# Involutes of a Cubical Parabola

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.

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…

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.

This picture shows part of a graph called the Balaban 11-cage. A **(3,11)-graph** is a simple graph where every vertex has 3 neighbors and the shortest cycle has length 11. A **(3,11)-cage** is a (3,11)-graph with the least possible number of vertices. The Balaban 11-cage is the unique (3,11)-cage.

This is the Harries graph. It is a graph with the minimum number of vertices such that each vertex is connected to 3 others and every cycle has length at least 10. Such graphs are called **(3,10)-cages**.

This is the Balaban 10-cage, the first known (3,10)-cage. An **\((r,g)\)-cage** is graph where every vertex has \(r\) neighbors, the shortest cycle has length at least \(g\), and the number of vertices is maximal given these constraints.