Symmetric Minimality

A symmetric minimal lattice trefoil knot, inspired by a conversation here with @11011110, who asked about minimal lattice trefoils with more symmetry and ended up finding the coordinates of this one.

Source code:


Start with the vertices of the regular icosahedron on the unit sphere. Make circles on the sphere consisting of those points whose spherical distance to one of the vertices is no bigger than \(\frac{\operatorname{arcsec} \sqrt{5}}{3}\). Stereographically project to the plane and this is the result.

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The Riemann mapping theorem guarantees a conformal map from the unit disk to any simply-connected planar domain; this shows an example of such a mapping to the equilateral triangle.

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The shortest-possible trefoil knot on the simple cubic lattice.

Source code and further explanation:

Anne Harding and I made a hand-drawn, hand-cranked version of Truncation (, showing cross sections of a hypercube.

Check it out in person at the Curfman Gallery (

Our paper "Random triangles and polygons in the plane"] – in which we give a novel answer to Lewis Carroll's question "What is the probability a random triangle is obtuse?" – was published recently in the American Mathematical Monthly:

Here's an animated version of Figure 2 from the paper, showing a geodesic in triangle space. The geodesic starts at the equilateral triangle shown, and the three curved paths show the tracks of the three vertices.

Dropped Call

Density histogram of Dirichlet-distributed barycentric coordinates on the square.

Source code and explanation:

Fitting In

Stereographic projection of putatively optimal packing of 124 points on the unit sphere.

Source code and explanation:


Chladni figures for linear combinations of the (20,21) and (21,20) vibration modes of the square.

Source code and explanation:


Combination of the (1,3) and (3,1) vibration modes of a square membrane.

Source code and explanation:

Double Projection

Vertices of a rotating 16-cell, projected the 2-sphere by the Hopf map, then stereographically projected to the plane. Also, a still image with frames composited together.

Source code and explanation:

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A Mastodon instance for maths people. The kind of people who make \(\pi z^2 \times a\) jokes.

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