# {5,3,5} Honeycomb

This is the {5,3,5} honeycomb, drawn by Jos Leys. A 3-dimensional honeycomb is a way of filling 3d space with polyhedra. It’s the 3-dimensional analogue of a tiling of the plane. Besides honeycombs in Euclidean space, we can also have honeycombs in hyperbolic space, which is a 3-dimensional Riemannian manifold with constant negative curvature. The {5,3,5} honeycomb lives in hyperbolic space.

# Weierstrass Elliptic Function

The Weierstrass elliptic function is built up as a sum of terms, one for each point in a lattice in the complex plane. Each term has a pole at one lattice point. The picture here shows the very first term, namely $1/z^2$. That’s why it’s bright in the middle and the colors go twice around the color wheel as you go around. If you continue reading, you’ll see a movie made by David Chudzicki where further terms are added one at a time!

# Pentagon-Hexagon-Decagon Identity

Suppose we inscribe a regular pentagon, a regular decagon, and a regular hexagon in circles of the same radius. If we denote the respective edge lengths of these polygons by $P$, $D$ and $H$, then these lengths obey $P^2=D^2+H^2$. So, the edges of a pentagon, decagon and hexagon of identical radii can fit together to form a right triangle! Recently Greg Egan gave a nice proof using the icosahedron.

# Truncated Hypercube

This is a truncated 4-dimensional cube, drawn in a curved style by Jos Leys. You can take an ordinary 3-dimensional cube, cut off its corners and get a truncated cube. Similarly, you can take a 4-dimensional cube, cut off its corners, and get a 4-dimensional uniform polytope with $2 \times 4 = 8$ truncated cubes as faces and $2^4 = 16$ tetrahedral faces! It’s called the truncated 4-cube.

# Deltoid Rolling Inside Astroid

A deltoid is a curve formed by rolling a circle inside a circle whose radius is 3 times larger. Similarly, an astroid is a curve formed by rolling a circle inside a circle whose radius is 4 times larger. The picture here, drawn by Greg Egan, shows a deltoid rolling inside an astroid. It fits in a perfectly snug way!

# Astroid as Catacaustic of Deltoid

This image, drawn by Xah Lee, shows a deltoid and its catacaustic. The deltoid is the curve traced by a point on the perimeter of a circle that is rolling inside a fixed circle whose radius is three times as big. It’s called a deltoid because it looks a bit like the Greek letter delta: $\Delta$. The catacaustic of a curve in the plane is the envelope of rays emitted from some source and reflected off that curve.

# Enneper Surface

This is the Enneper surface, as drawn by Greg Egan using Mathematica. It’s a minimal surface, meaning one that necessarily gets more area if you warp any small patch of it. A soap film will make a minimal surface if it doesn’t enclose any air. But the Enneper surface intersects itself: it’s immersed in 3d space, but not embedded. So, you can’t make it with soapy water!

# Atomic Singular Inner Function

This picture, drawn by Elias Wegert, uses colors to show the phase $f/|f|$ of the complex function

$$f(z) = \prod_{k=1}^5 \exp\left(\frac{z+\omega^k}{z-\omega^k}\right)$$

where $\omega$ is a nontrivial fifth root of unity. This is an ‘atomic singular inner function’. To understand what that means, it helps to start with some complex analysis.

# Catacaustic of a Cardioid

This image, drawn by Greg Egan, shows a cardioid and its catacaustic. The cardioid is a heart-shaped curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. The catacaustic of a curve in the plane is the envelope of rays emitted from some…

# {6,3,3} Honeycomb in Upper Half Space

A 3-dimensional honeycomb is a way of filling 3d space with polyhedra. It’s the 3-dimensional analogue of a tiiling of the plane. However, not only can we have honeycombs in Euclidean space, we can also have them in hyperbolic space. The {6,3,3} honeycomb lives in hyperbolic space.