# Small Stellated Dodecahedron

The **small stellated dodecahedron**, drawn here using Robert Webb’s Stella software, is made of 12 **pentagrams**, or 5-pointed stars, with 5 pentagrams meeting at each vertex.

The **small stellated dodecahedron**, drawn here using Robert Webb’s Stella software, is made of 12 **pentagrams**, or 5-pointed stars, with 5 pentagrams meeting at each vertex.

This image by Greg Egan shows the set of points \((a,b,c)\) for which the quintic \(x^5 + ax^4 + bx^2 + c \) has repeated roots. The plane \(c = 0 \) has been removed. This surface is connected to involutes of a cubical parabola and the 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.

This animation by Marshall Hampton shows the involutes of the curve \(y = x^3\). It lies at a fascinating mathematical crossroads, which we shall explore in a series of three posts.

A **sextic surface** is one defined by a polynomial equation of degree 6. The **Barth sextic**, drawn above by Craig Kaplan, is the sextic surface with the maximum possible number of **ordinary double points**: that is, points where it looks like the origin of the cone in 3-dimensional space defined by \(x^2 + y^2 = z^2\).

The **rectified truncated icosahedron** is a surprising new polyhedron discovered by Craig S. Kaplan. It has a total of 60 triangles, 12 pentagons and 20 hexagons as faces.

The Zamolodchikov tetrahedron equation, illustrated above by J. Scott Carter and Masahico Saito, is a fundamental law governing surfaces embedded in 4-dimensional space. It also arises purely algebraically in the theory of braided monoidal 2-categories.

This is an image of the Clebsch surface, created by Greg Egan. The Clebsch surface owes its fame to the fact that while all smooth cubic surfaces defined over the complex numbers contain 27 lines, for this particular example all the lines are real, and thus visible to the eye. However, it has other nice properties as well.

This animation by Greg Egan shows 27 lines on a surface defined by cubic equations: the Clebsch surface. It illustrates a remarkable fact: any smooth cubic surface contains 27 lines.

This is the Hoffman–Singleton graph, a remarkably symmetrical graph with 50 vertices and 175 edges. There is a beautiful way to construct the Hoffman–Singleton graph by connecting 5 pentagons to 5 pentagrams.

The Cairo tiling is a tiling of the plane by non-regular pentagons which is dual to the snub square tiling.

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.

This image by intocontinuum show how you can take 8 perfectly rigid regular tetrahedra and connect them along their edges to form a ring that you can turn inside-out.

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.

A little-known result in Newton’s *Principia Mathematica* is that by adding an inverse cube force to the usual force of gravity, you could change the angular motion of a planet while leaving its radial motion unchanged. This can have strange effects, shown in the animation here.