Discriminant of the Icosahedral Group

This image, created by Greg Egan, shows the ‘discriminant’ of the symmetry group of the icosahedron. This group acts as linear transformations of $\mathbb{R}^3$ and thus also $\mathbb{C}^3$. By a theorem of Chevalley, the space of orbits of this group action is again isomorphic to $\mathbb{C}^3$. Each point in the surface shown here corresponds to a ‘nongeneric’ orbit: an orbit with fewer than the maximal number of points. More precisely, the space of nongeneric orbits forms a complex surface in $\mathbb{C}^3$, called the discriminant, whose intersection with $\mathbb{R}^3$ is shown here.

Free Modular Lattice on 3 Generators

This is the free modular lattice on 3 generators, as drawn by Jesse McKeown. First discovered by Dedekind in 1900, this structure turns out to have an interesting connection to 8-dimensional Euclidean space.

Golay Code

The extended binary Golay code, or Golay code for short, is a way to encode 12 bits of data in a 24-bit word in such a way that any 3-bit error can be corrected, and any 7-bit error can at least be detected. The easiest way to understand this code uses the geometry of the dodecahedron, as shown in this image by Gerard Westendorp.

Heawood Graph

This is the Heawood graph. This graph can be drawn on a torus with no edges crossing in such a way that it divides the torus into 7 hexagons, each pair of which shares an edge. In 1890, Percy John Heawood proved that for any map drawn on a torus, it takes at most 7 colors to ensure that no two countries sharing a common boundary have the same color. The Heawood graph proves that the number 7 is optimal.

Icosidodecahedron from D6

The icosidodecahedron can be built by truncating either an icosahedron or a dodecahedron. It has 30 vertices. It is a beautiful, highly symmetrical shape. But it’s just a shadow of an even more symmetrical shape with twice as many vertices in twice as many dimensions!