon.com/Fibonacci-Resonance-other-Golden-discoveries/dp/0993216609</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/11-number-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11 Number theory</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/11b39" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11B39</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/11-00" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11-00</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a20</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a67" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a67</a></li></ul></span>Tue, 08 Dec 2015 09:38:20 +0000Adhemar Bultheel46596 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/fibonacci-resonance-and-other-new-golden-ratio-discoveries#commentsAnd yet it is heard. Musical, Multilingual and Polycultural History of Mathematics (2 vols.)
https://euro-math-soc.eu/review/and-yet-it-heard-musical-multilingual-and-polycultural-history-mathematics-2-vols
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In these two volumes of a thousand pages, Tonietti gives a very personal selection of the history of mathematics, and in particular the pieces where music and mathematics meet. He has some strong views on certain aspects that are not always mainstream and these are strongly put forward. One of his pet peeves is that many of his colleagues approached the subject with an eurocentric bias. Another one is that mathematics and science in general is too often considered to be an abstract universal entity besides or above the socio-cultural soil in which they are rooted. That means for example that also the language spoken and or even more so, the <em>lingua franca</em> used by the scientists and philosophers has had its influence which often goes hand in hand with cultural and religious foundations. This is his contribution to prove the Sapir–Whorf hypothesis in the case of mathematical sciences. Every culture generates its own science, and qualifications like superior and inferior or questions of precedence are often absurd. And of course, probably the main reason for writing this book, is his conviction that the mutual influence of music on the development of mathematics and vice versa is grossly underestimated. In this context he brings several contributions to the forefront that were wrongfully neglected (Aristoxenus, Vincenzo Galilei, Simon Stevin, Kepler,...). Thus there are many provocative viewpoints that some historians will disagree with, yet his arguments are extensively documented. Note however that this is not really a book on the history of mathematics, and neither is it a history of music. The reader is supposed to be familiar with music theory and should have some background in mathematics too. The book is a long plea and an extensive argumentation to underpin the viewpoints of the author, like those just mentioned. There are practically no formulas in the text, and relatively few illustrations, but the number of citations is overwhelming. These are almost always given in the original language with translation in brackets. The Chinese citations are written in pinyin, but full Chines characters are added in an appendix. This illustrates the importance that Tonietti is attaching to the language, since indeed, the translation is always an approximation and often an interpretation of what the original text is meant to say.</p>
<p>
Let's go quickly though some of the contents to illustrate what has been said above. Volume 1 contains Part I: The ancient world, and Volume 2 consists mainly of Part II: The scientific revolution, and a shorter Part III: It is not even heard.<br />
Part I treads the ancient cultures united around their language used: The Greek, Chinese, Sanskrit, Arabic, and Latin. For the Greek, music was part of the <em>quadrivium</em> and hence coexisted at the same level as mathematics and astronomy in the schools of Pythagoras, Euclid, Plato and Ptolemy. The search for harmony in music was reflected in the music of the spheres, all based on an orthodoxy of commensurability, hence integers and rationals. The music theory was developed on the basis of length of strings. The often neglected unorthodox outsiders here are Aristoxenus (who does not restrict to rationals) and Lucretius (although the latter wrote in Latin, he is Greek in spirit) who get special attention.<br />
The Chines on the other hand developed a theory of music studying the length of pipes (the <em>lülü</em>), bells, and chime stones. The cultural essense of <em>qi</em> is an energetic flow, a continuum which is apposite to the discrete orthodoxy of the Greek. Another difference is the lack of an equivalence for the verb "to be". This implies a different way of doing mathematics like for example the way in which they proved the Pythagoras theorem.<br />
Indian rules and regulations stem from religion. Precise prescriptions of how to build an altar show mathematical knowledge. Of course there was music, mostly by singing mathras, but most curiously, there is no trace of a music theory left. Musicians had `to trust their ears'.<br />
The Arabs are the saviors of the Greek culture. Most of what we know about the Greeks comes to us through them. The <em>Syntaxis mathematica</em> of Ptolemy came to us in Arabic as the <em>Almagest</em>: `the greatest' Greek collection of astronomical data. So they inherited the orthodoxy in music and mathematics from the Greek. They brought us our number system, but also terms like algorithm and algebra.<br />
Meanwhile in Europe, Latin had conquered the scientific scenary. This brought about a clash between the people, like Fibonacci promoting the introduction of the new Indo-Arabic number system against the Roman numerals. The Greek orthodoxy prevails, with Euclid being the reference for mathematics. Music theory florishes (Beothius, Guido D'Arezzo, Maurolico, Cardano,...). Tonietti gives special attention to Vicenzo Galilei, the father of Galileo, who picked up some ideas of Aristoxenus again.<br />
Besides the appendix with Chinese characters mentioned above, three other appendices are texts related to music translated from Chinese, Arabic, and Latin.</p>
<p>
In Part II chapters are named again afer the main languages (mostly European) used to disseminate scientific results. The interplay between geometry, astronomy, and music becomes explicit in work by Stevin and Galileo, but most of all in Kepler's <em>Harmonices Mundi Libri Quinque [Five books on the harmony of the world]</em> in which he completed Ptolemy's <em>APMONIKA</em> and interwaves geometry, astronomy, music and geometry, reflecting the music of the spheres. Tonietti does not shy away from critique on colleagues who had different interpretations of Kepler's work. People started using national languages besides Latin in their writings and (perhaps because of that) mathematical symbolism increases like writing music on staves was adopted before. Transcendent symbolism was mixed with music, God, and natural phenomena in work of Mersenne, Descartes, Wallis, and Huygens (Constantijn and Christiaan). The latter was not only a musician and composer, he used the newly invented logarithms and Leibnitz's differential calculus in his music theory. All this, according to Tonietti, shows that the status of music should be reinstalled as an essential element that contributed to the development of mathematics. Also Leibnitz and Newton worked on music for some time, but of course their main contribution here is the mathematical symbolism that allowed to deal with the infinite and the infinitesimal. With the use of the twelfth root of two in the equable temperament, the Pythagorean-Plato orthodoxy was definitely finished. The music of the spheres had degenerated and became intense discussions about God and creation. Again Tonietti analyzes interpretations of other historians, philosophers, or theologians on these topics sometimes rebutting them with his own vision. While in the 18th century French became the prominent language in Europe, music theory received its last flares. The French composer Rameau wrote a treatise on harmony using a theoreical basis, but he was opposed by Euler who declared tones more pleasing when they could be represented more simply. He based his analysis on prime numbers, with reminiscences of Pythagoras. Another opponent was d'Alembert discussing the harmonics of the vibrating string and also the other illuminists compiling the <em>Encyclopédie</em> had their explanation for musical terms. Still musicians wrote of science and scientists wrote of music. Entering the 19th century, Lagrange and Fourier, and later Riemann entered the discussion about vibrations, while von Helmholtz also did the physical experiments to analyze sound. Max Planck not only wrote about music but even composed some and also Einstein loved playing the violin and had correspondence with the composer Schönberg.</p>
<p>
Part III is very short. It gives a short round-up of things not discussed like Africa, Cental and South America, and more extensively the music and navigation skills of Polynesians. A last chapter briefly touches upon the science of acoustics, in which music is largely neglected. A quote from one of the final paragraphs that renders explicitly what Tonietti has allowed to emerge in his book:</p>
<p>
<em>The decision to move away from musicians and their music impoverished both natural philosophers, first of all, and then mathematicians and physicists. This influenced and facilitated the development of their research in those main directions which are known to everybody, but which continue to deserve to be criticised for their limitations and their (negative?) effects on our life.</em></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 history of music theory and its relation with mathematics, placed in a general cultural framework, and by doing so, giving a less usual, less eurocentric approach. Besides the well known historical framework, Tonietti selects and extensively discusses some less known sources and gives arguments for his critique on the views or interpretations of some of his colleagues. He comes to the conclusion that although mathematics and natural sciences has taken big steps forward in recent history, music theory was detached and has lost the interest of mathematicians, and this is a regrettable impoverishment for natural philosophers, mathematicians and physicists.</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/tito-m-tonietti" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Tito M. Tonietti</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/birkh%C3%A4user-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Birkhäuser Verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-3-0348-0667-0 (hbk - vols.)</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">253,34 €</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">1020</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.springer.com/birkhauser/mathematics/book/978-3-0348-0677-0" title="Link to web page">http://www.springer.com/birkhauser/mathematics/book/978-3-0348-0677-0</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/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</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/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</a></li></ul></span>Wed, 13 Aug 2014 07:09:30 +0000Adhemar Bultheel45674 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/and-yet-it-heard-musical-multilingual-and-polycultural-history-mathematics-2-vols#comments