I've been running a small ML agency for ages, since before it was cool, and recently I got the urge to refresh the brand. Somewhere in that itch I decided the logo shouldn't be a saved file at all. It should compute itself in the browser, redrawn every frame from a formula. The company is called Epoch8, so the math had to earn the name.
The name gives three hooks: the digit 8, the letter E, and epoch, which for an ML shop is a real word (one full pass over the training data). I wanted the mark to lean on all three.
The 8 was the easy half. A Bernoulli lemniscate (r² = a²·cos 2θ) is the sideways infinity sign; stand it upright and it reads as a clean figure eight. Infinity as a cycle, an epoch as a cycle. It also survives down to favicon size, which a fussier curve won't.
The E is where it got fun. No tidy formula draws a letter, so I went with a Fourier series: any closed outline can be redrawn by a stack of spinning circles, a big one for the gross shape and smaller ones adding detail. I sampled the E at 1024 points and ran a DFT. It hands the circles back in order of importance: the first few set the rough shape, and each extra one only adds finer detail. So if you draw just a handful you get a blobby E, and the more circles you add the more it sharpens into a clean letter. That sharpening is the convergence. Each circle is one harmonic, so I counted them off in eights and called every eight an epoch: the first 8 harmonics are epoch one (rough), 16 is epoch two (sharper), and the E tightens up epoch by epoch, the way accuracy climbs over training runs. That's the bit I got a kick out of.
The nicest accident is Gibbs. A partial Fourier sum always overshoots at a sharp corner, and the overshoot never shrinks to zero, it just squeezes tighter against the corner (holds around 9% of the jump). That behaves just like a training loss that keeps dropping but never quite hits zero. So I left the ripple in. The convergence "stalls" the way real training does.
One more piece: the name beside the mark needed a typeface, and since we are a shop full of programmers, a monospace font was the obvious call, the fixed-width kind you stare at all day in a code editor. I tried a handful and settled on JetBrains Mono, and the detail that sold me was its 8. Most fonts build an 8 from two stacked ovals that meet in a soft pinch. JetBrains Mono ExtraBold crosses it instead: the waist narrows to a sharp little beak, the same beak a lemniscate makes where it crosses itself in the middle. Lay the typed 8 over the math one and the two crossings land right on top of each other. So the wordmark's 8 rhymes with the logo's 8.
The convergence ladder earned its keep off the logo too.
When we scope a build in a proposal, it usually comes in stages. The first is a bare proof that the thing works; the last is a system that runs on its own.
A staged build is easy to describe badly: the client hears five phases and cannot tell well what changes between them. So we illustrate the stages with the mark itself, drawing each one as the E at a higher harmonic count. The early stage is the rough, low-harmonic E, blobby but already an E. The final stage is the crisp one. The client reads maturity as convergence: a letter you can recognize early, sharpening as it earns more of the work.
The ladder is also an argument. Each stage is useful on its own, so the client can stop after any one and keep what they have. That mirrors the math: a partial Fourier sum is already a readable E long before it is exact.
This tiny project is easily the most fun I have had with a logo :)
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