Banchoff and Dali
My love affair with art and music is deep and life-long. I had shown some talent in mathematics through K-12 schooling, but I was certain that I would spend my days doing something creative. Little did I know growing up that math was absolutely compatible with that...
In 2010, I attended the annual Daniel Bartlett Memorial Lecture with my office mate Victor. The talk combined two of my most favorite things: mathematics and surrealism. Tom Banchoff of Brown University was speaking about his correspondence with the eccentric artist Salvador Dali.
Banchoff and Dali
Both men were interested in visually representing four dimensional objects. By this I mean a fourth spatial dimension, not a dimension for time. One work of art mentioned in that evening dealt with precisely that. Seen below, this painting depicts a four dimensional "hypercube" unfolded in three dimensions.
"Corpus Hypercubicus" (1954) |
Folding a cross (2D) into a cube (3D) 
Folding a hypercross (3D) into a hypercube (4D) |
Victor and I were particularly taken with the painting since we both simultaneously realized that this "hypercross" arrangement of cubes was, in fact, the solution to a problem in algebraic topology that we were given in our of first year of graduate school. Let me explain. We were assigned the following standard exercise from Hatcher:
The problem itself is not so difficult. One builds up a 1-skeleton as a corner of a cube and then attaches faces (two-cells) according to the 1-quarter twists. Assuming the attaching was done carefully, one then uses a corollary of van Kampen to get a description of the fundamental group with generators and relations. The group is recognized to be the quaternion group Q8 and everyone's happy. However, we were also told to identify all the path connected covers of this space. D'oh! Since the size of the fundamental group was eight, we knew that the universal cover had to have eight sheets. This means we had to stack together eight cubes and wrap them up in a nice way. At the time, we didn't know which arrangements of cubes would give us the right cover and which would give us nonsense. As it turns out, there are exactly 261 ways of stacking the cubes correctly, and the most symmetric one seems to be Dali's hypercross.
For more info on the above discussion see the slides for my talk "Dalibraic Topology". I also compiled slides which outlines some other occurrences of mathematics in art.