Random Thoughts on Geometry
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Some pictures for a post on John Baez's weblog Azimuth:    Some pictures on the Kuramoto Sivashinsky equation:
Here is an electric circuit equivalent: The R3 resistors are non-linear: They are proportional to the gradient of the voltage (or a current in R1 and R2)

The equaitons generates stripe patterns: The stripes generally look like small streams flowing into a number of main rivers.. But, by chance i found, that if I initialise in a certain way, I get a regular strip pattern as below:   Model of Corona epidemic in Excel. Source file of BristorBrot (A 3D fractal)   Nice pictures of tiling and spiraled patterns here

Dodecahedral packing Regular dodecahedra almost pack space. (In a suitable neighbourhood of a black hole, they would form a perfect packing, due to the curvature of space)

A "Stewart Toroid" I discovered years ago, based on dodecahedra and "tri-diminished icosahedra": Note that icosa-dodecahedra nicely fit in the holes, forming a quasi crystalline packing.

Related to this, here are some funky structures you can build with rhombicosidodecahedra:    VRML versions:
Structure2
Structure3
Structure4
Structure5

Regular dodecahedra can be arranged in a cubic lattice, such that faces of the dodecahedra touch.
The arrangement leaves a gap, which can be filled with the shape below: The faces can be formed from intersecting pentagons.

This shape, together with regular dodecahedra, can pack space, in a cubical lattice
Below are 2 pictures on how the packing works.   Animation of the lattice D5. Variation on the Rossler attractor.
One of the simplest chaotic systems is the Rossler attractor:
`dx/dt = - (y+z)dy/dt= x+aydz/dt= b+z(x-c)Made a simution:` I made a variation that is a bit more similar to the Harmonic oscillator:
dx/dt = v
dv/dt = -x -R*v
dR/dt = -a+b*(x^2+v^2) If R=constant, we have the "ordinary" damped harmonic oscillator.
So we have damping as a 3rd dynamic variable, that depends non-linearly on x and v.

More variations: The strange pattern below was created as follows.
My sun was rubbing some children's paint across a paper.
Then he hit the painted  paper repeatedly with his hands.
After it dried,  it looked like this: Weird...

JuliaBrot fractal. Source:
For x = 1 To xpix
For y = 1 To ypix
xx = x_min + (x_max - x_min) * x / xpix
yy = y_min + (y_max - y_min) * y / ypix

c_re = c0_re + xx * A_re - yy * A_im
c_im = c0_im + xx * A_im + yy * A_re

gcount = 0
gstop = 0
z_re = xx
z_im = yy
Do
z_re_old = z_re
z_re = z_re * z_re - z_im * z_im + c_re
z_im = 2 * z_re_old * z_im + c_im
gcount = gcount + 1
If gcount > 100 Then gstop = 1
If (z_re * z_re + z_im * z_im) > 4 Then gstop = 1
Loop Until gstop = 1
If gcount > 100 Then Form2.Picture1.PSet (x, y)
Next y
Next x g

Torus with Farey sequence mod 6 double cover Circle packing animation of Ford circles modulo n.                 The above Pythogaras tree can be made by folding A4 paper into half repeatedly, and positioning as shown. Note that because of the 1:sqr(2) proportion, you always get right -angles. The branches are termnate once they touch another brang. Note that they touch exactly.

Animation of insect role in world food 