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 :)
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.
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.
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!
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 :)
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.
It's an ingroup joke but also a true warning: "Handwaving" Anyone who delves deeply into something reaches a part where they can't articulate a developing theory and hand-wave around it -- hoping you "get" it.
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 :)
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?
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.
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.
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!
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
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 :)
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.
I wonder if something similar can be applied to get a dither pattern with built-in level of detail adjustment.
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
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.
That was fun.
[flagged]
It's an ingroup joke but also a true warning: "Handwaving" Anyone who delves deeply into something reaches a part where they can't articulate a developing theory and hand-wave around it -- hoping you "get" it.
Are you wounded?