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.

@christianp Reminds me of this sign I saw in Cambridge several years ago.

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/

@christianp Not sure funding bodies would go for it, but an interesting hybrid would be for people to write, say, $500 or$1000 into their grants to go to arXiv.

@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: en.wikipedia.org/wiki/Smoothst

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.

@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.