agm-ized symmetric derivative

The symmetric derivative looks like there’s an average in it, so what about changing that average
with something else, like the arithmetic geometric mean:

\lim_{h\rightarrow 0} \frac{\mathrm{agm}(f(x+h),f(x-h))}{h}

new videos

First, I’ve rendered a 1024×1024 video of the Newton’s Method of the third Jacobi theta function ϑ3(z,q), for |q| = 0.90190165617299947 or thereabouts. This piece is entitled The Swift Luminescent Energy Drink of the Psyche, or When Goorialla Whirls and Whorls and Roars

Secondly, using GAP, I’ve made a video of all Cayley tables of finite groups of order 128:

And finally, A video showing Klein’s j-invariant j(\tau) under the transformation \tau\rightarrow -1/\tau in \mathbf{SL}(2,\mathbb{R}):