Symmetries and the 24-cell

by Greg Egan


The 24-cell is a 4-dimensional regular polytope that has:

In the image above, the 16 red vertices are the vertices of a hypercube, while the other 8 vertices (the 2 black and 6 yellow) are those of a cross-polytope. Exactly how this particular colouring was selected will be discussed below.

There are many ways to construct a 24-cell, but here we will focus on an approach in which features of this polytope, in four dimensions, correspond to symmetries of a very simple object in three dimensions.

The 24-cell from symmetries in three dimensions

As well as being a vector space, 4-dimensional Euclidean space can be treated as an algebraic system known as the quaternions, which allows the vectors to be multiplied and divided by each other as well as added, subtracted, and multiplied by ordinary numbers By convention, the vector (1,0,0,0) is usually chosen as the identity quaternion, 1, where multiplying by 1 leaves any other vector unchanged, while the other basis vectors are known as i = (0,1,0,0), j = (0,0,1,0) and k = (0,0,0,1). Multiplication is defined so that:

i2 = j2 = k2 = –1

i j = k
j k = i
k i = j

j i = –k
k j = –i
i k = –j

Vectors that are real multiples of 1 are called real, and vectors that lie in the 3-dimensional subspace spanned by {i, j, k} are called imaginary, by analogy with complex numbers, though now we have three independent square roots of minus one. If we divide a quaternion into real and imaginary parts, the multiplication rule can be written as:

(a 1 + v) (b 1 + w) = (a bv · w) 1 + a w + b v + v × w

Here a, b are real numbers, v and w are purely imaginary quaternions, and we have made use of the ordinary dot product and cross product between 3-dimensional vectors.

The length of a product of quaternions turns out to be simply the product of their lengths:

|a b| = |a| |b|

Suppose we have a quaternion r of length 1; this is called a unit quaternion. Consider the function R:

R(q) = r q r–1

This function preserves the length of q, and if applied to two vectors it preserves the angle between them. What’s more, if R is applied to any purely imaginary quaternion, the result remains purely imaginary. We can also check that R does not swap left- and right-handed objects. So R gives us a rotation of a 3-dimensional space: the space of imaginary quaternions.

This lets us treat any unit quaternion as a 3-dimensional rotation. But it is easy to see from the definition of R that r and –r will yield exactly the same rotation. So every 3-dimensional rotation corresponds to a pair of unit quaternions. Specifically, if u is a purely imaginary unit quaternion, we can obtain a rotation by an angle θ around u by setting:

r = ±[(sin θ/2) 1 + (cos θ/2) u]

We can use this correspondence to construct all the vertices of a 24-cell as the quaternions corresponding to the rotational symmetries of a simple 3-dimensional object: a regular tetrahedron. And at the same time, we can describe the centres of the octahedral cells of the 24-cell in terms of the symmetries of an associated cube.

Start with a regular tetrahedron, such as the red one in the image below. Then construct a second tetrahedron (blue) whose vertices are the opposite of the red ones; that is, if the origin of our coordinates is the centre of the red tetrahedron, and its vertices are the vectors v1, v2, v3, v4, then the vertices of the blue tetrahedron are –v1, –v2, –v3, –v4.

Compound of two tetrahedra inside a cube

The eight vertices we obtain this way are the vertices of a cube, whose edges are shown in white.

Now, every cube has 24 rotational symmetries, and each of these symmetries will either swap the red and blue tetrahedra, or leave them in place:

So, of the 24 symmetries of the cube, 12 leave the tetrahedra in place (which means they are also symmetries of the tetrahedra themselves), while the other 12 swap the tetrahedra.

In the image below, we show two 24-cells: the first is the same as the image at the top of the page, with black, red and yellow vertices corresponding to symmetries that leave the tetrahedra in place. The second has blue and cyan vertices, corresponding to symmetries that swap the tetrahedra. Note that there are twice as many vertices of each colour as there are symmetries, because every quaternion and its opposite correspond to the same symmetry.

The 24-cell is a self-dual polytope: the centres of its cells form the vertices of another polytope of exactly the same shape. And it turns out that if we rescaled the vectors pointing to the centres of the 24 octahedral cells of the first 24-cell by a factor of √2, they would coincide precisely with the 24 vertices of the second 24-cell (and vice versa).

The set of 24 quaternions comprising the vertices of the first 24-cell constructed this way is known as the binary tetrahedral group, while the set of 48 quaternions comprising the vertices of both 24-cells is known as the binary octahedral group. The ordinary tetrahedral group and octahedral group are just the groups of 3-dimensional rotational symmetries of a tetrahedron and a cube respectively (a cube and an octahedron have the same symmetry group); these “binary” variants, with two elements for every element of the original, are the subgroups of the unit quaternions that correspond to those rotations.

Binary Octahedral Group

Symmetries of the 24-cell

We have seen how the vertices, and the centres of the octahedral cells, of a 24-cell correspond to the rotational symmetries of a 3-dimensional cube. But what can we say about the symmetries of the 24-cell itself?

While unit quaternions can be used to describe 3-dimensional rotations, it turns out that we can describe 4-dimensional rotations with a very similar method, but we need to use pairs of unit quaternions instead. If we define the function R as:

R(q) = r q s–1

for some choice of unit quaternions r and s, then R will again preserve lengths and angles, but it will no longer be guaranteed to map the 3-dimensional space of imaginary quaternions into itself. We can actually obtain all 4-dimensional rotations this way, and just as we obtained the same 3-dimensional rotation from r and –r, we will obtain the same 4-dimensional rotation from (r, s) and (–r, –s).

If we choose (r, s) to be a pair of vertices of the first 24-cell we constructed in the previous section, there are 24×24/2 = 288 possible choices, and it turns out that all the rotations obtained this way are symmetries of the 24-cell itself.

But a 24-cell actually has 576 rotational symmetries ... and we can obtain the other 288 rotations by choosing (r, s) to be a pair of vertices from the second 24-cell, whose vertices are rescaled versions of the centres of the octahedral cells of the first 24-cell.

What if we choose r from the first 24-cell and s from the second one, or vice versa? The rotation we obtain then swaps the two 24-cells, mapping every vertex of the first to a vertex of the second, and every vertex of the second to a vertex of the first.

That these 24-cells yield symmetries of themselves this way is not really suprising, given that their vertices are derived from the symmetry groups of a tetrahedron and a cube, and as a consequence they are closed under quaternion multiplication (to be precise, the first 24-cell is closed under multiplication, and the 48 vertices of the two 24-cells combined is also closed, but the second 24-cell by itself is not closed, as it comes from only some of the cube’s symmetries, not all of them).

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