most recent 30 from http://mathoverflow.net 2013-05-25T12:07:35Z http://mathoverflow.net/feeds/question/2949 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams Elisha Peterson 2009-10-28T00:10:17Z 2012-12-15T00:27:05Z

<p>Most math papers have few figures, if any, although sometimes a well-chosen figure can be a tremendous help in understanding mathematical concepts. Does anyone have any examples of notable uses of figures in mathematical writing and/or texts that make great use of figures/diagrams/illustrations?</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/2953#2953 Grétar Amazeen 2009-10-28T00:17:00Z 2009-10-28T00:17:00Z

<p>Mumford's picture of Spec(Z[x]) comes to mind.</p> <p>You can read a discussion of it <a href="http://www.neverendingbooks.org/index.php/mumfords-treasure-map.html" rel="nofollow">here</a>.</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/2963#2963 Andy Putman 2009-10-28T00:54:07Z 2009-10-28T16:40:47Z

<p>I am a huge fan of the drawings of Anatoly Fomenko (for some of his drawings, see <a href="http://www.mathe.tu-freiberg.de/~hebisch/cafe/fomenko/fomenko.html" rel="nofollow">here</a>; for a description of his odd historical theories, see his wikipedia pages <a href="http://en.wikipedia.org/wiki/Anatoly_Fomenko" rel="nofollow">here</a> and <a href="http://en.wikipedia.org/wiki/New_Chronology_(Fomenko)" rel="nofollow">here</a>).</p> <p>In particular, his book "Algorithmic and Computer Methods for Three-Manifolds" with Matveev (which really has nothing to do with computers) is IMHO one of the best intro books on 3-manifolds available largely because of its drawings. The mathscinet review of the Russian version is worth reading; see</p> <p>MR1162113 (93f:57002) Matveev, S. V.; Fomenko, A. T. {\cyr Algoritmicheskie i kompʹyuternye metody v trekhmernoĭ topologii}. (Russian) [Algorithmic and computer methods in three-dimensional topology] Moskov. Gos. Univ., Moscow, 1991. 303 pp. ISBN: 5-211-01743-9 </p> <p>EDIT : To give an idea of how amazingly cool the above book is, I have posted two pages of it <a href="http://www-math.mit.edu/~andyp/Figures/FIGURE1.pdf" rel="nofollow">here</a> and <a href="http://www-math.mit.edu/~andyp/Figures/FIGURE2.pdf" rel="nofollow">here</a>. It's a crying shame that it is not better known...</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/2979#2979 John D. Cook 2009-10-28T02:03:58Z 2009-10-28T02:03:58Z

<p>Here's one of my favorite diagrams, based on a paper by Lawrence Leemis:</p> <p><a href="http://www.johndcook.com/distribution%5Fchart.html" rel="nofollow">Probability distribution relationships</a></p> <p>Also, here are some diagrams due to Robert Bartle relating various modes of convergence:</p> <p><a href="http://www.johndcook.com/blog/2008/05/25/diagramming-modes-of-convergence/" rel="nofollow">Diagramming modes of convergence</a></p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/3002#3002 Theo Johnson-Freyd 2009-10-28T05:34:39Z 2009-10-28T05:34:39Z

<p>A paper that used figures to, in my view, revolutionize the understanding of an area of mathematics is: R. Penrose. Applications of negative dimensional tensors. In D.J.A. Welsh, editor, Combinatorial mathematics and its applications, pages 221–244. Mathematical Institute, Oxford, London, New York, Academic Press, 1971.</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/3014#3014 David Lehavi 2009-10-28T08:02:59Z 2009-10-28T08:02:59Z

<p>Turning a sphere inside out</p> <p><a href="http://www.treeincarnation.com/images/centerfold.gif" rel="nofollow">http://www.treeincarnation.com/images/centerfold.gif</a></p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/3089#3089 David Hansen 2009-10-28T17:02:03Z 2009-10-28T17:02:03Z

<p>There's a chapter in Littlewood's "Mathematician's Miscellany" entitled The Zoo, which has many charming examples of pictoral proofs inspired by various animals.</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/7271#7271 Alasdair McAndrew 2009-11-30T13:22:57Z 2009-11-30T13:22:57Z

<p>The figures in <a href="http://en.wikipedia.org/wiki/John%5FStillwell" rel="nofollow">John Stillwell</a>'s books are always superbly drawn, and really enhance the exposition.</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/7273#7273 Kevin Lin 2009-11-30T13:38:11Z 2009-11-30T13:38:11Z

<p>When I was an undergraduate I read the book <a href="http://books.google.com/books?id=tqK6VZ9xnqkC&amp;printsec=frontcover&amp;source=gbs%5Fv2%5Fsummary%5Fr&amp;cad=0#v=onepage&amp;q=&amp;f=false" rel="nofollow">Intuitive Topology</a> by Prasolov. It's a wonderful illustrated guide to low-dimensional topology (mostly knots/links and surfaces). If I recall correctly, almost all of the "proofs" are by pictures.</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/7274#7274 Kevin Lin 2009-11-30T13:50:54Z 2009-11-30T13:50:54Z

<p><a href="http://www.edwardtufte.com/tufte/" rel="nofollow">Edward Tufte's books</a> are quite beautiful, though they do not focus so much on mathematical figures/diagrams per se. However, via Tufte, I did come across <a href="http://www.sunsite.ubc.ca/DigitalMathArchive/Euclid/" rel="nofollow">this version of Euclid's Elements by Oliver Byrne</a>, which presents the propositions and proofs of the Elements using colored diagrams and symbols. I'm not sure whether Byrne's edition is clearer or better to learn from than the original Euclid, but it sure is pleasing to look at.</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/7278#7278 Sam Nead 2009-11-30T14:40:16Z 2009-11-30T14:59:56Z

<p>Knots, links, braids, and 3-manifolds, by Prasolov and Sossinsky (you can look at it on <a href="http://books.google.co.uk/books?id=znCLtJKnZXQC&amp;dq=Knots,+links,+braids,+and+3-manifolds&amp;printsec=frontcover&amp;source=bl&amp;ots=VYF6LQhJ5j&amp;sig=PhIeG6%5Fjoq1PE5HiJqr1mOiEZxM&amp;hl=en&amp;ei=19cTS5LrMN2ZjAfnp7nJAw&amp;sa=X&amp;oi=book%5Fresult&amp;ct=result&amp;resnum=1&amp;ved=0CAgQ6AEwAA#v=onepage&amp;q=&amp;f=false" rel="nofollow">Google books</a>) essentially has a picture on every page. They are very pretty. </p> <p>Indra's pearls by Mumford, Series, and Wright has some breath-taking pictures. There are also cartoons by Gonick, which improves any book. Here's the copy at <a href="http://books.google.co.uk/books?id=kC5kdUirHHoC&amp;printsec=frontcover&amp;dq=Indra%27s+pearls#v=onepage&amp;q=&amp;f=false" rel="nofollow">Google books</a>.</p> <p>Edit: A topological picturebook, by Francis is wonderfully illustrated. There are directions for reproducing the figures, as well as their mathematical meaning. Chapter eight of the <a href="http://books.google.com/books?id=L0DhG2%5FDb3oC&amp;dq=A+topological+picturebook&amp;source=gbs%5Fnavlinks%5Fs" rel="nofollow">book</a> deals with the figure eight knot. :)</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/7285#7285 Scott Morrison 2009-11-30T16:07:26Z 2009-11-30T16:07:26Z

<p>Bar-Natan's <a href="http://www.math.toronto.edu/~drorbn/papers/Categorification/" rel="nofollow">first paper on Khovanov homology</a> included a great figure (visible immediately if you follow that link) that summarised the entire construction. A great improvement over the <a href="http://front.math.ucdavis.edu/9908.5171" rel="nofollow">previous epic</a>. </p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/7289#7289 ivane 2009-11-30T16:47:19Z 2009-11-30T16:47:19Z

<p>see my starting effort to help see 3-manifolds at <a href="http://commons.wikimedia.org/wiki/Category:3-manifolds" rel="nofollow">http://commons.wikimedia.org/wiki/Category:3-manifolds</a> </p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/24915#24915 coudy 2010-05-16T19:16:10Z 2010-05-16T19:16:10Z

<p>The book by G. K. Francis entitled "A topological picturebook" is beautiful. Have a look at this <a href="http://books.google.com/books?id=L0DhG2_Db3oC&amp;printsec=frontcover&amp;dq=a+topological+picturebook" rel="nofollow">snapshot</a>, starting with page 16.</p> <p>The book explains how to draw and visualize pictures of low dimensional famous topological spaces: the dunce hat, a tetrahedral hyperbolic manifold, the Withney bottle, the Hopf fibration, and so on.</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/24931#24931 Qiaochu Yuan 2010-05-16T20:45:55Z 2010-05-16T20:45:55Z

<p>Reid's <a href="http://books.google.com/books?id=mUL1us0mRrAC&amp;printsec=frontcover&amp;dq=miles+reid+commutative+algebra&amp;cd=1#v=onepage&amp;q&amp;f=false" rel="nofollow">Undergraduate Commutative Algebra</a> has a lot of great figures, most notably the frontispiece depicting the statement "let $A$ be a ring and $M$ an $A$-module" geometrically.</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/60289#60289 Denis Serre 2011-04-01T13:39:09Z 2011-04-01T13:39:09Z

<p>The following is a wondeful candidate: <strong>Jos Leys</strong>, <em>Lorenz and Modular Flows: A Visual Introduction</em>, <a href="http://www.ams.org/samplings/feature-column/fcarc-lorenz" rel="nofollow">Feature Column</a> of the AMS web site, November 2006.</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/116384#116384 Rodrigo A. Pérez 2012-12-14T16:53:24Z 2012-12-14T16:53:24Z

<p>Arnol'd was famous for his pictures (that poor $\ $ s-t-r-e-t-c-h-e-d $\ $ cat...), but the Award for Best Pictures must surely go to <i>"Geometry and the Imagination"</i> by D. Hilbert and S. Cohn-Vossen.</p>

http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/116397#116397 Kristal Cantwell 2012-12-14T20:35:11Z 2012-12-14T20:35:11Z

<p>H.S.M. Coxeter's books include <strong>Regular Polytopes</strong>. This book deals with the classification of regular polytopes. In this book there are Coxeter diagrams which are closely related to Dynkin diagrams. In his works are many diagrams,figures and illustrations. They influenced M.C Escher. Many of Escher's works reflect his ideas. In 1996 Coxeter published a paper on one of these "Circle Limit III." For more information see <a href="http://www-history.mcs.st-and.ac.uk/Obits2/Coxeter_NYTimes.html" rel="nofollow">here</a> and <a href="http://en.wikipedia.org/wiki/Dynkin_diagram" rel="nofollow">here</a>. </p>