That fractal that's been up on my wall for years(chriskw.xyz)
chriskw.xyz
That fractal that's been up on my wall for years
https://chriskw.xyz/2025/05/21/Fractal/
45 comments
Outstanding work and a delightful read.
Thanks Cliff, it means a ton coming from you! The videos from you and all the other folks on Numberphile always inspired me to see the beauty in math growing up :)
Please you two, make an awesome YouTube vídeo out of this. It is fascinating and beatiful and deserves a chance to viralize a little :)
Wow
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
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
Curious if it would work in 3D
Very interesting! I wonder what that would look like
Right now, roughly, the algorithm recursively divides the image by doing decimation (ie. picking every other pixel), and keeps the decimated pixels as a second image
Not sure how that algorithm would apply to a 3d data structure
Do you know how 3d objects/images are usually represented?
It would be cool to recursively decompose a 3d object into smaller versions of itself :)
Right now, roughly, the algorithm recursively divides the image by doing decimation (ie. picking every other pixel), and keeps the decimated pixels as a second image
Not sure how that algorithm would apply to a 3d data structure
Do you know how 3d objects/images are usually represented?
It would be cool to recursively decompose a 3d object into smaller versions of itself :)
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.
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.
Found a space-filling curve for the wallflower:
https://onlinetools.com/math/l-system-generator?draw=ABCD&sk...
The previous one fills out the Koch island.
https://onlinetools.com/math/l-system-generator?draw=ABCD&sk...
The previous one fills out the Koch island.
That's really cool! I tried to get something to work last week on pen and paper but couldn't get anything to stick. Is there a strategy you used or did you just go by feel?
Edit: just noticed how you encoded a flip (AB <--> CD) between iterations like how the matrix flips the orientation of space. Super neat!
Edit: just noticed how you encoded a flip (AB <--> CD) between iterations like how the matrix flips the orientation of space. Super neat!
> noticed how you encoded a flip (AB <--> CD)
Exactly! There is also a less obvious relationship between A and B too: B is a A "backwards" (A rotated 180°, starting the curve from the opposite end).
The strategy was to put 5 lines on the plus sign on the sides of the 5 cells, with the idea that each line eventually fills out a neighboring cell in subsequent iterations. I found one such path that had a chance of working. Not sure if this makes sense.
Exactly! There is also a less obvious relationship between A and B too: B is a A "backwards" (A rotated 180°, starting the curve from the opposite end).
The strategy was to put 5 lines on the plus sign on the sides of the 5 cells, with the idea that each line eventually fills out a neighboring cell in subsequent iterations. I found one such path that had a chance of working. Not sure if this makes sense.
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?
Question to the author: what would you recommend to hang on my kid’s wall today?
I'm by no means a parenting expert, but my answer would be anything related to something they feel passion or wonder for in the moment. I snuck in a paragraph near the end about burnout. At the root of the problem for me was that I lost the feeling of fascination and curiosity I had for math and programming, and doing this write-up helped me tap into that feeling of childlike wonder that used to come easily.
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
It still works in the alternative system (14+21 should equal 12)
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
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
==12Well 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.
The post presents it a bit out of order, but it was mostly from realizing at some point that the way the fractal grows by a factor of 5, base 5 number systems, and the "spiral" mentioned in the post can all fit together. I also thought a lot about how to programmatically draw the fractal and a natural way would be to start from the middle and zoom out.
There's an apocryphal story about Richard Feynman about how he used to keep a dozen or so random problems in the back of his mind and made a little bit of progress on them every time he saw a connection, until finally he'd solve one and everyone would think he magically figured it out instantly. This was a bit similar except I'm not nearly at that level and I've only been able to do that for one problem instead of a dozen.
There's an apocryphal story about Richard Feynman about how he used to keep a dozen or so random problems in the back of his mind and made a little bit of progress on them every time he saw a connection, until finally he'd solve one and everyone would think he magically figured it out instantly. This was a bit similar except I'm not nearly at that level and I've only been able to do that for one problem instead of a dozen.
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. :)
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. :)
> 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.
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.
Really cool and in-depth, thank you!
I believe that there is a typo in the pattern formula (right after "Looking closely you might pick up on the pattern"): it should read
Also, regarding Knuth's mistake: Youtube comments point out that his fractal is in fact correct; he just mistook the beginning point with the end point. Loosely speaking, the fractal is symmetrical about its middle turn, which is precisely the one Knuth believed to be incorrect. All in all, he still made a fractal-related mistake, so the conclusion holds.
I believe that there is a typo in the pattern formula (right after "Looking closely you might pick up on the pattern"): it should read
5**(n/2) instead of 5**n
5**((n-1)/2) instead of 5**(n-1)
Also, regarding Knuth's mistake: Youtube comments point out that his fractal is in fact correct; he just mistook the beginning point with the end point. Loosely speaking, the fractal is symmetrical about its middle turn, which is precisely the one Knuth believed to be incorrect. All in all, he still made a fractal-related mistake, so the conclusion holds.
Good catch, corrected the formula!
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.
Isn’t that it on the left in the last image?
Yeah, maybe it is. It would be cool to make it much bigger, frame it and put it on the wall. Or create a mosaic tiled artwork, similar to Knuth’s dragon curve wall.
Yeah, it's in the last image and in the thumbnail at very top (which I realize now is really hard to spot on mobile), intentionally not in the spotlight to leave space for the twist at the end.
https://chriskw.xyz/images/fractal/thumbnail.jpg
I think it would work perfectly as a mosaic eventually, but for the time being I'm perfectly content with the "rustic" 8x11 graph paper sized one taped to the wall. Currently planning to put up a slice of the orthotopeflower as a companion piece once I find matches for the colored pencils I used back then.
https://chriskw.xyz/images/fractal/thumbnail.jpg
I think it would work perfectly as a mosaic eventually, but for the time being I'm perfectly content with the "rustic" 8x11 graph paper sized one taped to the wall. Currently planning to put up a slice of the orthotopeflower as a companion piece once I find matches for the colored pencils I used back then.
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.
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.
Nice writeup. The Heighway dragon of Jurassic Park fame is pretty neat too.
https://en.m.wikipedia.org/wiki/Dragon_curve
https://en.m.wikipedia.org/wiki/Dragon_curve
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.
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.
I wonder if something similar can be applied to get a dither pattern with built-in level of detail adjustment.
well, that escalated beautifully
This is beautiful. Thank you.
Kinda looks like a propeller
Things with four arms that all curve the same way unfortunately tend to look swastika-ish.
The the arms of the author's "wallflower" fractal don't seem to curve, as opposed to the other, similar fractal (quadratic von Koch island). Which can be explained by each iteration adding a mirroring.
The unfortunate thing here is that the swastika was appropriated by a genocidal regime. The symbol still has a totally different life in India and Japan.
That was fun.
Too much math.
Now make a tiling game engine that uses these!