# Sphere in Mirrored Spheroid

A **spheroid** is an ellipsoid with an axis of rotational symmetry. This image created by Robert Figura shows a sphere inside a mirrored spheroid, reflected almost endlessly.

A **spheroid** is an ellipsoid with an axis of rotational symmetry. This image created by Robert Figura shows a sphere inside a mirrored spheroid, reflected almost endlessly.

A Schwarz triangle is a spherical triangle that can be used to generate a tiling of a branched covering of the sphere by repeatedly reflecting this triangle across its edges. Sometimes we get an actual tiling of the sphere, but in general we get a branched covering, because the same point can lie in the…

This is the **small cubicuboctahedron**, as drawn by Robert Webb’s Great Stella software. It looks simple enough, but it conceals some interesting mathematics.

This picture drawn by Katherine Stange shows what happens when we apply fractional linear transformations $ z \mapsto \frac{a z + b}{c z + d} $ to the real line sitting in the complex plane, where $a,b,c,d$ are **Eisenstein integers** that is, complex numbers of the form $ m + n \exp(2 \pi i/3) $ with $m,n$ being integers.

Two regular pentagons and a regular decagon fit snugly at a point: their interior angles sum to 360°. Despite this, you cannot tile the plane with regular pentagons and decagons. However, there is a branched covering of the plane tiled with pentagons and decagons, which map to regular pentagons and decagons on the plane. Here Greg Egan has drawn a portion of this branched covering.

Two regular pentagons and a regular decagon meet snugly at a vertex: their interior angles sum to 360°. However, they can’t tile the plane. However, they come fairly close, as shown in this picture by Greg Egan.

The **moving sofa problem** asks: what is the shape of largest area that can be maneuvered through an L-shaped hallway of width 1? This animated image made by Claudio Rocchini shows one attempt to solve this problem.

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!

This picture by Toby Hudson shows the densest known packing of the regular pentagon.

This picture by Toby Hudson shows the densest known packing of the regular heptagon. Of all convex shapes, the regular heptagon is believed to have the lowest maximal packing density.

Which shape is worst of all for packing the plane? That is, which has the lowest maximal packing density? Suppose we demand that our shape be convex and also centrally symmetric: that is, a subset $S \subseteq \mathbb{R}^2$ such that $x \in S$ implies $-x \in S$. Then a certain ‘smoothed octagon’ is conjectured to be the worst. Amazingly, this shape has a 1-parameter family of maximally dense packings, shown in this image created by Greg Egan.

This is the densest packing of regular octagons in the plane, drawn by Graeme McRae. It is interesting because it is a counterexample to the 2-dimensional analogue of a conjecture made in 3 dimensions by Stanislaw Ulam.

This image created by Christopher Culter shows the compact abelian group of 2-adic integers (black points), with selected elements labeled by the corresponding character on the Pontryagin dual group (colored discs).

This is the **Prüfer $2$-group**, the subgroup of the unit complex numbers consisting of all $2^n$th roots of unity. It is also called $\mathbb{Z}(2^\infty)$.

The **{3,3,7} honeycomb** is a honeycomb in 3d hyperbolic space. It is the dual of the {7,3,3} honeycomb shown last time. This image, drawn by Roice Nelson, shows the ‘boundary’ of the {3,3,7} honeycomb: that is, the set of points on the ‘plane at infinity’ that are limits of points in the {3,3,7} honeycomb.