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

Master Control Program

A minimal \(7_4\) knot on the simple cubic lattice

Source code: https://community.wolfram.com/groups/-/m/t/1644214

Home on the Range

Produced from a minimal \(7_7\) knot on the BCC lattice.

Source code: https://community.wolfram.com/groups/-/m/t/1641626

BCC

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

Source code: https://community.wolfram.com/groups/-/m/t/1640932

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.

Source code: https://community.wolfram.com/groups/-/m/t/1639178

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.

Buy it here: https://shonkwiler.org/store

@11011110 Yeah, that does seem like the right question, doesn’t it?

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.

Buy it: https://shonkwiler.org/store

@11011110 Nice! That totally works! It’s in Diao’s class (c), but surprisingly it’s not the one he shows. It also looks to me like Scharein et al.’s 2–17 (https://doi.org/10.1088/1751-8113/42/47/475006), but they don’t remark on it either.

I wonder whether one should expect highly symmetric minimal lattice knots in general; my guess is probably not, but all the evidence seems to suggest that there are lots of minimal examples for each knot type.

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Math and art

Joined May 2017