The math here wanders into territory I can’t describe without putting us both to sleep, so I’m giving the words a rest. If you want me to take a swing at explaining it, you can always email me — but for now, the images get the spotlight.

For this part of the Vulcan series, I set a boundary for myself: one layer, flat colors, and a repeating pattern that moves like a quiet rhythm under the surface. Within that boundary I’m testing its strength — how far it can be stretched, where it warps, how it responds to pressure. These pieces are the early hammer‑strikes in Vulcan’s workshop, learning the metal before shaping it into something more elaborate.

Today’s image didn’t come from a formula in the way the recent pattern posts did. If there’s math inside it, it’s buried under many layers of decisions. The real work happened in the process — choosing a tool, applying it, seeing what changed, and deciding what to try next.

The program I wrote has a library of tools that each manipulate pixels in a specific way. Some distort, some blend, some shift values around. None of them create a finished image on their own. They’re more like brushes or knives in a physical studio: you pick one up, make a pass across the canvas, and see whether the piece moves in the right direction.

For this artwork, I started with a simple base and then applied tools one at a time. Each pass altered the image in a way I couldn’t fully predict, so the piece grew through iteration — adjust, evaluate, repeat. The math behind the tools is real, but it isn’t the point here. What matters is the sequence of choices that shaped the final result.

Today’s work is a variation of Vulcan #1. I started from the same base, tilted the background away from the viewer, and added a fade‑to‑black. The steps I used to build that first piece — plus the extra passes that led to this one — are all reflected in the process I described above.

It’s worth saying, quietly, that this isn’t AI‑generated. The computer executes the transformations, but the decisions — which tool to use, when to stop, what “finished” looks like — are mine. The creative work lives in that sequence of choices, not in the machinery underneath.

This one has the four-way symmetry, but without the center star.

Nerd talk: (Non-nerds please skip.) The star is still there, it is the little diagonal thorn near the center. First I do the rounded saw Vulcan #21 with a longer, 10x, period. This "copies" the center across the whole plane. Then $z = z^2 / \lvert z \rvert$ creates the four-way (2 axis+2 diagonal) symmetry, with many 'centers'.

Math for the curious, not required to enjoy the post.

I use a rounded version of the saw function. $$ a = \lvert \sin(x) \rvert\\ b = \lvert \sin(y) \rvert\\ pixel.value = a + b $$

The next set of images introduces some symmetry around the diagonal axes in addition to the orthogonal axes.

The pattern in the earlier images could be repeated forever. A vast plane in which wherever you are standing, it looks exactly the same as anywhere else. These new ones, add a unique point at the center. There is an eight pointed star at the center. Other than some small distortion, everywhere else looks like everywhere else.

I am going to dial back on the math. But if you are curious, the additional symmetry is the result of $$z = z^2 /\lvert z \rvert$$ The squaring maps the diagonal axes to the vertical axis. Dividing by the absolute value keeps things the same size.

I am running out of things to say. That is normal, 50% of my posts are only pictures. I am falling back into that mode now.

For those who are interested, and for images where the math is visible, I will include some brief code/formula math snippets.

$pixel.value = \sqrt{a}+\sqrt{b}$

The rest of the setup is similar to the previous posts.

Why is this series called Vulcan? See Wikipedia and Altar Gods. He is often depicted with a hammer and a forge. Unlike his lazier counterparts who simply conjured things into existence, Vulcan actually built things the hard way. I’ve already set a precedent for mixing mythologies, and I chose Vulcan because it’s easier to read and say than Hephaestus.

This series is about building images from basic patterns — and also about building and using my new art program and the new website. I’m not sure where this series is headed, only that wherever it ends up, it will get there through construction and craft, not by conjuring a finished product.

Today’s piece is a picture of the frac/saw mix with distance coloring.

var z = pixel.point;
var a = saw(z.Real);
var b = frac(z.Imaginary);
pixel.value = Math.Sqrt(aa + bb);