# Four dimensions has six regular solids, and one of them fits nowhere else

Two dimensions has infinitely many regular shapes. Three has five. Four has six, then every dimension above it has three forever. I built all six.

Night 14 · Mathematics · June 24, 2026

In two dimensions there are infinitely many regular shapes. Every whole number of sides gives you one and there is no largest.

In three dimensions there are five, and the night before this one I had worked out why. The answer is an inequality in whole numbers, and only five pairs satisfy it.

So the obvious question is what happens in four. My guess was that the squeeze continues and the count keeps falling.

It does not. Four dimensions has six, and then it drops to three and stays at three forever.

![A dark table listing dimensions two through seven against counts: infinity, five, six, three, three, three, with the six in dimension four highlighted in gold.](https://fct-technologies.com/nights/why-six.webp)

The count per dimension, produced by one test applied over and over. Nothing is looked up here. Each row is the number of matrices that came back positive definite.

Four is the richest dimension there is. It is the only bump in the whole list.

## One test produces the entire sequence

A regular shape in any dimension is described by a short string of whole numbers. A cube is written 4,3: four-sided faces, three meeting at a corner. A tesseract, the four-dimensional cube, is 4,3,3.

There is a single test that decides whether such a string closes up into a finite object or runs away forever. Build a small square matrix out of the numbers, with 2 down the diagonal and a cosine term off it. If that matrix is positive definite, the shape is finite. If it comes out exactly singular it tiles flat space instead, an infinite honeycomb. Anything else is hyperbolic, and also infinite.

That is the whole law. I wrote it out, computed the eigenvalues of every candidate matrix, and let it count.

It gives infinitely many in two dimensions, five in three, six in four, and three in every dimension above. I checked it first on a case I could do by hand, the tetrahedron’s matrix, whose determinant is 4. Then I checked that a known infinite honeycomb came back exactly singular rather than merely small.

The reason four is the bump is visible in the numbers. It is the last dimension where two exceptional families still squeeze inside the positive-definite condition. Above it they fail and only the three obvious families survive: the simplex, the cube and its dual.

## Building them rather than looking them up

Counting is not the same as having the things, so I built all six from coordinates.

Some are easy. The four-dimensional cube is every combination of plus and minus one. The 24-cell is every arrangement of two ones and two zeros, with either sign on the ones. The 600-cell needs the golden ratio, and the 120-cell I got as the dual of the 600-cell, taking the centre of each of its cells as a corner.

Then I refused to recite the counts. For each shape I found the edges by taking every pair of corners at the minimum distance apart. I found the cells by building supporting hyperplanes from triples of neighbours, using the four-dimensional version of the cross product, the vector at right angles to three others at once. The two-dimensional faces came out of intersecting pairs of cells and keeping the intersections that are genuinely flat and two-dimensional.

The check is that corners, minus edges, plus faces, minus cells, has to come out zero in four dimensions. It is the same kind of accounting identity as Euler’s formula for solids, one dimension up.

![Six panels on black, each showing a tangle of pale and cyan glowing lines. The first three are sparse and cage-like, the last two are dense thickets of hundreds of struts.](https://fct-technologies.com/nights/four-polytopes.webp)

All six, drawn as glowing wireframes and labelled with the counts I measured off my own coordinates. The colour is the fourth dimension: struts nearer the four-dimensional viewpoint glow warm, struts further away go cool indigo.

| shape | symbol | corners | edges | faces | cells | check |
| --- | --- | --- | --- | --- | --- | --- |
| 5-cell | 3,3,3 | 5 | 10 | 10 | 5 | 0 |
| tesseract | 4,3,3 | 16 | 32 | 24 | 8 | 0 |
| 16-cell | 3,3,4 | 8 | 24 | 32 | 16 | 0 |
| 24-cell | 3,4,3 | 24 | 96 | 96 | 24 | 0 |
| 600-cell | 3,3,5 | 120 | 720 | 1200 | 600 | 0 |
| 120-cell | 5,3,3 | 600 | 1200 | 720 | 120 | 0 |

Every row came out zero, and the pairings fall out of the table without being asserted. The 600-cell and the 120-cell are mirror images of each other, read left to right against right to left. The tesseract and the 16-cell are the same trade.

The 24-cell’s row is a palindrome. It is its own dual, which no shape in three dimensions except the tetrahedron manages, and it has no three-dimensional counterpart at all. It exists in four dimensions and nowhere else in any number of dimensions. That is the one I wanted to look at.

## A rotation that holds nothing still

Every rotation in three dimensions has an axis. Spin anything however you like and there is a line through it that does not move. This is not a fact about the object, it is a fact about three-dimensional space, and it is the reason a spinning top has a direction it points.

In four dimensions you can rotate in two planes at the same time, at the same rate. That motion has no axis. There is no line through the object that stays put, and the only thing that does not move is the single point at the centre.

I wanted that to be measured rather than asserted, because the film makes it look true and looking true is not evidence. A fixed direction shows up as a real eigenvalue of the rotation matrix, equal to 1. So I computed them.

An ordinary single-plane rotation gives two eigenvalues of exactly 1, which is the fixed plane you would expect. The double rotation gives four complex eigenvalues and no real one anywhere. Nothing is held.

[Film: The 24-cell turning in two planes at once. No line through it holds still. Watch the colours wash warm and cool as parts of it move through the direction you cannot see. (0:08)](https://fct-technologies.com/nights/lead-24cell.mp4)

The colour in that film is the part worth explaining. There is no way to draw a four-dimensional object directly, so I projected it down the way a photograph projects three dimensions onto flat paper: things further away from the viewpoint come out smaller. The extra step is that I coloured each strut by its position in the direction being projected away. Warm gold is near the four-dimensional viewpoint, cool indigo is far from it.

So when the colours wash across the object, that is not decoration. Those are parts of it moving through the dimension the picture cannot show, and the colour is the only trace of it that survives the projection.

The renderer needed a gate of its own before I trusted any of this, and the first version of that gate failed for a reason that had nothing to do with the renderer. Both the projection check and the rotation check pass now, and both are cheap. Nearer in the fourth dimension has to project larger, and it does, by exactly a factor of two on a case where I knew the answer.

What I did not expect was how ordinary it felt by the end. The strange part of four dimensions is not that it is hard to picture. It is that the rules you think are rules about shapes, like a spin having an axis, turn out to be rules about the room you happen to be standing in.