Sierpiński Triangle? In My Bitwise and?

I’d like to share some little demos here.

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.

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!

Just a heads up, all (binary?) logical operators produce fractals. This is pretty well-known[1].

[1] https://icefractal.com/articles/bitwise-fractals/

Try this one liner pasted into a Unix shell:

  cc -w -xc -std=c89 -<<<'main(c){int r;for(r=32;r;)printf(++c>31?c=!r--,"\n":c<r?" ":~c&r?" `":" #");}'&&./a.*

  main(c,r){for(r=32;r;)printf(++c>31?c=!r--,"\n":c<r?" ":~c&r?" `":" #");}

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Apologies for a comment not related to the content, but it makes it difficult to read the article on mobile.

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!

Rendering proof: https://lambda-screen.marvinborner.de/?term=ERoc0CrbYIA%3D

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.

I reached a similar result when researching all possible "binary subpixel" configurations that would give a pixel its fuzzy value. Arranging the configurations in ascending order row-wise for one pixel and column-wise for the other, performing an intersection between the two pixels, and plotting against their resulting fuzzy value results in a sierpinski triangle.

(if interested, see fig 4.3, page 126 of my thesis, here: https://ora.ox.ac.uk/objects/uuid:dc352697-c804-4257-8aec-08...)

Cool stuff. Especially the bottom right panel, you might not have expected that kind of symmetry in the intersection when looking at the individual components.

Ah. Is that why LFSRs (linear feedback shift registers) and specifically PRBS generators (pseudo-random binary sequences) produce Sierpinski triangles as well?

PRBS sequences are well-known, well-used "pseudo-random" sequences that are, for example, used to (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.

> the magic is the positional numeral system

— of course. In the same way the (standard) Cantor set consists of precisely those numbers from the interval [0,1] that can be represented using only 0 and 2 in their ternary expansion (repeated 2 is allowed, as in 1 = 0.2222...). If self-similar fractals can be conveniently represented in positional number systems, it is because the latter are self-similar.

There are so many ways to produce sierpinski gaskets.

It you specify n points and the pick a new point at random, then iteratively randomly select (uniformly) one of the original n points and move the next point to the mid point of the current point and the selected point. Coloring those points generates a sierpinski triangle or tetrahedron or whatever the n-1 dimensional triangle is called

I draw these with paper and pen when I am extremely bored in meetings.