https://arxiv.org/abs/1806.00079

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: xhttp://community.wolfram.com/groups/-/m/t/1350667

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: http://community.wolfram.com/groups/-/m/t/1338150

Correlations

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: http://community.wolfram.com/groups/-/m/t/1335544

In Balance

Mercator projection of concentric circles on a rotating sphere.

Source code and explanation: http://community.wolfram.com/groups/-/m/t/1333964

Map

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

Source code and explanation: http://community.wolfram.com/groups/-/m/t/1333054

Unfold

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

Source code: http://community.wolfram.com/groups/-/m/t/1328434

Follow Through

A morphing 11-pointed star.

Source code: http://community.wolfram.com/groups/-/m/t/1320708

Fall In

Stereographic projection of dots on the sphere.

Source code: http://community.wolfram.com/groups/-/m/t/1299870 https://mathstodon.xyz/media/Es-C9H3rK6pOov5yc5E

Twist

Just some spinning line segments centered at points on the circle.

Source code: http://community.wolfram.com/groups/-/m/t/1296052 https://mathstodon.xyz/media/ZC0QaoaI6n96HgdczyU

Loxo

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 http://community.wolfram.com/groups/-/m/t/1291902 https://mathstodon.xyz/media/OsWh_UoCkHrl6fm7nkg

Oof. Another one destroyed by over-aggressive video compression. See the source code link for a better version: http://community.wolfram.com/groups/-/m/t/1286395

Caught

Voronoi diagram of stereographic projection of some spherical spirals.

Source code and slightly more explanation: http://community.wolfram.com/groups/-/m/t/1286395 https://mathstodon.xyz/media/Emwp_r0RuMC22N-elhw

Platonic II

Stereographic projection of the 5 Platonic solids

Source code: http://community.wolfram.com/groups/-/m/t/1283065 https://mathstodon.xyz/media/6Bm5npaO1gN7gfPhMho https://mathstodon.xyz/media/lzl8tlf8nSZQbH7NYus

Platonic I

Stereographic projection of the 5 Platonic solids

Source code: http://community.wolfram.com/groups/-/m/t/1283065

https://mathstodon.xyz/media/7tTbZ2Ly4RoowHCAOtQ https://mathstodon.xyz/media/a-PkGx2K0vVAqVct7bg https://mathstodon.xyz/media/2U3Lcij6HJ1TMEHQeII

Our paper on the least symmetric triangle has been published by Geometriae Dedicata: http://rdcu.be/GIkL

Here’s another fun image from the paper, which comes up in the course of determining the least symmetric acute triangle: https://mathstodon.xyz/media/3p4i0S_rVw08Xt3K6is

Small Changes

Stereographic projection of a Hamiltonian cycle on the great rhombicosidodecahedron.

Source code and more explanation: http://community.wolfram.com/groups/-/m/t/1282077 https://mathstodon.xyz/media/uh-1e9hg3ZY7p5Fq4YY

Throw

A Hamiltonian cycle on the vertices of the 120-cell.

Source code and more explanation: http://community.wolfram.com/groups/-/m/t/1273027 https://mathstodon.xyz/media/Kzas7Me2PkF5Dztcl_M

How Does That Work?

Hamiltonian cycle on the 5-cell.

Source code: http://community.wolfram.com/groups/-/m/t/1269759 https://mathstodon.xyz/media/13SjLIpd8RXPw-6E03I

All Day

Stereographic projection of a Hamiltonian cycle on the 24-cell.

Source code: http://community.wolfram.com/groups/-/m/t/1265322 https://mathstodon.xyz/media/nTRJw3IYnt0CNbIeBsc