Can we find 5 regular hypersolids in 4D? Surprise answer; and introducing Ludwig Schläfli

New readers:

• At this point you may like to click here. A new window will open where you can find my definition of the 4th dimension. This links to a simple explanation of a 4D cube ... how it relates to a point, a line, a square and an ordinary cube ... and how to draw it ... and how it rolls.

Now, continuing our search for regular hypersolids in 4D, we can consider whether there are any other possible building blocks, which we might be able to use to construct regular 4D shapes.   Is it a bit sad, for example, not to have an equivalent in 4D, of some kind, for the dodecahedron in 3D?   Could the dodecahedron, that strange 12-faced shape, be usable as a building block, however unlikely that may seem?   What's the dihedral angle?

It turns out, perhaps surprisingly, that the dihedral angle between two adjacent faces of a regular dodecahedron where they meet is just less than one third of a 360 degree revolution, i.e. less than 120 degrees.   It follows that we can – just – fit three dodecahedra together at one edge.   The 3D dodecahedron could, in principle, be used as a building block to build a 4D shape.

We now have, therefore, five candidates for building regular 4D hypersolids.   Three of them use the tetrahedron as a building block, using respectively three, four and five tetrahedra clustering around an edge.   One uses a cubical building block, with three around each edge.   And one uses dodecahedra, again with three around each edge.

I can now reveal to you that all five of these candidates do "really" (whatever that means) exist.   They are collectively known as polytopes, specifically in fact as 4D polytopes.   I can show you their properties later in terms of the numbers of corners, edges, faces and so on which they possess.   But in essence we have firstly the 4-simplex (corresponding to the 3-simplex or tetrahedron in 3D) with three tetrahedra meeting at each edge.   As we shall see, this 4-simplex consists of a total of 5 tetrahedral "cells" (as they are usually known).   So the 4D simplex itself is called the 5-cell.

Secondly we have the hypercube (corresponding to the ordinary 3D cube) with three cubes meeting at each edge.   We shall examine it in detail later and we shall find that it consists in total of 8 cubical cells.   It is therefore (sometimes) known as the 8-cell.

Thirdly we have the so-called "cross polytope", which corresponds to the octahedron in 3D.   It has four tetrahedra meeting at each edge.   In all, this 4D cross polytope has 16 tetrahedral cells.   We shall show how it is built up later.   It is known as the 16-cell.

Fourthly we have a polytope which corresponds to the icosahedron in 3D.   It is built up, once again, from tetrahedral cells and it has five of them meeting at each edge.   It is complicated to draw (so I won't try) because in all it turns out to have 600 of these tetrahedral cells.   It is a magnificent object and is known as the 600-cell.

There is no room to have 6 tetrahedra meeting at an edge, so fifthly you will recall that, rather surprisingly, we found that three dodecahedra could just about manage to find room to fit round an edge and could therefore, in principle at least, be used as building blocks.   There is indeed a 4D polytope which uses such dodecahedral building blocks, with three of them meeting at each edge.   It corresponds, as a matter of fact, to the dodecahedron itself in 3D.   Again it is a magnificent and surprising object, very difficult to draw, and it has 120 cells.   Each cell consists of one of these dodecahedral building blocks.   It is known as the 120-cell.

Is that all, do you suppose?   Recall that in 2D space there is an infinite number of possible regular polygons, in 3D space we know that there are just 5 regular polyhedra and we have already found 5 regular polytopes in 4D space.   Jumping ahead a bit, it turns out that in 5D space, and in all higher spaces from 6D to infinity-D, the number of regular shapes goes down to just three.   In all these spaces, there are regular shapes corresponding (a) to the 3D tetrahedron and the 4D 5-cell, (b) to the 3D cube and the 4D 8-cell or hypercube, and (c) to the 3D octahedron and the 4D 16-cell.   That is all that are possible in 5 or more dimensions.

There's one more surprise coming however, which doesn't correspond to anything in any of the other dimensions.   Can you guess what it might be?

There is in fact one extra 3D building block we haven't considered.   It's the octahedron.   Could we use that as a building block?   What's its dihedral angle?

An octahedron's dihedral angle turns out to be more than that of the cube, but less than that of the dodecahedron.   It follows, at least in theory, that you could fit three of the things round an edge – but not four of them.   And in practice this octahedral building block comes up trumps.   There is, as it turns out, a polytope with octahedral cells.   It has 24 of them and is known as the 24-cell.   It is the only one of our 4D polytopes which doesn't seem to correspond to any equivalent in 3-space.   The person who discovered it therefore (much longer ago than you might think) had a particular soft spot for it.

Who was he?

In the 1880s, many mathematicians around the world had worked out that there must be six regular polytopes in 4D.   The race was on.   Competition was intense.   Who could be the first to enumerate and describe them all correctly?

It was all in vain however.   A ton of humility was forcibly squirted at the contestants when it was found that the problem had been completely solved some thirty years earlier in 1852 by the remarkable, but then almost unknown, self-taught Swiss mathematician Ludwig Schläfli (1814 - 1895).   His exposition of the problem and its solution, incredibly, contains not a single diagram.