This image, which I made a few years ago, is based on a tiling of the hyperbolic plane with triangles whose angles are π/2, π/3, π/7. Other than the 7, which can be changed to any higher integer, I couldn’t vary these numbers without ruining the effect. Recently I thought of a simpler, and thus more general, way to generate the ribbons.

( . . more . . )

I put a series of tilings of the hyperbolic plane on Wikimedia Commons.

I’ve long had an idea to design “outline” typefaces which, at appropriate low resolution, would mimic certain bitmap fonts that have sentimental resonance.

The orange discs are the original dots, of course. The blue arcs are least-squares fits (linear, quadratic) to subranges of the dots. The arcs are blended with a weighting function that favors longer arcs, as well as the middle of each arc. Finally, the stroke is thickened by adding ±i/2 to the parametric variable.

This is the first version in which the stroke-ends are neither brutally stiff nor (in some cases) grotesquely exuberant. I don’t know yet whether the lumpiness, here and there, reflects a flaw an opportunity to improve the blending function or a limitation of the cubic splines used to simplify the final curve.

(I previously made a TrueType version of Apple’s “Los Angeles” font, by a much more ad hoc approach.)

For my own future reference, in case I lose the bit of paper on which I jotted it.

In a function of period 2π, a unit step discontinuity in the nth derivative at phase α contributes this to the Fourier series:

i (iⁿ e^-ik(t-α) – i^-n e^ik(t-α)) / (2πkⁿ⁺¹)

I haven’t the skill to prove this for general n, but then I’m unlikely to need it for n>2.

A few people will recognize immediately how and why I did this.

My strip representation of the hyperbolic plane inspired Vladimir Bulatov to explore weirder conformal mappings thereof. (Conformal means angles are preserved.)