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There are many examples of artistic works which have some mathematical aspects. A high school or undergraduate math teacher can use them as interesting examples in his/her teaching. e.g. In a combinatorics course when a teacher introduces the notion of a magic square, he/she can add examples form paintings and literature as follows:

(1) Albrecht Dürer's magic square in his famous painting Melancolia I

enter image description here

(2) Goethe's description of a magic square in Faust:

$$ \begin{array}{l|l} \text{Du musst verstehen!} & \text{You shall see, then! } \\ \text{Aus Eins mach Zehn,} & \text{From one make ten!} \\ \text{Und Zwei lass gehen,} & \text{Let two go again,} \\ \text{Und Drei mach gleich} & \text{Make three even,} \\ \text{So bist du reich.} & \text{You’re rich again.} \\ \text{Verlier die Vier!} & \text{Take away four! } \\ \text{Aus Fünf und Sechs,} & \text{From five and six,} \\ \text{So sagt die Hex,} & \text{So says the Witch,} \\ \text{Mach Sieben und Acht,} & \text{Make seven and eight,} \\ \text{So ist's vollbracht:} & \text{So it’s full weight:} \\ \text{Und Neun ist Eins,} & \text{And nine is one, } \\ \text{Und Zehn ist keins.} & \text{And ten is none.} \\ \text{Das ist das Hexen-Einmaleins!} & \text{This is the Witch’s one-times-one!} \\ \end{array}$$ (See Faust, line 2540.)

Note that the above phrases are describing a magic square as follows:

enter image description here

Question: What are interesting examples of using mathematical notions in arts which are useful to introduce these notions to students at high school/undergraduate level?

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    $\begingroup$ I remember seeing an interesting picture of the $3$-adic disk interpreted in Koblitz's book on $p$-adic numbers: i.imgur.com/ffcYMej.jpg This is not really a subject broached by high school or (in general) undergraduate students, and I can't speak to the image's pedagogical value (hence the mere comment). $\endgroup$
    – Benjamin Dickman
    Aug 1, 2014 at 3:03
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    $\begingroup$ I'm reading Remmert's Theory of Complex Functions. I have no picture, but, it is an art. Math, good math, is art. (someday, I want to be an artist) $\endgroup$
    – James S. Cook
    Aug 1, 2014 at 4:24
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    $\begingroup$ @JamesS.Cook, I am agree with you that good maths is art. In fact "mathematical beauty" is really a kind of artistic beauty. Quoted from Halbeisen in his book, Combinatorial Set Theory: "I tried to write this book like a piece of music, not just writing note by note, but using various themes or voices." See also: "DAVID J. BENSON: Music: A Mathematical Offering. Cambridge University Press, Cambridge (2007)" $\endgroup$
    – user230
    Aug 1, 2014 at 4:57
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    $\begingroup$ Anything based upon fractals tends to make for both good art and good mathematical discussion. $\endgroup$
    – aroth
    Aug 1, 2014 at 6:37
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    $\begingroup$ NYU Poly had a conference devoted to the mathematical connections with the Durer painting where many prominent mathematicians (Conway, Richard Stanley, Ziegler, etc.) spoke. The videos from these talks are available here: youtube.com/watch?v=O3glV75mhF0 $\endgroup$
    – Joseph Malkevitch
    Aug 1, 2014 at 13:26

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Some crazy mathematicians have managed to calculate the missing centre of the Escher artwork below.

Go to their website to learn all about it. They have some pretty cool animations of it.

Also, this article gives a pretty good overview of the project.

Here is a link to a paper about Escher and symmetry.

Original

Filled in

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    $\begingroup$ Nice Example! Thanks! $\endgroup$
    – user230
    Aug 1, 2014 at 3:49
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    $\begingroup$ This is just plain cool. $\endgroup$
    – JPBurke
    Aug 1, 2014 at 21:36
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Henderson & Taimina (2006) point out that models in hyperbolic geometry are aesthetically compelling, and that the artist M. C. Escher made us of this in a representation of infinity within his works:

Repeating patterns on the sphere have an aesthetic appeal through their simplicity and finiteness. However, in these various hyperbolic models, the patterns have an aesthetic appeal for us because of their connections with infinity--there are infinitely many such patterns and each also draws us to the infinity at the edge of the disc, leaving sufficient space for our imagination. (p. 69)

M.C. Escher's Circle Limit III (based on the Poincaré disc model) M.C. Escher's Circle Limit III (based on the Poincaré disc model)

Henderson, D. W., & Taimina, D. (2006). Experiencing meanings in geometry. In N. Sinclair, D. Pimm, & W. Higginson (Eds.), Mathematics and the aesthetic: New approaches to an ancient affinity (pp. 58–83). Springer Verlag.

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Salvador Dali's 1954 Crucifixion (Corpus Hypercubus) "deviates from traditional portrayals of the Crucifixion by depicting Christ on the polyhedron net of a hypercube [...]." It could be used to introduce the idea of a fourth spatial dimension.

Crucifixion (Corpus Hypercubus)

From the Wikipedia entry:

Instead of painting Christ on a wooden cross, Dalí depicts him upon the net of a hypercube, also known as a tesseract. The unfolding of a tesseract into eight cubes is analogous to unfolding the sides of a cube into six squares. The use of a hypercube for the cross has been interpreted as a geometric symbol for the transcendental nature of God. Just as God exists in a space that is incomprehensible to humans, the hypercube exists in four spatial dimensions, which is equally inaccessible to the mind. The net of the hypercube is a three-dimensional representation of it, similar to how Christ is a human form of God that is more relatable to people.

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    $\begingroup$ (+1) Nice Example. Thanks. $\endgroup$
    – user230
    Aug 1, 2014 at 1:46
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MAA has a special interest group devoted to mathematics and the arts. Here is the link to its web page which includes galleries and resources:

http://sigmaa.maa.org/arts/exhibits.html

AMS sponsors an exhibit of mathematical art at the Joint Mathematical Meetings and here is a link for that and related things:

http://www.ams.org/mathimagery/

There is also an annual conference dealing with connections between mathematics and the arts which generates lots of materials both of research and pedagogical interest:

http://bridgesmathart.org/

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Since this question has been bumped up, I recently learned (via an AMS tweet) about the work done by Crockett Johnson -- best known as the creator and illustrator of Harold and the Purple Crayon -- at the intersection of "artistic works" and mathematics.

More information can be found in the write-up from Atlas Obscura; here are two sample excerpts, along with images, that I found particularly interesting. The first is on the Pythagorean Theorem, and the second is on an alternative proof [to, among others, that of Archimedes] that the regular heptagon is constructible, which was published in a 1975 issue of The Mathematical Gazette:

Johnson, C. (1975). A construction for a regular heptagon. The Mathematical Gazette, 59(407), 17-21.

enter image description here

enter image description here

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I haven't looked at many of these in a while, but there are several links to the math behind patterns, perspective, etc.

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    $\begingroup$ (+1) Surely perspective is one of the most important mathematical aspects of paintings. Do you have any particular example to introduce here? Please add its picture if it is possible. $\endgroup$
    – user230
    Aug 1, 2014 at 2:07
  • $\begingroup$ Rennaissance paintings are the usual examples. Some actually have pencil lines visible to see how the artist did the math. Also it was really a development at that time. So fits into a discussion (similar to the learnings on anatomy by dissection for art purpose). see here for a few: google.com/… $\endgroup$
    – guest
    Jun 25, 2017 at 20:15
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Here's a good example to show how beautiful math is.

http://touch-geometry.karazin.ua/sitemap

enter image description here

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Popular fractal images (paisley looking things.)

Also this is maybe more applied math or data science, but a lot of the Edward Tufte book images are compelling.

https://en.wikipedia.org/wiki/Edward_Tufte#/media/File:Minard.png

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https://en.wikipedia.org/wiki/Octacube_(sculpture)

Also check out the woodcuts for De re Metallica.

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  • $\begingroup$ Link-only answers really don't work well here. Mind adding the image with attribution, and perhaps a few sentences? $\endgroup$
    – JTP - Apologise to Monica
    Jun 27, 2017 at 22:29

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