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Three symbol cyclic system

Sun, 2007-07-29 17:42 — Mark Dow

A three symbol 2-D cyclic system:
a -> [ a b c; b c a; c a b ], b -> [ b c a; c a b; a b c ], c -> [ c a b; a b c; b c a ]

(right click for full image)

This system is a close relative of the Three symbol cyclic and two symbol Thue-Morse system, although my construction method is somewhat different.

This is an analogue of the Thue-Morse system:
a -> [ a b; b a ], b -> [ b a; a b ]

This example has had a gaussian blur (FWHM = 1 px.) applied:

‹ Three symbol cyclic and two symbol Thue-MorseupBrigid's cross tesselations ›

Comments

Mon, 2007-07-30 08:23 — PSR

parameter animation

Nice. If you were to cycle each point through a gray scale gradient would this make a rolling wave pattern...Or was the point to be able to make smooth transitions on higher orders based on the sequence pattern?

Would you be willing to provide a paragraph or so on what motivated you do it and what it means?

The smoothness evokes thoughts of Touring reaction diffusion eqns...but this is very ordered.


Mon, 2007-07-30 23:06 — Mark Dow

animated phase waves

Hey, I did this animation for a four symbol system (see bottom of post), and it's cool. Try blurring your eyes a bit, or viewing from a distance. Good idea. Can't do it for the 3 symbol systems, as they have unequal numbers of different values.
Mon, 2007-07-30 10:39 — Mark Dow

waves

No, I wasn't trying to make a rolling wave, and I don't think it would get smoother with more gradations. The cyclic animation would only have three frames, I'll try that sometime.

The motivation is that it is simple, it's a recursive subtitution system (fractal), it's new to me, and it's cool! I added some comments and will integrate it with other things as I get time. It came up while thinking about combining these simple systems, math on substitution systems. I don't think it means anything, although the waves within waves within waves... is evocative. 

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