Ian Sammis
Creating mathematical art gives me a wonderful excuse to explore parts
of mathematics well outside my specialty. The effort to make a
mathematical idea aesthetically pleasing forces me to understand the
underlying mathematics differently and more completely than I
otherwise would.
The image of an endless braided cord can be represented as a
function $f:\mathbb{R}_2\to\mathbb{R}_3$ from a location to an RGB
value by finding the nearest point on each strand, shading each
point within some thickness $T$ of a strand with a brightness that
drops to zero at distance $T$ linearly, then writing rules to
handle overlaps. This gives us an endless braid that can easily be
composed with other maps. Composing with the conformal map
$g(z)=z^2/2$ creates four braids in the
$\frac{N\pi}{2}+\frac{\pi}{4}$ directions. Cropping this we arrive
at a unit cell that can create a diagonal braided net throughout
the plane. The displayed movie is that braided net mapped through
$\frac{1}{z^\alpha}$ for steadily increasnig values of $\alpha$.