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.

Involute of Cubical Parabola - Marshall Hampton

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.

Barth Sextic

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 $x^2 + y^2 = z^2$ .

Rectified Truncated Icosahedron

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.

Zamolodchikov Tetrahedron Equation

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.

Clebsch Surface

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.

27 Lines on a Cubic Surface

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.

Hoffman–Singleton Graph

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.

Cairo Tiling

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

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.

Visual Insight

Mathematics Made Visible

Discriminant of the Icosahedral Group

Involutes of a Cubical Parabola

Barth Sextic

Rectified Truncated Icosahedron

Zamolodchikov Tetrahedron Equation

Clebsch Surface

27 Lines on a Cubic Surface

Hoffman–Singleton Graph

Cairo Tiling

Free Modular Lattice on 3 Generators

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