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

Rise

Conformal image of parallel lines.

Versions that haven't been crushed by video recompression: shonkwiler.org/image-feed/rise

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 $$z \mapsto \frac{-4i}{z}$$.

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/-

2500 Right Angles

Limited edition of 2.

🌲

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.

Cube Life

Stereographic projection of the vertices of a rotating cube.

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/-

Catalan

A pen plotter print showing orthogonal projections of the 13 Catalan solids.

Limited edition of 5.