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Can you think of an image, whether technical or nontechnical, available for viewing online that says a lot about what you think mathematics or a particular field of mathematics is all about?

For instance, some look at Hokusai's "Great Wave" as evoking a notion of fractals. http://en.wikipedia.org/wiki/File:Great_Wave_off_Kanagawa2.jpg

There is some interesting discussion of this here, http://www.squarecirclez.com/blog/math-in-art-hokusais-the-wave/595

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closed as off-topic by HJRW, j.c., Carlo Beenakker, Chris Godsil, Ryan Budney Oct 24 '13 at 12:05

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – j.c., Carlo Beenakker, Chris Godsil
If this question can be reworded to fit the rules in the help center, please edit the question.

    
I'm voting to close; this question is now several months old and is rather stale. Moreover, none of the answers so far are good. –  Kevin H. Lin Jun 3 '10 at 3:42
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I don't know whether MO questions can get stale, as long as there are new MO users who haven't seen them before. Or are new users supposed to familiarize themselves with all previous questions? –  John Stillwell Jun 3 '10 at 4:38
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Moreover, who is really capable of judging if "none of the answers is good" when talking about a soft-question (and be confident that he could convince the rest of the people about him being right)? I learned some things from these answers, and I am pretty sure that there probably is at least one person that found good any particular answer - namely, its poser! –  Jose Brox Jul 22 '10 at 13:03
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If it wasn't stale in 2010, it's certainly stale now. –  stankewicz Oct 24 '13 at 8:17
    
@stankewicz, I completely agree, and have voted to close accordingly. –  HJRW Oct 24 '13 at 9:41

25 Answers 25

up vote 17 down vote accepted

"Epitomizing mathematics" is a tall order, and even representing what a single field is about (rather than just giving a cool glimpse of the subject matter) is pretty hard to imagine.

Still, given your example, one of course thinks of Escher, for example his hyperbolic plane tilings:

Escher's hyperbolic plane tiling

and the less well-known art of Anatoly Fomenko. I believe one of Fomenko's drawings is on the first page of one of Springer's GTM books - now I just need to remember which one...

Another picture that is frequently associated with mathematical research is Durer's Melancolia. I can't say it really epitomizes anything for me, but it's a pretty picture.

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re: Fomenko, you're thinking of Shiryaev's "Probability" –  Erik Davis Nov 23 '09 at 3:28
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Or perhaps GTM 58, Koblitz, $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions, which features Fomenko's conception of the 3-adic unit disk. Or GTM 97, Koblitz, Introduction to Elliptic Curves and Modular Forms, with a Fomenko drawing depicting the family of elliptic curves that arises in the congruent number problem. –  Gerry Myerson Jun 2 '10 at 4:21
    
Curiously, in the Russian translations of Koblitz's books, they have been transposed. –  Victor Protsak Jun 2 '10 at 6:00
    
I'm quite partial to this variation by Henry Segerman and Paul-Olivier Dehaye: segerman.org/autologlyphs/Poincaredisk_small.gif –  Mark Meckes Jun 2 '10 at 13:45
    
Victor, the Russian translation of Koblitz's book on p-adic analysis has Fomenko's drawing of the 2-adic solenoid (both on the cover and inside the book), not a family of elliptic curves. That I checked directly. By a web search I find that the picture in the ell. curves book is the same in the translated volume as in the original one too. –  KConrad Jul 7 '10 at 23:01

I think this sums it up most of the time.

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The top of page 1199 (page 4 of 17 of the PDF) in the article "Comme Appelé du Néant" in the Notices of the AMS.

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It's probably not a good thing that I knew what you were referring to before I even clicked the link... –  Harrison Brown Oct 28 '09 at 17:26

hyperbolic space

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Wow, this takes me back. <i>Not Knot</i> completely changed my impression of mathematical visualization and what it could do; to my mind it still sets the standard for expository mathematics video, and it's a shame that more aren't trying to reach for the bar it sets. –  Steven Stadnicki Jul 7 '10 at 0:14

Raphael's School of Athens

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Link appears to be broken; try en.wikipedia.org/wiki/The_School_of_Athens –  JBL Jun 2 '10 at 15:51
    
Fixed by imbedding the image. –  Victor Protsak Jun 2 '10 at 16:09

Mandelbrot set

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Check out the mandelbulb too. skytopia.com/project/fractal/mandelbulb.html –  Dan Piponi Jun 2 '10 at 23:30
    
The Mandelbulb is just great! –  Jose Brox Jul 7 '10 at 0:09

What about the five platonic solids?

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Apart from the glorious, mountaintop pictures, this comic captured my initial experience reading proofs.

enter image description here

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Hm, would a picture of Serre count?

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:) $ $ –  Mariano Suárez-Alvarez Jun 2 '10 at 13:48

Some History

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I've reduced the size to fit in a page. Feel free to roll back if inappropriate. –  Victor Protsak Jun 2 '10 at 16:15
    
An easier way to do that would have been to change the 600px parameter in the original url. –  Jon Awbrey Jun 2 '10 at 16:45

For category theory, from Abstract and Concrete Categories: The joy of cats, by Adámek, Herrlich and Strecker, page 12:

(http://katmat.math.uni-bremen.de/acc/acc.pdf)

Categorists have developed a symbolism that allows one quickly to visualize quite complicated facts by means of diagrams.

For me, this represents the fact that most, if not all of mathematics, is about structures and relations: even the simplest of them, when combined and interrelated, can give birth to fairly complex behaviour.

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José, please take a look at the meta thread tea.mathoverflow.net/discussion/489/retagging/#Item_3 –  Mariano Suárez-Alvarez Jul 7 '10 at 2:04

The Lorenz attractor would be the canonical image for chaos theory, and with only a little verbal explanation it demonstrates the sensitivity to dependence on initial conditions very well.

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Interesting! My impression was that the bifurcation diagram for the logistic map, en.wikipedia.org/wiki/File:LogisticMap_BifurcationDiagram.png, was more canonical. –  Victor Protsak Jun 2 '10 at 6:06
    
@Victor: Good point! I was also thinking about the Mandelbrot set. Chaos theory/fractals have many of the most interesting images. –  Henry Segerman Jun 2 '10 at 15:38
    
I confess: you've given me the idea of M set. –  Victor Protsak Jun 2 '10 at 16:20

A proof without words for the Pythagorean theorem (Zhou Bi Suan Jing).

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You could add a link to this (or even copy the answer there!) to the question "Proofs without words", for completeness' sake. –  Mariano Suárez-Alvarez Jun 2 '10 at 18:01

... and months after this question was posted, no mention of Sidney Harris?

Start with his "... and then a miracle occurs" and his "You want proof? I'll give you proof!" . Continue from there.

Gerhard "That's Enough Subconscious For Today" Paseman, 2010.07.06

The first one available for sale as prints, notecards, etc. http://www.newyorkerstore.com/All-Industry-Cartoons/I-think-you-should-be-more-explicit-here-in-step-two/invt/118181

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Bernar Venet's paintings [http://www.bernarvenet.com/ ] - those with colored commutative diagrams (if I remember properly, there was also a Notices article about that).

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Here is a more mathematical rendition of Richard Kent's answer:

http://math-art.net/2007/12/03/eternal-scream-a-droste-effect/

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Looks more like a mathematical rendition of "This is Spartaaaaaaaaa"... –  darij grinberg Jun 2 '10 at 10:20

There is a painting by Diego Rivera of René Paresce (1866-1937) (gotten from web address http://www.vf.utwente.nl/~jagersaa/D_R.html)

alt text El Matemático

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No image. Oh, well. –  Will Jagy Jul 6 '10 at 19:54
    
I added the image. Great painting, btw. –  Mariano Suárez-Alvarez Jul 7 '10 at 0:36
    
Thanks, Mariano. How did you do that? –  Will Jagy Jul 10 '10 at 18:34

Voronoi diagrams and their respective Delaunay triangulations.

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I really like combinatorics because of it's ability to formalize basic notions like comparison (being greater than, less than, or equal to some amount) and counting. For example here is a proof without words (from Ferrar's Diagram - Wikipedia) that the number of partitions of n objects with distinct odd parts is the same as the number of self-conjugate partitions. Math allows us to see patterns like this simply by moving colored dots around. How epic is that! To me, that's what combinatorics is all about.

bijection between the set of partitions with distinct odd parts and the set of self-conjugate partitions

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Mathematics sometimes evokes emotion, as does music, and sometimes other forms of art. The strongest emotion I remember was the first time I saw Euler's Identity. So my image is just Euler's Identity written on a chalk board.

Epi-graph

Chalk this up to Benjamin Peirce

Benjamin Peirce's script for e^(i pi)

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