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Comment by vanderZwan | Hacker News Reader

A few years ago, I accidentally found this way of creating a Sierpinski Gasket-like structure:

Lines "flow" from left to right (horizontally) or bottom to top (vertically), and all lines start out white. When two lines cross, they might switch color depending on which color they come across, as follows:

* white (h) + white (v) = black (h) + black (v)

* black (h) + black (v) = black (h) + black (v)

* white (h) + black (v) = white (h) + white (v)

* black (h) + white (v) = white (h) + white (v)

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What you've got there is a truth table for the XOR bitwise operation.

    HV | O
    ---+---
    11 | 0
    00 | 0
    10 | 1
    01 | 1

You can do it very similarly with the coordinates of the pixels on the screen[1]. It's really interesting how often that shape comes out of places. Which is why I think that this page got started.

[1] http://www.stilldreamer.com/mathematics/sierpinskis_triangle...

I designed a system of cellular automata based around this idea of colliding lines. You can play with it at http://sciencevsmagic.net/logicgrid

The XOR Sierpinksi's triangle is at http://sciencevsmagic.net/logicgrid/#282800 but there are other ways make it too. For example XNOR makes an inverse http://sciencevsmagic.net/logicgrid/#828200

If you keep hitting random you'll find plenty of surprising analogues to Sierpinksi's.

XOR is addition mod 2, which is the same reason you get Pascal's triangle mod 2 is sierpinski gasket.

Note that where binary operations on screen coordinates tend to produce Sierpiński's triangle, ternary operations will produce Sierpiński's carpet (I don't know what happens as you increase bases beyond that, the number of possible operations does explode).

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