Icosa

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.

Conformal

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.

Minimal

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

Source code and further explanation: community.wolfram.com/groups/-

Because It’s There

Parametric plot of the surface $(x, y, (1-|x|-y^2)^3)$.

Anne Harding and I made a hand-drawn, hand-cranked version of Truncation (shonk.tumblr.com/post/12997043), showing cross sections of a hypercube.

Check it out in person at the Curfman Gallery (lsc.colostate.edu/campus-activ).

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: doi.org/10.1080/00029890.2019.

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: community.wolfram.com/groups/-

Fitting In

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

Source code and explanation: community.wolfram.com/groups/-

Dance

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

Source code and explanation: community.wolfram.com/groups/-

Bounce

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

Source code and explanation: community.wolfram.com/groups/-

Rolling

Two circles rolling inside a circle.

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: community.wolfram.com/groups/-

J34

Hopf map image of a rotating 600-cell.

Source code and explanation: community.wolfram.com/groups/-

This is Only a Test

Decagons formed from stereographic projection of points on concentric spherical circles.

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: community.wolfram.com/groups/-

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: community.wolfram.com/groups/-

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