Dali and the Unfolded Hypercube

Figure 1

In April of 1984, as a birthday present from my father, I acquired a TR-80 computer from Tandy with 32Kb of RAM and a keyboard but no disk drive. Data input and output were via a cassette tape. Display was via a connected TV. Using this I began my BASIC programming and one of my first creations was a simulation of a projectile being launched to hit a target. The essential formulae for the projectile's motion are:$$x(t)=v_0 \cos(\theta) \cdot t \text{ and } y(t)=v_0 \sin(\theta) \cdot t - 0.5 g t^2$$where we have:

• \(x(t)\) is horizontal position of projectile after time \(t\) with \(x=0\) when \(t=0\)
• \(y(t)\) is vertical position of projectile after time \(t\) with \(y=0\) when \(t=0\)
• \(v_0\) is initial velocity at time \(t=0\)
• \(\theta\) is angle of projection
• \(g\) is deceleration due to gravity \(\approx \) -9.8 \(ms^{-2}\)

Figure 1 shows the situation with an angle of 60° and an initial velocity fo 6 \(ms^{-1}\). What has all this to do with unfolding hypercubes? Well, April of 1984 marked my return to Mathematics after a long absence. In August of that year, I began my training as a Mathematics teacher and one of the assigned projects was an assignment on some aspect of Mathematics. I chose Mathematics and Art, a choice which led to me to the unfolded hypercube. In the University library, I came across Salvator Dali's 1954 Crucifixion (Corpus Hypercubus). See Figure 2.

Figure 2: source

It is a large oil painting, with dimensions of 194.3 cm × 123.8 cm (76.5 in × 48.7 in). Figure 3 shows the net of a hypercube (or tesseract).

Figure 3: source

What reminded me of my youthful investigation into the realm of the hypercube was a Mathologer video that I watched today titled The Iron Man hyperspace formula really works.

Figure 4 shows a screenshot that demonstrates information about the number of vertices, edges and faces in an \(n\)-dimensional cube where \(n\) varies from 0 to 4. Looking at it, we see that a zero dimensional "cube" consists of only a single point or vertex. The number of vertices is shown by the coefficients of the \(x^0\) terms, coloured red. A one dimensional "cube", corresponding to a line, has two vertices and one edge. The number of edges is shown by the coefficients of the \(x^1\) terms, coloured green. A two dimensional "cube", corresponding to a square, has four vertices, 4 edges and one face. The number of faces is shown by the coefficients of the \(x^2\) terms. The familiar three dimensional cube has 8 vertices, 12 edges and 6 faces, plus one cube. The number of cubes is shown by the coefficients of the \(x^3\) terms. The four dimensional hypercube has 16 vertices, 32 edges, 24 faces, 8 cubes and one hypercube. The vertices, edges, faces and cubes can be counted using the net shown in Figure 2.

Figure 4: source

Figure 5 shows another screenshot where a more general version of Euler's polyhedron formula is displayed. It begins with one dimension where a line of finite length does indeed have two end points or vertices. Thus V=2. For two dimensions, the number of vertices equals the number of edges e.g. consider a square. Thus V=E. For three dimensions, we have the familiar V-E+F=2 and for the hypercube we have V-E+F-C=0 because as can be seen in Figure 4, V=16, E=32, F=24 and C=8.

Figure 5: source

However, getting back to Dali's hypercube, I just finished watching an interesting talk about Dali from an American mathematician who met the artist on a number of occasions. The video is titled Math Encounters -- Encountering Salvador Dali in the Fourth Dimension. 

The video was uploaded in March of 2014 and the mathematician, Tom Banchoff, is still alive and now aged 83. He born on April 7th 1938. He has written a book, titled Beyond the Third Dimension Geometry, Computer Graphics, and Higher Dimensions, a digital copy of which I've managed to acquire. It was first published in 1990. The cover is shown in Figure 6.

Figure 6

The author makes mention of his book in this talk and mentions also Flatland which is a classic that I remember reading many years ago but exactly when I can't recall. Here is a little more about this 1884 book:

This masterpiece of science (and mathematical) fiction is a delightfully unique and highly entertaining satire that has charmed readers for more than 100 years. The work of English clergyman, educator and Shakespearean scholar Edwin A. Abbott (1838-1926), it describes the journeys of A. Square, a mathematician and resident of the two-dimensional Flatland, where women-thin, straight lines-are the lowliest of shapes, and where men may have any number of sides, depending on their social status. 

Through strange occurrences that bring him into contact with a host of geometric forms, Square has adventures in Spaceland (three dimensions), Lineland (one dimension) and Pointland (no dimensions) and ultimately entertains thoughts of visiting a land of four dimensions—a revolutionary idea for which he is returned to his two-dimensional world. Charmingly illustrated by the author, Flatland is not only fascinating reading, it is still a first-rate fictional introduction to the concept of the multiple dimensions of space. "Instructive, entertaining, and stimulating to the imagination." — Mathematics Teacher.

Figure 7 shows the frontispiece to the original publication. Dali was probably familiar with this book as he was very interested in such dimensional adventures. 

Figure 7

Speaking of Flatland's two dimensions, what would a hypercube look like in Flatland? One way it could appear is shown in Figure 8. I got this image from an interesting web page about hypercubes.

Figure 8: source

Figure 9 gives a clearer idea of why this is so. The shadow of the tesseract, itself the 3D shadow of the hypercube, is shown falling on a two dimensional surface.

Figure 9: source

To end this post, I'll return to Mathologer and his YouTube channel. In July of 2017, he posted an interesting video about hypercube shadows that's well worth a look.