Rotation Redux

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

Source code and more explanation: https://community.wolfram.com/groups/-/m/t/1857532

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: https://community.wolfram.com/groups/-/m/t/1850408

🌲

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

Explanation and source code: https://community.wolfram.com/groups/-/m/t/1844404

🌐

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: https://community.wolfram.com/groups/-/m/t/1842947

👀

\(z \mapsto \frac{4}{z}\) 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: https://community.wolfram.com/groups/-/m/t/1838263

https://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.

https://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.

Source code: https://community.wolfram.com/groups/-/m/t/1752892

Framing

This is an animated version of a figure in my Bridges paper (http://archive.bridgesmathart.org/2019/bridges2019-187.html), 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: https://community.wolfram.com/groups/-/m/t/1720829

Off the End

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

Source code and further explanation: https://community.wolfram.com/groups/-/m/t/1710318

Catalan

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

Limited edition of 5.

Buy one here: https://shonkwiler.org/store/catalan

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: https://community.wolfram.com/groups/-/m/t/1696628

Pathways

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

Source code: https://community.wolfram.com/groups/-/m/t/1686259

@jsiehler Haversine is actually surprisingly useful if you want to smoothly interpolate between \(f(0)=0, f'(0)=0\) and \(f(a)=1,f'(a)=0\).

I also sometimes use the polynomial smoothstep/smootherstep/smootheststep functions: https://en.wikipedia.org/wiki/Smoothstep

Five Easy Pieces

Rotating truncation of the tetrahedron.

Source code: https://community.wolfram.com/groups/-/m/t/1660413

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Math and art

Joined May 2017