cc -w -xc -std=c89 -<<<'main(c){int r;for(r=32;r;)printf(++c>31?c=!r--,"\n":c<r?" ":~c&r?" `":" #");}'&&./a.*
It used to be cooler back when compilers supported weird K&R style C by default. I got it under 100 characters back then, and the C part was just 73 characters. This version is a bit longer but works with modern clang. The 73-character K&R C version that you can still compile today with GCC is:
Hey, at least it's not doing `curl | bash` like some people's installers do. It's only 109 characters, you can review that right? :-P
gjm11 · 2h ago
Here's a possibly-too-highbrow explanation to complement the nice simple one in the OP.
"As everyone knows", you get a Sierpinski triangle by taking the entries in Pascal's triangle mod 2. That is, taking binomial coefficients mod 2.
Now, here's a cute theorem about binomial coefficients and prime numbers: for any prime p, the number of powers of p dividing (n choose r) equals the number of carries when you write r and n-r in base p and add them up.
For instance, (16 choose 8) is a multiple of 9 but not of 27. 8 in base 3 is 22; when you add 22+22 in base 3, you have carries out of the units and threes digits.
OK. So, now, suppose you look at (x+y choose x) mod 2. This will be 1 exactly when no 2s divide it; i.e., when no carries occur when adding x and y in binary; i.e., when x and y never have 1-bits in the same place; i.e., when x AND y (bitwise) is zero.
And that's exactly what OP found!
marvinborner · 2h ago
Very cool! This basically encodes a quad-tree of bits where every except one quadrant of each subquadrant recurses on the parent quad-tree.
The corresponding equivalent of functional programming would be Church bits in a functional quad-tree encoding \s.(s TL TR BL BR). Then, the Sierpinski triangle can be written as (Y \fs.(s f f f #f)), where #f is the Church bit \tf.f!
Ah. Is that while LFSR (linear feedback shift registers) and specifically PRBS (pseudo-random binary sequences) produce Sierpinski triangles as well?
PRBS sequences are well-known, well-used "pseudo-random" sequences that are, for example, used for (non-cryptographically!) scramble data links, or to just test them (Bit Error Rate).
I made my own PRBS generator, and was surprised that visualizing its output, it was full of Sierpinski triangles of various sizes.
Even fully knowing and honoring that they have no cryptographic properties, it didn't feel very "pseudo-random" to me.
jcul · 3h ago
I can't dismiss the cookie popup on this page. After rejecting or accepting cookies it reloads and reappears.
Apologies for a comment not related to the content, but it makes it difficult to read the article on mobile.
jcul · 2h ago
Really interesting, and surprising article though!
IceDane · 2h ago
Same problem here. Firefox on Android.
kragen · 2h ago
The 31-byte demo "Klappquadrat" by T$ is based on this phenomenon; I wrote a page about how it works a few years ago, including a working Python2 reimplementation with Numpy: http://canonical.org/~kragen/demo/klappquadrat.html
I should probably update that page to explain how to use objdump correctly to disassemble MS-DOG .COM files.
If you like making fractal patterns with bitwise arithmetic, you'll probably love http://canonical.org/~kragen/sw/dev3/trama. Especially if you like stack machines too. The page is entirely in Spanish (except for an epilepsy safety warning) but I suspect that's unlikely to be a problem in practice.
userbinator · 52m ago
Sierpinski triangles are definitely a common sight in demoscene productions, to the point that they're acceptable in the smaller sizes, but others will think you're not good enough if that's all you do for a 64k or above entry.
But this is not using bitwise AND, just the Pascal's triangle approach. (Interestingly, you can reformulate that as a neighborhood-2 2-state 1-dimensional cellular automaton pretty easily; it occurs in a couple of different guises in Wolfram's catalog.)
Here's an ASCII-art version that uses AND as Michał describes:
32 value size : line cr size 0 do dup i and if bl else [char] # then dup emit emit loop drop ;
: pasand size 0 do i line loop ;
That looks nicer than my version. But you should put the `cr` before the inner loop, not after it. That way you can remove the `cr` before the outer loop.
peterburkimsher · 2h ago
Wolfram did a lot of research into cellular automata, and the Sierpinski Triangle kept showing up there too:
Bitwise XOR modulo T: https://susam.net/fxyt.html#XYxTN1srN255pTN1sqD
Bitwise AND modulo T: https://susam.net/fxyt.html#XYaTN1srN255pTN1sqN0
Bitwise OR modulo T: https://susam.net/fxyt.html#XYoTN1srN255pTN1sqDN0S
Where T is the time coordinate. Origin for X, Y coordinates is at the bottom left corner of the canvas.
You can pause the animation anytime by clicking the ‘■’ button and then step through the T coordinate using the ‘«’ and ‘»’ buttons.
[1] https://icefractal.com/articles/bitwise-fractals/
"As everyone knows", you get a Sierpinski triangle by taking the entries in Pascal's triangle mod 2. That is, taking binomial coefficients mod 2.
Now, here's a cute theorem about binomial coefficients and prime numbers: for any prime p, the number of powers of p dividing (n choose r) equals the number of carries when you write r and n-r in base p and add them up.
For instance, (16 choose 8) is a multiple of 9 but not of 27. 8 in base 3 is 22; when you add 22+22 in base 3, you have carries out of the units and threes digits.
OK. So, now, suppose you look at (x+y choose x) mod 2. This will be 1 exactly when no 2s divide it; i.e., when no carries occur when adding x and y in binary; i.e., when x and y never have 1-bits in the same place; i.e., when x AND y (bitwise) is zero.
And that's exactly what OP found!
The corresponding equivalent of functional programming would be Church bits in a functional quad-tree encoding \s.(s TL TR BL BR). Then, the Sierpinski triangle can be written as (Y \fs.(s f f f #f)), where #f is the Church bit \tf.f!
Rendering proof: https://lambda-screen.marvinborner.de/?term=ERoc0CrbYIA%3D
PRBS sequences are well-known, well-used "pseudo-random" sequences that are, for example, used for (non-cryptographically!) scramble data links, or to just test them (Bit Error Rate).
I made my own PRBS generator, and was surprised that visualizing its output, it was full of Sierpinski triangles of various sizes.
Even fully knowing and honoring that they have no cryptographic properties, it didn't feel very "pseudo-random" to me.
Apologies for a comment not related to the content, but it makes it difficult to read the article on mobile.
I should probably update that page to explain how to use objdump correctly to disassemble MS-DOG .COM files.
If you like making fractal patterns with bitwise arithmetic, you'll probably love http://canonical.org/~kragen/sw/dev3/trama. Especially if you like stack machines too. The page is entirely in Spanish (except for an epilepsy safety warning) but I suspect that's unlikely to be a problem in practice.
https://m.youtube.com/watch?v=tRaq4aYPzCc
Just kidding. This was a fun read.
Here's an ASCII-art version that uses AND as Michał describes:
Running `pasand` then yields this:https://www.wolframscience.com/nks/