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Clayton Shonkwiler

Open and closed random walks with fixed edgelengths
by Jason Cantarella, Kyle Chapman, Philipp Reiter, and Clayton Shonkwiler

In this paper, we show that a random walk is (with overwhelming probability) surprisingly close to a closed loop with the same step sizes, and that closing up has very little impact on local features.

In particular, this suggests that local knots should occur at essentially the same rate in loops as in open chains.

Omnes Pro Uno

Yet more Mercator projections of level sets of sums of dot products, now with the vertices of a triangular bipyramid.

Source code and explanation: x

Power Surge

More Mercator projections of level sets of sums of dot products, this time with the vertices of a regular tetrahedron.

Source code and explanation:


Mercator projection of the contour plot of the sum of the absolute values of the dot products of a point on the sphere with the vertices of a rotating octahedron.

Source code and explanation:

In Balance

Mercator projection of concentric circles on a rotating sphere.

Source code and explanation:


Mercator projection of rotating great circles on the sphere. Inspired by what @roice3 is doing with TilingBot on Twitter.

Source code and explanation:


Linearly interpolating (in space, though not in time) between a double-covered octagon and a regular 16-gon

Source code:


Start with a bunch of points on loxodromes. Now rotate in space, then stereographically project to the plane. Finally, form the Voronoi cells of the resulting point set.

The aggressive video compression does this one no favors; see a better version along with source code at

Oof. Another one destroyed by over-aggressive video compression. See the source code link for a better version:


Voronoi diagram of stereographic projection of some spherical spirals.

Source code and slightly more explanation:

Our paper on the least symmetric triangle has been published by Geometriae Dedicata:

Here’s another fun image from the paper, which comes up in the course of determining the least symmetric acute triangle:

Small Changes

Stereographic projection of a Hamiltonian cycle on the great rhombicosidodecahedron.

Source code and more explanation: