Cubes and Tesseracts on Paper

A tesseract is a ‘four dimensional cube’, that is projected into three space. The qualifier may be unnecessary, but of course you’ll never see an example of a tesseract for which this isn’t the case. We can’t draw a tesseract accurately because it requires further projection onto a plane. Of course we can still determine what is and isn’t a tesseract after this projection. In this context, ‘projection’ is the standard ‘photographic projection’ shown. It is worth noting that I refer exclusively to perspectives (or choices of photographic projection) under which the far side is not eclipsed by nearer sides. In particular, if we have a nice projection (luckily almost all photographic projections are nice), we can distinguish a tesseract, from a cube, from a square, from a line, from a point by looking at their two dimensional projections. It might seem odd that I mention lines and points, but these are exactly ‘cubes’ in one and zero dimensions. We might even call these a 1-cube or a 0-cube.

While playing around with these shapes ( $n$-cubes) on paper I came across an algorithm for generating projections of the $n$ cubes up to the 4-cube, or tesseract. I haven’t verified this algorithm for higher dimensional projections onto a plane due to the drawing quickly becoming cumbersome, I do think the algorithm is illustrative of the relationships between these shapes. I also do not claim that I am the first to come up with this, I am almost certainly not.

The algorithm involves drawing two copies of the $n$-cube, for convenience they are drawn parallel to one another, and then drawing lines between corresponding vertices (single points that terminate and connect line segments) to obtain the $n+1$ cube. Which vertices correspond should be obvious.

The process works fine if you decide randomly which vertices correspond, but it is difficult to illustrate and interpret the resulting pictures. Note that this description of the algorithm is not at all rigorous.

For example, a line is two points (connected by a line). A cube is two squares with a line joining each of their corresponding vertices. A tesseract is drawn as two cubes with a line joining each vertex. When it is said that a ‘tesseract is to a cube as a cube is to a square’. This is exactly what can be inferred from that statement.

A pattern to observe, among others, is that a line has 2 vertices, a square has 4 line sides, a cube has 6 square faces, a tesseract has 8 cubic cells. A $5$-cube presumably contains 10 regular four polytope (polytopic) cells.

The surprising thing about this is the linear growth of this particular quantity. Although quantities such as the “number of lines” or “the number of faces” do have growth of order $2^n$, which would be expected.

This procedure is hopefully illustrative of the nature of higher dimensions, particularly for students. Provided it is clear that the picture is a projection onto a 2 dimensional object and not actually a higher dimensional object.

I intend to update this post upon finding sources that reference this procedure or generalisation to higher dimensions, and with actual drawings up to at least 4 dimensions.