# Menger Sponge

Take a cube. Chop it into 3×3×3 = 27 small cubes. Poke holes through it, removing 7 of these small cubes. Repeat this process for each remaining small cube. Do this forever! The result is the Menger sponge.

# 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.

# Algebraic Numbers

This is a picture of the algebraic numbers in the complex plane, made by David Moore based on earlier work by Stephen J. Brooks. Algebraic numbers are roots of polynomials with integer coefficients. In this picture the color indicates the degree of the polynomial.

# Tübingen Tiling

A systematic way to generate quasiperiodic tilings of the plane is to take a lattice in higher dimensions and slice it at a funny angle. Greg Egan has created an applet that generates quasiperiodic tilings by projecting selected triangles from an $n$-dimensional lattice called the $\mathrm{A}_n$ lattice onto a plane. This particular picture comes from the $\mathrm{A}_4$ lattice.