New paper: “Distributions of distances and volumes of balls in homogeneous lens spaces” with Chris Peterson and our student Brenden Balch

arxiv.org/abs/2004.13196

Vanishing Point

Conformal image of the square grid in the infinite strip \(\{z \in \mathbb{C} : 1 \leq \operatorname{Re}(z) \leq 2\}\) under the map \(z \mapsto \frac{-4i}{z}\).

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

// //

Nines

Minimal-stick examples of the knots \(9_{35}\), \(9_{39}\), \(9_{43}\), \(9_{45}\), and \(9_{48}\).

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

Square Grid

Schwarz–Christoffel mapping from the square grid to the unit disk.

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

“Gaussian random embedding of multigraphs”, by Jason Cantarella, Tetsuo Deguchi, Clayton Shonkwiler, and Erica Uehara: arxiv.org/abs/2001.11709

I’m very proud of this one: we give a principled mathematical model of topological polymers of arbitrary complexity which both recovers the standard physics model and is simple to both compute and prove theorems with.

Rotation Redux

A grid of circles, inverted in the unit circle, then mapped by the inverse Cayley transform.

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

Fourth Power

A square grid in the first quadrant under the map \(z \mapsto z^t\) as \(t\) ranges from \(1\) to \(4\), reflected across the line \(y=-x\) (plus some other stuff).

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

🌲

A Schwarz–Christoffel mapping of the upper half-plane to an equilateral triangle.

Explanation and source code: community.wolfram.com/groups/-

🌐

Level sets of the real and imaginary parts of \(\frac{1}{\pi i} \log z\), mapped from the upper half plane to the unit disk by the Cayley transform.

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

👀

\(z \mapsto \frac{4}{z}\) applied to an infinite stack of circles between the lines \(\{z:\operatorname{Re}(z)=1\}\) and \(\{z:\operatorname{Re}(z)=2\}\)

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

arxiv.org/abs/1909.06947

In which we show that the knots K13n592 and K15n41127 (pictured) both have stick number 10. These are the first non-torus knots with more than 9 crossings for which the exact stick number is known.

SIAM

Made for the local Society for Industrial and Applied Mathematics (SIAM) chapter.

arxiv.org/abs/1909.00917

In which my student Tom Eddy and I give improved bounds on stick number of more than 40% of the knots up to 10 crossings for which it was not previously known.

Here's a 10-stick 10_16.

Framing

This is an animated version of a figure in my Bridges paper (archive.bridgesmathart.org/201), showing a geodesic in the Grassmannian \(G_2(\mathbb{C}^n)\) between a \(+3\)-framed regular 200-gon and a 0-framed regular 200-gon.

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

Off the End

Each row shows the stereographic projection of a rotating regular polygon to the line.

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

School’s Out

100 random points in the first orthant on the unit sphere, undergoing two simultaneous random rotations and being stereographically projected to the plane.

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

Show more
Mathstodon

The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon!