Tesseract

The tesseract in this visualization

The tesseract is the set of points $(x, y, z, w)$ with every coordinate between $-1$ and $1$: a cube with one more direction. Sliding a point gives a segment, sliding the segment a square, the square a cube, and the cube — in a fourth direction perpendicular to the other three — a tesseract. Each step doubles the vertices; the $n$-dimensional cube has $\binom{n}{k}2^{n-k}$ faces of dimension $k$, so the tesseract has 16 vertices, 32 edges, 24 squares and 8 cubes, its cells. The 8 cells form its boundary, as the 6 squares form a cube’s surface.

What the picture shows

The large picture is a shadow. A lamp sits on the $w$ axis at distance $d$ from the centre, and every point of the tesseract casts its shadow into our space $w = 0$:

This is a central projection, the four-dimensional version of a lamp above a wire cube throwing its shadow on the table — the screen at the top of the panel shows that cube. The shadow is a three-dimensional object; the screen then shows it from one side, a second projection. Dragging the picture only changes that second one.

The panel turns the tesseract itself. In four dimensions a rotation turns a plane: of the six coordinate planes, the three that contain $w$ (XW, YW, ZW) change the shadow’s shape; turns in XY, XZ and YZ only turn the shadow, which is what dragging does. XY+ZW turns two planes at once.

The blue cell is the one you track. Its name, such as w−, is the object’s own label (the cell where its own $w$ is $-1$), kept however it turns. The red key Half turn turns the tesseract 180° in the chosen plane at a steady rate and leaves the paths of the tracked cell’s vertices, with a tick every 15°, so equal gaps in time.

The mode dial gives two more ways to look. Slice: our space $w = t$ cuts the tesseract in its current pose. The cut is a solid drawn at its true size, inside the faint shadow cast by parallel light along $w$ (the cut lies in the hyperplane $w = t$, so dropping $w$ moves it without changing it). It is a real solid in our space, so it is drawn the way we would see one: glass lit by an ordinary light here. The red key Pass through sweeps $t$ across at a steady rate and leaves a row of cuts at equal steps. The cuts’ volumes $V(t)$ integrate to the tesseract’s four-dimensional volume: $\int V\,dt = 2^4 = 16$.

Net: eight cubes laid out in our space as a cross, the shape Salvador Dalí painted in Corpus Hypercubus (1954), fold about their shared squares into the fourth dimension. The flat cross lies in the hyperplane $w = 1$, so its shadow is the cross enlarged $d/(d-1)$ times: true in shape and proportion, not in size. At 90° the cubes close into the tesseract, and the bottom cube of the cross becomes the small cube inside the shadow.

Why the inner cube is small

In the standard pose two cells lie flat to the lamp: $w = 1$, nearer, and $w = -1$, farther. A cell at height $w$ is enlarged $d/(d - w)$ times, so their shadows are cubes of edge $2d/(d-1)$ and $2d/(d+1)$: the famous cube in a cube. The other six cells reach from one to the other and their shadows look like slanted boxes; all eight are equal cubes.

A turn in the XW plane cycles four cells — $w-$ to $x+$ to $w+$ to $x-$ — and turns the other four in place, as a cube turned in front of the lamp brings its bottom face round to the side and then to the top. Each vertex moves on a circle; the shadow of a circle is an ellipse — or a segment, when the light is parallel or the circle’s plane passes through the lamp.

Turning XY and ZW together at the same rate moves every point but the centre, each by the same angle (up to 180°). Half a turn takes every point to the opposite side of the centre — the map $-I$. In three dimensions $-I$ is a reflection; every rotation there keeps an axis fixed.

The lamp’s light

The lamp also lights the tesseract, as a lamp lights a cube. Lambert’s cosine law holds in any dimension: a piece of surface takes light in proportion to the cosine of the angle between its outward normal and the direction to the lamp. A tesseract’s surface is its 8 cells, each with a normal; a cell facing the lamp is lit, more brightly as it faces it more squarely, and a cell facing away is dark. The glow around bright places is display; the brightness itself is this light.

A point lamp’s light spreads over spheres around it. In our space a sphere’s area grows as $r^2$, so light falls off as $1/r^2$; in four dimensions the sphere around the lamp is three-dimensional, its size grows as $r^3$, and light falls off as $1/r^3$. That is why the part nearest the lamp glows and the light fades so quickly across a cell. With the lamp at infinity there is no falloff.

What the lamp lights is exactly what an eye at the lamp would see: the cells facing away lie behind the lit ones. Lit cells only keeps just those (the tracked cell, when it is behind, stays as a faint trace). A cube shows at most 3 faces at once; a tesseract shows at most 4 cells, at most one for each of its own axes — the cell on that axis’s side where the lamp lies beyond it. Cell first there is 1, the large outer cube; face first 2; edge first 3; vertex first 4, the four keys of the slice mode. The lit cells’ shadows fill the whole outline of the shadow without overlapping, as the top of a cube with a lamp straight above it casts the whole of its shadow.

The net is lit by the same lamp. Lying flat in $w = 1$ every cube faces it; folding, each cube but the centre turns its normal away from $w$ and darkens; closed, only $w+$ is lit — the picture of the shadow with only the lit cells kept.

Easy to misread

• The inner cube is not smaller than the outer one; it is farther from the lamp.
• A dark cell has not shrunk or gone: it faces away from the lamp.
• Edges that cross in the picture need not meet in the shadow, and edges that meet in the shadow need not meet in four dimensions: there are two projections between the tesseract and the screen.
• Nothing turns inside out or bends: every turn here is rigid. What changes is how far each part is from the lamp.
• The fourth dimension here is a direction in space, not time; the red key Pass through only uses time to sweep that direction.
• A slice is not a shadow: it is the part of the tesseract lying in one hyperplane, at its true size, while the shadow flattens the whole object.

History

Charles Howard Hinton named the tesseract in 1888, in a book on how one might learn to imagine four dimensions [1]; Edwin Abbott’s Flatland (1884) had already used beings of two dimensions meeting a third to make the analogy [2]. Thomas Banchoff’s computer films and his book Beyond the Third Dimension showed the tesseract’s shadows, slices and unfoldings turning on a screen [3].

Related concepts

Linear transformation · Orthogonal projection (page planned)

References

1. C. H. Hinton, A New Era of Thought, Swan Sonnenschein (1888).
2. E. A. Abbott, Flatland: A Romance of Many Dimensions, Seeley & Co. (1884).
3. T. F. Banchoff, Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Scientific American Library (1990).

Further reading: Wikipedia: Tesseract.