Outstanding work and a delightful read. Show 2 replies

Fun post! I drew the first 5 iterations by hand myself and I'm finding it easiest to think of as a self-similar coloring of a square tesselation.

If you start with the shape of iteration 3, it tessellates as a 5x5 square tile. Make an infinite grid of those tile shapes with one iteration 3 version in the center. Treat that center tile as the center square in the iteration 3 pattern and color the tiles around it according to how the 2nd and 3rd iterations were built of squares. This gives you the 4th and 5th iteration and you can continue to iterate on the coloring outwards to color the grid of tiles in the wallflower pattern.

Amazing insightful and thoughtful write up, thank you!

Loved the 3d visualizations

It reminds me of this thing I built some time ago while playing with recursive decimation to generate effects similar to fractals from any image

You can play with it here: https://jsfiddle.net/nicobrenner/a1t869qf/

Just press Blursort 2x2 a couple of times to generate a few frames and then click Animate

You can also copy/paste images into it

There’s no backend, it all just runs on the browser

Don’t recommend it on mobile

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Thought I'd check the arithmetic for 2 two-digit numbers, and it works!

I expect 41+14 to be 12 (two right plus two up equals two right and two up).

Long addition in long form below uses:

'=' to show equivalent lines (reordering of terms (1+2=2+1), spliting numbers (41=40+1), adding single digits (1+4=22))

'->' for when the algorithm gives a digit

'<' for when we move over a column

    41+14
    = (40+1)+(10+4)
    = 40 + 10 + (1+4)
    = 40 + 10 + 22
    -> 1s digit = 2
    < 4 + 1 + 2
    = 22 + 2
    = 20 + 2 + 2
    = 20 + 41
    -> 10s digit = 1
    < 2 + 4
    = 0
    -> done
    == 12

[edit] Just noticed the article has two different numbering systems, one where 10, 20, 30, 40 are clockwise and one where they are anticlockwise. In both, 1, 2, 3, 4 are clockwise. My addition is on the second, where 10s are anticlockwise (this is what is used in the addition table).

It still works in the alternative system (14+21 should equal 12)

    14+21
    =10+20+42
    ->2
    <1+2+4
    =13+4
    =10+3+4
    =10+31
    ->1
    <1+3
    =0
    ==12

Holy cow, I was expecting a quick read. Wound up having to skim some, as I need to get some work today. Will be coming back to this to play with some. Really well done!

This went much deeper and harder than expected. One has to admire the dedication.

Question to the author: what would you recommend to hang on my kid’s wall today?

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Got a bit nerd-sniped by this and came up with an L-system that fills out (I think) "the wallflower":

https://onlinetools.com/math/l-system-generator?draw=AB&skip...

edit: On second thought, this probably generates the other fractal, but I'm not sure.

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This is so much better than reading the news.

Favorited—I'll be coming back to absorb more, as my aging semi-fluency in engineering physics and SQL doesn't help much with the notation I last saw in the 1980s.

Well written! Would you mind sharing how you came up with the "middle out" numbering system? I can never seem to come up with something this inspired when I'm doing math problems by myself. Show 1 reply

> Deciding to delegate to a future version of me that knows more math

Relatable. Huge part of my decision on what degree to pursue was a list of problems (mostly linear algebra) I needed to solve, but didn't have the guidance (and internet connection) to.

Nice writeup! I was hoping to see a photo of the fractal on your wall.. Nice link to Knuth video that I somehow have missed. Show 1 reply

I wonder if something similar can be applied to get a dither pattern with built-in level of detail adjustment.

This is beautiful. Thank you.

well, that escalated beautifully

Nice writeup. The Heighway dragon of Jurassic Park fame is pretty neat too.

https://en.m.wikipedia.org/wiki/Dragon_curve

Kinda looks like a propeller Show 1 reply

I had this one up the wall (giant print) at a place I worked:

https://raw.githubusercontent.com/cies/haskell-fractal/refs/... [17MB, sorry Github]

It contains the Haskell code that produced it: https://github.com/cies/haskell-fractal

Especially the `sharpen` function was interesting to come up with (I used some now-offline tool to do curve fitting for me): https://github.com/cies/haskell-fractal/blob/master/fractal....

Fun little project. :)

That was fun.

Now make a tiling game engine that uses these!

Too much math.