Rotation Redux

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

Source code and more explanation:

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:


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

Explanation and source code:


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:


\(z \mapsto \frac{4}{z}\) 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:

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.


Made for the local Society for Industrial and Applied Mathematics (SIAM) chapter.

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.


This is an animated version of a figure in my Bridges paper (, 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:

Off the End

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

Source code and further explanation:

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:

@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:

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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.