European Mathematical Society - 05-03
https://euro-math-soc.eu/msc-full/05-03
enIslamic Design: a Mathematical Approach
https://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="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/brian-wichmann" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Brian Wichmann</a></li><li class="vocabulary-links 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="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-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><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="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="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/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</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/01a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A30</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/51-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-03</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/05-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05-03</a></li></ul></span>Mon, 02 Jul 2018 08:59:48 +0000Adhemar Bultheel48569 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/islamic-design-mathematical-approach#commentsHow Euler Did Even More
https://euro-math-soc.eu/review/how-euler-did-even-more
<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>
Sandifer is a renowned expert on Euler. In 2001 he was co-founder and later secretary of the Euler Society. He wrote several books and contributed and promoted the Euler centenary year in 2007. Since November 2003, he also had a monthly column called <em>How Euler Did It</em> hosted by the Euler Archive website in collaboration with the Mathematical Association of America (MAA). These columns are stand-alone items elucidating a particular aspect of Euler's work. They are available online at the <a href="http://eulerarchive.maa.org/hedi/" target="_blank">Euler Archive</a>. A first collection of these columns appeared as a book <em>How Euler Did it</em> (MAA, 2007). Unfortunately in 2009, Sandifer had to recover from a severe stroke, but some colleagues filled up some of the gap so that the column kept appearing until early 2010. This book contains 35 of these columns from March 2007 till February 2010. September 2009 does not exist for the reason just explained. October, November, and December 2009 were filled up by guest author Rob Bradley.</p>
<p>
In the previous collection, the items were ordered chronologically, but in this volume, they are grouped by topic: Geometry, Number Theory, Combinatorics, Analysis (the largest part), Applied Mathematics, and some miscellaneous part called <em>Euleriana</em>. The columns are explaining indeed how Euler did prove some of his results. Euler got his problems for example from marginal notes by Fermat who, as we know, announced theorems by scribbling some notes in the margin of a book, or some problems were formulated by Euler himself and there are many other sources. Some historical background is given, but the main contribution is just explaining how Euler indeed constructed his proof. Sometimes original drawings are reproduced. These columns are certainly welcomed by readers not familiar with the original language of the papers or the correspondence which was often Latin, German, or French. Some translations of the original texts by Euler can be found at the <a href="http://eulerarchive.maa.org/" target="_blank">Euler Archive</a> but many are not translated yet.</p>
<p>
There are too many different topics to be discussed in detail in this review. They include prime numbers, trigonometry, probability theory, mortality tables and actuarial science, the zeta and gamma functions, formulas to approximate π, partial fractions, complex analysis, optics, fluid dynamics, gravity and many more. The <em>Euleriana</em> part deals with Euler as a teacher, but there are also contributions about Euler and the hollow earth, errors that Euler made, and about Euler and pirates. I leave it to your imagination what these are all about. You will have to read the column to know.</p>
<p>
Let me take just one example from the applied mathematics chapter to illustrate how the items are treated by Sandifer in his blog. In 1756 Euler while working in Berlin publishes a paper on modeling of saws. The column starts by explaining that Euler worked in 1728 as a physician for the Russian Navy in St. Petersburg where he learned the importance on lumber and the operation of sawmills. When 30 years later the Prussian King Frederik II was about to embark in a war, Euler, remembering the importance of lumber, wrote his paper. The saw model is about a vertical blade that cuts when moving down. Each tooth should cut the same amount of wood, which means the teeth side of the saw should be slanted, the next tooth cutting another peel where the previous one had just removed its part. This models the shape of the saw. Then the motion is formalized, the middle part of the saw blade that really cuts the wood, the speed and the energy needed, and finally the manpower needed to lift the saw (it was supposed to move down by gravity), and the amount that was cut per worker and per hour. It beautifully illustrates the genius of Euler in an easily understandable mathematical language as it is brought to us by Sandifer.</p>
<p>
As you can see from my previous enumeration, there are enough topics to make the book of interest to many. Simple mathematics suffice to illustrate the brilliant mind of Euler. They are also mathematical gems as a column: well written and documented, mainly addressing mathematicians or teachers, but understandable with a minimum of training. If fits perfectly well in the MAA Spectrum series that targets the general mathematically-interested reader. Given the dynamics and volatility of websites, it is a good initiative that the MAA has made these columns available as a book.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of monthly columns that Sandifer wrote on the work of Euler. It contains 35 of the last columns that appeared in 2007-2010 after previously a similar collection of <em>How Euler Did It</em> was published in book form by the MAA in 2007.</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/c-edward-sandifer" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">C. Edward Sandifer</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/maa-cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MAA; Cambridge 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">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">9780883855843 (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">£ 23.00</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">247</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="http://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/how-euler-did-even-more" title="Link to web page">http://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/how-euler-did-even-more</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/01-06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-06</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/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/05-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05-03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/11-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11-03</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/26-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/51-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-03</a></li><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>Mon, 20 Jul 2015 16:06:22 +0000Adhemar Bultheel46314 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/how-euler-did-even-more#comments