🧊 The 4D hypercube

What you’re seeing

On the left is an ordinary cube turning, drawn on a flat screen. On the right is a tesseract, or 4D hypercube, turning in four dimensions and drawn the same way, one step further.

Look at the cube first. Its drawing is a square inside a square, with the corners joined. The back face looks smaller because it is farther away. That’s perspective.

Now look at the hypercube. It looks like a cube inside a cube, with the corners joined. The inner cube is the “far” cube in the fourth direction. As it turns in a plane that uses the fourth direction (we call it $w$), the inner cube swells and slides out while the outer one shrinks and slides in, until they trade places. Nothing is bending: every edge keeps its true length in 4D. Only the shadow changes.

The colors show direction: green edges run along $x$, blue along $y$, gold along $z$, and the thinner red edges along $w$.

Why it works

Building up one dimension at a time. Start with a point. Slide it 1 unit and it sweeps out a segment (2 ends). Slide the segment sideways and it sweeps out a square (4 corners). Slide the square up and you get a cube (8 corners). Slide the cube in a fourth direction, at right angles to all three, and you get the tesseract: 16 corners. Each slide doubles the corners and joins each old corner to its copy.

Coordinates make it exact. Put the corners at every point whose coordinates are all $\pm 1$. A cube has corners $(\pm1, \pm1, \pm1)$: $2^3 = 8$ of them. A tesseract has corners $(\pm1, \pm1, \pm1, \pm1)$: $2^4 = 16$. Two corners share an edge when they differ in exactly one coordinate. Each corner has 4 neighbors, so there are $16 \cdot 4 / 2 = 32$ edges. Counting faces the same way, an $n$-cube has

$$\binom{n}{k}\, 2^{\,n-k}$$

pieces of dimension $k$. For $n = 4$: 16 corners, 32 edges, 24 squares and 8 cubes.

Turning in four dimensions. In 3D we say a turn happens “about an axis”, but it is really a turn in a plane. Turning in the $x$–$y$ plane moves $x$ and $y$ and leaves the rest alone:

$$x' = x\cos\theta - y\sin\theta, \qquad y' = x\sin\theta + y\cos\theta.$$

In 4D there are six planes to turn in ($xy$, $xz$, $xw$, $yz$, $yw$, $zw$). The picture turns in the planes you tick, the ones that use $w$, because those are the turns that look strange.

Two shadows. To draw a 3D point $(x, y, z)$ with perspective, divide by the distance from your eye: the screen point is about $(x, y)/(d - z)$. To draw a 4D point, do it twice. First divide by $(d - w)$ to get a 3D point, then divide by the depth again to reach the screen. That is why the “far” cube in $w$ looks small and sits inside.

Why we can’t just look at it. Our eyes and our space give us three directions at right angles. A fourth one has nowhere to fit. But nothing in the algebra above cares: a fourth coordinate works just like the other three. That is how mathematicians “see” in 4, 10 or 1,000 dimensions.

Try this

• Turn off all three turning planes except $x$–$w$. Watch the inner cube trade places with the outer one. How long until it comes back?
• In the cube on the left, count the squares you see. Some are squashed into other shapes by perspective. Find all 6.
• In the hypercube, find all 8 cubes. Hint: the inner cube, the outer cube, and 6 “slanted” cubes that join a face of one to a face of the other.
• Using $\binom{n}{k} 2^{n-k}$, how many corners, edges and squares does a 5D cube have?