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The Sierpinski triangle page to end most Sierpinski triangle pages

oftenpaper.net

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Re: The Sierpinski triangle page to end most Sierpinski triangle pages

#31

Post by simcop2387 »

vanderZwan wrote: post #9

A few years ago, I accidentally found this way of creating a Sierpinski Gasket-like structure: https://vimeo.com/7690310 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) + bla…

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...

Re: The Sierpinski triangle page to end most Sierpinski triangle pages

#33

Post by pleion »

simcop2387 wrote: post #31
vanderZwan wrote: post #9

A few years ago, I accidentally found this way of creating a Sierpinski Gasket-like structure: https://vimeo.com/7690310 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) + bla…

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.

Re: The Sierpinski triangle page to end most Sierpinski triangle pages

#34

Post by Camillo »

pflats wrote: post #18
hansjorg wrote: post #5

That's amazing. It just goes on and on with beautiful plots. About halfway down, there's a section on understanding the Riemann sphere: > From what I can tell, one of the settings used to deal with division by 0 is the so-called Riemann sphere, which is where we take a space shuttle and use it to fly over and drop a cow on top of a biodome, and then have the cow indiscriminately fire laser beams at the grass inside a…

I'll attempt a layman's explanation of a Riemann sphere for anyone who has no idea what this is about: The Riemann sphere lets you deal with dividing by 0 by adding one more point to the complex plane: ∞[1]. Imagine putting it in the air above the origin, 0, and folding the midpoints of the four sides of your graph paper to meet there. (Yup, put all 4 points of the arrows in the same spot!) That's the Reimann sphere[…

That sounds similar to the projective space used in computer vision.

Re: The Sierpinski triangle page to end most Sierpinski triangle pages

#35

Post by spot »

GhotiFish wrote: post #30
spot wrote: post #28

http://draves.org/pix/frame3.cgi?zoom=1&dir=flames&file=205.... one he missed, from the early 90s

He did not. Fractal Flam3's are the chaos game. to quote: 1 start at any point. call it p 2 pick one of the three vertices at random 3 find the point halfway between p and that vertex 4 call that point p and draw it 5 goto 2 He goes into great depth about this strategy.

i don't mean flames in general, i mean that particular image, which is a spherical version of the sierpinski with rotation.

Re: The Sierpinski triangle page to end most Sierpinski triangle pages

#36

Post by xyzzyz »

Camillo wrote: post #34
pflats wrote: post #18

Earlier quoted context omitted.

I'll attempt a layman's explanation of a Riemann sphere for anyone who has no idea what this is about: The Riemann sphere lets you deal with dividing by 0 by adding one more point to the complex plane: ∞[1]. Imagine putting it in the air above the origin, 0, and folding the midpoints of the four sides of your graph paper to meet there. (Yup, put all 4 points of the arrows in the same spot!) That's the Reimann sphere[…

That sounds similar to the projective space used in computer vision.

Maybe it's because Riemann sphere is nothing else but a complex projective line, but I guess the reason you feel they're related is that both of these are manifolds which are created by performing certain identifications of points inside flat prototypes.

Re: The Sierpinski triangle page to end most Sierpinski triangle pages

#39

Post by judk »

simcop2387 wrote: post #31
vanderZwan wrote: post #9

A few years ago, I accidentally found this way of creating a Sierpinski Gasket-like structure: https://vimeo.com/7690310 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) + bla…

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...

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

Re: The Sierpinski triangle page to end most Sierpinski triangle pages

#40

Post by judk »

RockofStrength wrote: post #10

I was thinking about Sierpinski triangles recently, and trying to relate it to the 3 4 5 triangle. The 3 side has a Sierpinski triangle, with 9 segments, equaling 3^2. The 4 side has an equivalent "Sierpinski square", with 16 segments, equaling 4^2 (basically four squares touching at the corners with an empty square space in the middle, forming a cross shape). Then I was going to show how the two shapes could be comb…

Regular Pentagon does not tile the plane.
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