Gaurav Singh

Hypercube

The hypercube, more often called the tesseract, is a shape that lives in four-dimensional space. If you remember the tesseract scene in the movie "Interstellar", you've already seen a version of it. But how do we visualise a 4D object in our 3D world, and then draw it on a 2D screen?

Mathematically, a point on a line can be represented by a single number $x$, on a plane by a pair $(x, y)$, and in space by a triplet $(x, y, z)$. The fourth dimension is a bit trickier. Imagine a new axis, $w$, that's perpendicular to the other three. This gives us a 4D point represented as $(x, y, z, w)$. Just as a square is to a cube, a cube is to a hypercube. It's the 4D analog of our familiar 3D cube. If you've ever wondered how a cube would cast a shadow in 4D, it would look like a hypercube!

Each vertex of a hypercube in 4D space can be represented by a 4-bit binary number. Mathematically, a vertex in n-dimensional space can be represented as:

$$V = (b_1, b_2, \ldots, b_n)$$

where each $b_i$ is a bit (0 or 1).

With a bit of combinatorics, a hypercube has $2^4 = 16$ vertices and $4 \times 2^3 = 32$ edges.

Projection maps points from a higher-dimensional space to a lower one. A 3D object casting a 2D shadow is a projection, and seeing a 4D object in 3D works the same way: you "slice" it so its structure shows up in three dimensions.

In 4D there are hyperplanes like the XY and ZW planes, and a point can be rotated around them using rotation matrices. A rotation in the XY plane looks like:

$$\begin{bmatrix} \cos(ax) & -\sin(ax) & 0 & 0 \\ \sin(ax) & \cos(ax) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

After rotating, we need to fake a sense of depth. Perspective projection does this by dividing by the fourth dimension, $w$:

$$\text{projected}[x,y,z] = \left[ \frac{pt[x]}{2 - pt[w]}, \frac{pt[y]}{2 - pt[w]}, \frac{pt[z]}{2 - pt[w]} \right]$$

The last piece is the difference between orthographic and perspective projection, which is really just how we choose to view the 3D object on a 2D screen.

Modern computer graphics handle all of this with a series of matrix multiplications. The viewport transformation fits the 3D object within the bounds of the screen, and the camera sets the angle we view it from.