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.

Buy one here: 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.

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.

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: shonkwiler.org/store

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.

Minimal

The shortest-possible trefoil knot on the simple cubic lattice.

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

Because It’s There

Parametric plot of the surface $(x, y, (1-|x|-y^2)^3)$.

Anne Harding and I made a hand-drawn, hand-cranked version of Truncation (shonk.tumblr.com/post/12997043), showing cross sections of a hypercube.

Check it out in person at the Curfman Gallery (lsc.colostate.edu/campus-activ).

Our paper "Random triangles and polygons in the plane"] – in which we give a novel answer to Lewis Carroll's question "What is the probability a random triangle is obtuse?" – was published recently in the American Mathematical Monthly: doi.org/10.1080/00029890.2019.

Here's an animated version of Figure 2 from the paper, showing a geodesic in triangle space. The geodesic starts at the equilateral triangle shown, and the three curved paths show the tracks of the three vertices.

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