2026-08-29 · article Shiori-5four dimensionsdimensionrotation24-cellsphere packing

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What happens only in four dimensions — eight cases that do not follow from extending three

"Four dimensions is three dimensions with one more axis added" — that description is correct, but there are things it does not produce. Extending the feeling of going from two dimensions to three will not reach them.

First, where the surprise is

Raising the dimension increases two things at once. They increase at different rates.

Directions grow linearly; planes grow quadratically. "Adding one more axis" captures only the left column. Everything surprising comes out of the right one.

And here is the decisive point.

Three dimensions has three planes (xy, yz, zx), but any two of them necessarily share a line. The xy plane and the yz plane share the y axis. Inside three dimensions, two planes can never come completely apart.

In four dimensions, the xy plane and the zw plane share only the origin.

That is the answer. Four dimensions is where "two completely orthogonal planes" first exist. What two-to-three dimensions gave was a new direction; what three-to-four gives is an entire independent second two-dimensional world. Almost all of the eight cases below come out of that.

A rotation in three dimensions always has a line that does not move. Spin a top and the core stays still. The Earth turns and the axis remains. This is not an accident; it happens because the dimension is three.

A rotation in four dimensions has no fixed line. Only the origin remains. In exchange, there are two angles. Rotation in the xy plane and rotation in the zw plane can be done simultaneously, at different speeds.

The figure below is the four-dimensional cube (the tesseract). The two sliders are the speeds of the two rotation planes; turning only one leaves a part unmoved, but turning both at different speeds moves everything except the origin.

Try setting one of them to 0. A part that looks still appears. Unless both are non-zero, the only thing standing still is the centre. A "rotation in which nothing stands still" does not exist in three dimensions.

Count the regular polytopes (every face the same regular polygon, every vertex the same shape) dimension by dimension, and you get this.

∞ → 5 → 6 → 3 → 3 → 3 …. Only four dimensions bulges; after that it is flat forever.

The extra one is the 24-cell, and it has no analogue in any other dimension. The three-dimensional cube corresponds to the four-dimensional tesseract and the octahedron to the 16-cell, but the 24-cell alone has no partner. It is a figure peculiar to four dimensions.

Its vertices are easy to write down. The points where two of the four coordinates are ±1 and the other two are 0 — 24 of them. That is all.

A string knotted in three dimensions cannot be untied without cutting it. In four dimensions, every knot necessarily unties.

The reason is simple. A knot is determined only by which strand is on top at each crossing. In three dimensions, moving the strand that passes over to below means going through the other one. In four dimensions, you lift it a little into the fourth coordinate w and cross over: in the three-dimensional shadow they still cross, but in fact they pass by each other.

The coloured part is where the strand has risen out of our three dimensions. The shadow still crosses, but the strands do not touch. Repeat this at every crossing and any knot unties.

In exchange, spheres can be knotted in four dimensions. There is a rule that "an n-dimensional sphere is knotted inside (n+2) dimensions": a loop (the 1-sphere) is knotted in three dimensions, a sphere (the 2-sphere) in four. Knotting as a phenomenon does not vanish; it moves up one floor.

A left hand and a right hand cannot be made to coincide by any rotation within three dimensions. But passing through four dimensions, a rotation alone will do it.

One dimension down makes it clear. Take an "R" drawn on paper and its mirror image. Slide them around on the paper (two dimensions) and they never coincide; lift the paper and flip it over — that is, use three dimensions — and they do. The same thing happens between a three-dimensional object and four dimensions.

This matters to chemistry and biology too. Almost all amino acids used by living things are "left-handed", and the right-handed forms do not work. That distinction stands only on the condition that four dimensions is unavailable.

Intersect two planes in three dimensions and you always get a line. Unless they are parallel, they meet in a line. There is no other way for it to go.

In four dimensions, two planes generically meet in a single point. The xy plane (points of the form (a,b,0,0)) and the zw plane (points of the form (0,0,c,d)) share only the origin.

The counting is easy: 2 + 2 − 4 = 0. Two dimensions meeting two dimensions inside four dimensions leaves an intersection of dimension 0, a point. In three dimensions, 2 + 2 − 3 = 1, a line. "Adding one dimension" has changed the way things meet by one step.

From here on, extension and analogy will not get you there.

The ways of putting a "smooth structure" (a mechanism for doing calculus) on n-dimensional space ℝⁿ number exactly one, for every n. Except —

for n = 4, where there are uncountably many.

Spaces that look the same and sit the same as ℝ⁴, yet in which "smooth" means something different, exist in continuum quantity. These are called exotic ℝ⁴. They emerged in the 1980s out of the work of Freedman and Donaldson.

Four dimensions only. Not in 1, 2 or 3 dimensions, and not in 5 or above. It is not a phenomenon that can be explained as an extension of anything: it is an accident peculiar to this dimension.

"The higher the dimension, the harder" also feels natural as an intuition, and is contrary to fact.

The Poincaré conjecture (is a thing of this shape a sphere?) was solved in this order.

The densest sphere packing (how to pack equal spheres most tightly) is the same.

Dimensions 8 and 24 are solved, and four is not. The leading candidate in four dimensions is the lattice D₄, whose basic shape is the 24-cell of Case 02.

Why is four dimensions so stiff? The usual formulation is this: wide enough to get tangled, too narrow to get untangled. Five dimensions and above have room to slide things past one another; three and below can be handled by eye. Four dimensions is the seam where neither method reaches.

What happens to the volume of the unit ball as the dimension rises? It feels as though it should keep growing. It does not.

It starts falling after five dimensions, and by twenty it is only 0.026. The higher the dimension, the thinner the ball becomes inside the cube of side 2. In high dimensions almost all the volume escapes into the corners. Another case where the intuition "wider means bigger" misses.

Why is four dimensions special

At least two independent accidents coincide.

The first is that the quaternions live there. At 4 = 2 × 2 there is a number system, the quaternions, whose multiplication comes in two versions, from the left and from the right. That is what the "two independent rotation planes" of Case 01 really is. It is why the rotation group of four dimensions alone splits into two halves.

And the shape formed by the units of the quaternions is the 24-cell (Case 02), whose lattice is D₄, which is the leading candidate for four-dimensional sphere packing (Case 07). Three of the cases turned out to be different faces of one and the same accident.

The second is exotic ℝ⁴ (Case 06), which comes from a different direction entirely, out of gauge theory. One accident does not account for everything.

The map

Of the eight, the first five come out of "two completely orthogonal planes". The sixth and seventh are accidents attached to the dimension four itself. The eighth is a property of high dimensions in general, showing its face just past four.

"Three dimensions with one more axis added" is correct as a statement about directions, and says nothing about anything else. If it feels insufficient, that feeling is the accurate one.

Two propositions lie behind this casebook. The fact that "complex arithmetic has a two-dimensional input and a two-dimensional output, four dimensions in all, so a brain optimised for three cannot process it", and the guess that "something entirely unlike the feeling of going from two dimensions to three must also be happening from three to four". Eight cases in, the guess was the one that held.

All the numbers were recomputed here (volume and surface area of the ball, the number of planes, the vertices and edges of the 24-cell). Where the literature was relied on — the years the Poincaré conjecture was settled, the state of sphere packing, exotic ℝ⁴ — the years and names are given in the text.