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.

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.

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.

Pathways

20 random points on the sphere, rotating around 2 random axes. Inspired by work of Caleb Ogg: instagram.com/p/BxXPRSmnc5S/

Five Easy Pieces

Rotating truncation of the tetrahedron.

Master Control Program

A minimal $$7_4$$ knot on the simple cubic lattice

Home on the Range

Produced from a minimal $$7_7$$ knot on the BCC lattice.

BCC

A minimal figure-eight knot on the body-centered cubic lattice (coordinates taken from Andrew Rechnitzer’s web page: math.ubc.ca/~andrewr/knots/min)

Symmetric Minimality

A symmetric minimal lattice trefoil knot, inspired by a conversation here with @11011110, who asked about minimal lattice trefoils with more symmetry and ended up finding the coordinates of this one. A Mastodon instance for maths people. The kind of people who make $$\pi z^2 \times a$$ jokes. Use $$ and $$ for inline LaTeX, and $ and $ for display mode.