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Sierpinski's Triangle Fractal Yellow-Red Flame.
A random chaos game!

Fractals produced by systems of affine transformations are indeed very awesome. John Hutchinson and Michael Barnsley called them "IFS fractals" back in the 1980s, IFS standing for "iterated function system". Not sure if that's still a commonly recognized term, but "chaos game" seems to have survived for the general method of tracking a point as it wanders through a constrained domain.

There doesn't seem to be much overlap between IFS fractals and cellular automata, though, beyond the fact that they can both produce things like Sierpinski triangles. In response to a post report I've moved this thread to the Sandbox.

Excel doesn't seem like a particularly efficient way to do IFS transformations or display the results. These days it seems like dedicated programs like IFStile or Larry Riddle's IFS Construction Kit might be more efficient exploration tools -- especially if you don't explain your Excel-based methods for generating that animated GIF.

dvgrn wrote: October 24th, 2026, 6:16 pm Fractals produced by systems of affine transformations are indeed very awesome. John Hutchinson and Michael Barnsley called them "IFS fractals" back in the 1980s, IFS standing for "iterated function system". Not sure if that's still a commonly recognized term, but "chaos game" seems to have survived for the general method of tracking a point as it wanders through a constrained domain.

There doesn't seem to be much overlap between IFS fractals and cellular automata, though, beyond the fact that they can both produce things like Sierpinski triangles. In response to a post report I've moved this thread to the Sandbox.

Excel doesn't seem like a particularly efficient way to do IFS transformations or display the results. These days it seems like dedicated programs like IFStile or Larry Riddle's IFS Construction Kit might be more efficient exploration tools -- especially if you don't explain your Excel-based methods for generating that animated GIF.

EDIT: It seems like my browser glitched out and ended up posting a verbatim quote of dvgrn's response instead of what I was actually trying to post. Let me fix that hold on.
Edit 2: What I was *trying* to say:

dvgrn wrote: October 24th, 2026, 6:16 pm There doesn't seem to be much overlap between IFS fractals and cellular automata, though, beyond the fact that they can both produce things like Sierpinski triangles. In response to a post report I've moved this thread to the Sandbox.

While I don't think IFSes are CA, I don't think they're completely different either.

dvgrn wrote: October 24th, 2026, 6:16 pm Fractals produced by systems of affine transformations are indeed very awesome. John Hutchinson and Michael Barnsley called them "IFS fractals" back in the 1980s, IFS standing for "iterated function system". Not sure if that's still a commonly recognized term, but "chaos game" seems to have survived for the general method of tracking a point as it wanders through a constrained domain.

There doesn't seem to be much overlap between IFS fractals and cellular automata, though, beyond the fact that they can both produce things like Sierpinski triangles. In response to a post report I've moved this thread to the Sandbox.

Excel doesn't seem like a particularly efficient way to do IFS transformations or display the results. These days it seems like dedicated programs like IFStile or Larry Riddle's IFS Construction Kit might be more efficient exploration tools -- especially if you don't explain your Excel-based methods for generating that animated GIF.

I'd like to clarify one point: is your objection specifically to the method I used to generate the Sierpinski triangle in Excel, or do you consider the Sierpinski triangle itself, regardless of how it is generated, unrelated to cellular automata?

I understand that IFS fractals and cellular automata are distinct computational approaches, even when they produce similar patterns. However, the same pattern can emerge from different underlying mechanisms. The fact that a Sierpinski triangle can be generated by an IFS does not, by itself, mean that every method of generating it must also be classified as an IFS process rather than a cellular automaton.

My approach is based on a row-by-row algebraic recurrence derived from the Khayyam–Pascal triangle, with each cell determined by values in the preceding row. Could you clarify which specific feature of this process, in your view, places it outside the scope of cellular automata?

There is one further point I'd like to raise. With all due respect, I believe your assessment of Excel's capabilities deserves a second look.
You describe Excel as a less efficient tool for exploring fractals and suggest that dedicated software would be more suitable. That may well be true for certain tasks, but I don't think such a broad judgment is justified without first examining the specific implementation and considering what Excel can actually do.

Excel is far more than a conventional spreadsheet. Its computational capabilities, formula system, conditional formatting, and animation possibilities can be combined in ways that may not be immediately apparent, especially when it is approached as a computational environment rather than merely a tool for tabulating data.

I'm not suggesting that Excel is the best tool for every task, or that specialized software has no advantages. My point is that a tool's limitations should be assessed on the basis of its demonstrated capabilities and the specific task at hand, rather than assumptions about what it can or cannot do.

I would therefore encourage you to reconsider your characterization of Excel, particularly before dismissing an implementation you have not examined in detail. A more informed assessment would be fairer both to the tool and to the work being presented.

dvgrn wrote: October 24th, 2026, 6:16 pm

?
I just wanted to follow up on my reply, as I'd genuinely be interested in hearing your thoughts on the points I raised.
I realize you may be busy, and I don't expect an immediate response. However, since the discussion touches on both the classification of cellular automata and the capabilities of Excel, I'd appreciate knowing your perspective when you have a chance.
Thanks!