The n-Category Café

To understand all this, start with the group of rotational symmetries of the icosahedron. This is a 60-element subgroup of the rotation group SO(3), so it has a double cover, called the binary icosahedral group, consisting of 120 unit quaternions. With a suitable choice of coordinates, we can take these to be

$$\displaystyle{ \pm 1 , \quad \frac{\pm 1 \pm i \pm j \pm k}{2}, \quad \frac{\pm i \pm \phi j \pm \Phi k}{2} }$$

together with everything obtained from these by even permutations of $1, i, j,$ and $k,$ where

$$\displaystyle{ \phi = \frac{\sqrt{5} - 1}{2}, \quad \Phi = \frac{\sqrt{5} + 1}{2} }$$

are the ‘little’ and ‘big’ golden ratios, respectively. These 120 unit quaternions are the vertices of a convex polytope in 4 dimensions. In fact this is a regular polytope, called the 600-cell since it has 600 regular tetrahedra as faces.

If we slice the 600-cell with halfway between two of its opposite vertices, we get an icosidodecahedron. This is easiest to see by intersecting the 600-cell with the space of purely imaginary quaternions

$$\{ a i + b j + c k : \; a,b,c \in \mathbb{R} \}$$

Of the 600-cell’s vertices, those that lie in this 3-dimensional space are

$$\pm i, \pm j, \pm k$$

which form the corners of an octahedron, and

$$\displaystyle{ \frac{\pm i \pm \phi j \pm \Phi k}{2} , \quad \frac{\pm j \pm \phi k \pm \Phi i}{2} , \quad \frac{\pm k \pm \phi i \pm \Phi j}{2} }$$

which form the corners of three ‘golden boxes’. A golden box is the 3d analogue of a golden rectangle: its three sides are in the proportions $\phi,$ 1 and $\Phi.$

It is well-known that these points are the vertices of an icosidodecahedron. Here are the three golden boxes and octahedron inscribed in an icosidodecahedron, as drawn by Rahul Narain:

But we are not done with the binary icosahedral group—far from it!

Integer linear combinations of these 120 elements of the quaternions form a subring of the quaternions, which Conway and Sloane [CS] call the icosians. Since any icosian can be written as $a + bi + cj + dk$ where the numbers $a,b,c,d \in \mathbb{R}$ are of the form $x + y \sqrt{5}$ with $x,y$ rational, any icosian gives an 8-tuple of rational numbers. However, we do not get all 8-tuples of rationals this way, only those lying in a certain lattice in $\mathbb{R}^8.$ And there is a way to think of this lattice as a rescaled copy of the famous E₈ lattice! To do this, Conway and Sloane put a new norm on the icosians as follows. The usual quaternionic norm is

$$\|a + b i + c j + d k\|^2 = a^2 + b^2 + c^2 + d^2$$

But for an icosian this norm is always of the form $x + \sqrt{5} y$ for some rationals $x$ and $y.$ Conway and Sloane define a new norm on the icosians by setting

$$|a + b i + c j + d k|^2 = x + y$$

With this new norm, Conway and Sloane show the icosians are isomorphic to a rescaled version of the E₈ lattice in $\mathbb{R}^8.$

The 240 shortest nonzero vectors in this lattice are the vertices of an 8-dimensional convex polytope called the E₈ root polytope:

However, if we remember that each of these 240 vectors came from a quaternion, we can also think of them as 240 quaternions. These turn out to be the vertices of two 600-cells in the quaternions! In the usual quaternionic norm, one of these 600-cells is larger than the other by a factor of $\Phi.$

In fact, there is an orthogonal projection from $\mathbb{R}^8$ down to $\mathbb{R}^4$ that maps the E₈ root polytope to the 600-cell. So, in a very real sense, the 600-cell is the ‘shadow’ of a polytope with twice as many vertices, living in a space whose dimension is twice as large. And as a spinoff, this fact gives the same sort of relationship between the icosidodecahedron and a 6-dimensional polytope.

The key is to look at pure imaginary icosians: those of the form $a i + b j + c k$ for real $a,b,c.$ Since $a,b$ and $c$ are each of the form $x + \sqrt{5}y$ with $x$ and $y$ rational, any pure imaginary icosian gives a 6-tuple of rational numbers. We do not get all 6-tuples of rationals this way, but only those lying in a certain lattice. We have

$$\|a i + b j + c k\|^2 = a^2 + b^2 + c^2$$

For a pure imaginary icosian this is always of the form $x + \sqrt{5} y$ for some rationals $x$ and $y.$ So, we can define a new norm on the pure imaginary icosians by

$$|a i + b j + c k|^2 = x + y$$

With this new norm, the pure imaginary icosians are isomorphic to a rescaled version of a familiar lattice in $\mathbb{R}^6,$ called the ‘D₆ lattice’.

The 60 shortest nonzero vectors in the D₆ lattice are called the roots of D₆, and they are the vertices of a 6-dimensional convex polytope called the D₆ root polytope. There is an orthogonal projection from $\mathbb{R}^6$ to $\mathbb{R}^3$ that maps this polytope to an icosidodecahedron. In fact 30 vertices of the D₆ root polytope map to the vertices of this icosidodecahedron, while the other 30 map to vertices of a second, smaller icosidodecahedron.

Here is an image of the setup, created by Greg Egan:

Let’s see some details! The usual coordinatization of the D₆ lattice in Euclidean $\mathbb{R}^6$ is

$$\mathrm{D}_6 = \left\{ (x_1, \dots, x_6) : \; x_i \in \mathbb{Z}, \; \sum_i x_i \in 2\mathbb{Z} \right\} \subset \mathbb{R}^6$$

The roots of D₆ are

$$(\pm 1, \pm 1, 0, 0, 0, 0)$$

and all vectors obtained by permuting the six coordinates. We shall see that these vectors are sent to the vertices of an icosidodecahedron by the linear map $T \colon \mathbb{R}^6 \to \mathbb{R}^3$ given as a 3 × 6 matrix by

$$\left( \begin{array}{cccccc} \Phi & \Phi & -1 & -1 & 0 & 0 \\ 0 & 0 & \Phi & -\Phi & -1 & 1 \\ -1 & 1 & 0 & 0 & \Phi & \Phi \end{array} \right)$$

The rows of this matrix are orthogonal, all with the same norm, so after rescaling it by a constant factor we obtain an orthogonal projection. The columns of this matrix are six vertices of an icosahedron, chosen so that we never have a vertex and its opposite. For any pair of columns, they are either neighboring vertices of the icosahedron, or a vertex and the opposite of a neighboring vertex.

The map $T$ thus sends any D₆ root to either the sum or the difference of two neighboring icosahedron vertices. In this way we obtain all possible sums and differences of neighboring vertices of the icosahedron. It is easy to see that the sums of neighboring vertices give the vertices of an icosidodecahedron, since by definition the icosidodecahedron has vertices at the midpoints of the edges of a regular icosahedron. It is less obvious that the differences of neighboring vertices of the icosahedron give the vertices of a second, smaller icosidodecahedron. But thanks to the symmetry of the situation, we can check this by considering just one example. In fact the vectors defining the vertices of the larger icosidodecahedron turn out to be precisely $\Phi$ times the vectors defining the vertices of the smaller one!

The beauties we have just seen are part of an even larger pattern relating all the non-crystallographic Coxeter groups to crystallographic Coxeter groups. For more, see the work of Fring and Korff [FK1,FK2], Boehm, Dechant and Twarock [BDT] and the many papers they refer to. Fring and Korff apply these ideas to integrable systems in physics, while the latter authors explore connections to affine Dynkin diagrams. For more relations between the icosahedron and E₈, see [B2].

Acknowledgements

I thank Greg Egan for help with developing these ideas. The spinning icosidodecahedron was created by Cyp and was put on Wikicommons with a Creative Commons Attribution-Share Alike 3.0 Unported license. The 600-cell was made using Robert Webb’s Stella software and is on Wikicommons. The icosidodecahedron with three golden boxes and an octahedron inscribed in it was created by Rahul Narain on Mathstodon. The projection of the 240 E₈ roots to the plane was created by Claudio Rocchini and put on Wikicommons with a Creative Commons Attribution 3.0 Unported license. The spinning pair of icosidodecahedra was created by Greg Egan and appears in an earlier blog article on this subject [B1]. The article here is an expanded version of that earlier article: the only thing I left out is the matrix describing a linear map $S \colon \mathbb{R}^8 \to \mathbb{R}^4$ that when suitably rescaled gives a projection mapping the E₈ lattice in its usual coordinatization

$$\{ x \in \mathbb{R}^8: \, all \; x_i \in \mathbb{Z} \; or \; all \; x_i \in \mathbb{Z} + \frac{1}{2} \; and \; \sum_i x_i \in 2\mathbb{Z} \}$$

to the icosians, and thus mapping the 240 E₈ roots to two 600-cells. For completeness, here is that matrix:

$$\left( \begin{array}{cccccccc} \Phi+1 & \Phi -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \Phi & \Phi & -1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & \Phi & -\Phi & -1 & 1 \\ 0 & 0 & -1 & 1 & 0 & 0 & \Phi & \Phi \end{array} \right)$$

The image at the bottom of this post was also created by Greg Egan, on Mathstodon.

References

[B1] John Baez, Icosidodecahedron from D₆, Visual Insight, January 1, 2015.

[B2] John Baez, From the icosahedron to E₈, London Math. Soc. Newsletter 476 (2018), 18–23.

[BDT] Celine Boehm, Pierre-Philippe Dechant and Reidun Twarock, Affine extensions of non-crystallographic Coxeter groups induced by projection, J. Math. Phys. 54, 093508 (2013). [CS] John H. C. Conway and Neil J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, Berlin, 2013.

[FK1] Andreas Fring and Christian Korff, Affine Toda field theories related to Coxeter groups of non-crystallographic type, Nucl. Phys. B729 (2005), 361–386.

[FK2] Andreas Fring and Christian Korff, Non-crystallographic reduction of generalized Calogero–Moser models, J. Phys. A39 (2006) 1115–1132.