Enzymind Labs

Thirds waves

Large versions, up to 8000 x 1000 px.

Large animation, 512 x 512 px

This is a coloring of a regular square tessellation with a motif that includes a recurrent similarity transformation. The coloring is vaguely naturalistic, as ocean and atmosphere are suggested by the wave-like spiral elements. Like 'Mouse teeth weave', the motif basis pattern is derived from the largest linear features of the Bound 2x2 5s 1 fractal boundary:

The eighth generation of the Bound 2x2 5s 1 L-system.	The longest linear features of the fractal pattern (left). The pattern can be formed by combining rotations of integral scalings of this figure.	A union of the outline (center) with itself rotated by 180 degrees (pi radians). Note that the "dart" shape of the center figure is recapitulated, half size and rotated, at the top-right and bottom left of this figure.

In the right figure (above) the line segments and exensions of segments intersect the square tile at corners or 1/3 or 2/3 of the side length from a corner. The slopes of all line segments, in all the figures above, is +-1:3 or +-2:3. It can easily be constructed from scratch using these points and slopes.

Four copies of the union (above right) rotated with respect to each other in a 2x2 array form the exterior boundaries (below) of each quadrant of the motif.

Partial motif boundary template.

The central 1/3 x 1/3 and corners are then replaced recursively by a scaled (x 1/3) copy of the whole:

'Thirds waves' motif template

Each 'wave' figure is based on one of the many ways that this motif can be divided into a binary fractal, symmetric with respect to a half rotation about the center:

These are a kind of discrete logarithmic spiral; related sets of vertices on the boundaries lie on logarithmic spirals. The recursive similarity transform (a scaling) combined with quarter rotations (inherent in symmetries of the motif boundaries) maps points in these sets to one another.

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Terms for use: Please get written permission (Mark Dow, dow[at]uoregon.edu) before using the final images ('Thirds waves'), outlining how it will be modified, reproduced and/or used. There are no restrictions on the use of the other images on this page. Claiming to be the originator or owner, explicitly or implicitly, is bad karma. A link (if appropriate), a note to dow[at]uoregon.edu, and credit are appreciated but not required.