Random Thoughts on Geometry
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(3,3,3,3,3,3) tiling sliding to (4,4,4,4) tiling

Model of Corona epidemic in Excel.

Link to openscad file for generating ruled surface knots
Ruled surface knot


Spidron folding animation


Spidron animation



Nice pictures of tiling and spiraled patterns here


Dodecahedral packing
Three dimensional packing with regular dodecahera
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":
3 dimensional structure with 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:
Rhombicosadodecahedral structure 2 Rhombicosadodecahedral structure 3
Rhombicosadodecahedral structure 4 Rhombicosadodecahedral structure 5

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:
Connector shape for dodecahedra in cubic lattice
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.
Cubic packing with dodecahedra  cubic packing of 3D space with dodecahedra

Pentagrams on cubically stacked dodecahedra


Animation of the lattice D5.
Animation of D5 lattice

Variation on the Rossler attractor.
One of the simplest chaotic systems is the Rossler attractor:
dx/dt = - (y+z)
dy/dt= x+ay
dz/dt= b+z(x-c)
Made a simution:

Rosler attractor

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)
Gerard attractor
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:
Gerard attractor

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:
Strange drying pattern

Weird...

JuliaBrot fractal.
JuliaBrot type 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

Another Juliabrot

Youtube clip of stirling engine

Usenet statisticsg


Torus with Farey sequence mod 6 double cover

Torus with Farey sequence mod 6


Circle packing animation of Ford circles modulo n.


Ford circles modulo n


Frame n=4Frame n=5Frame n=6
Frame n=6_5Frame n=7Frame n=8
Frame n=9Frame n=10
Frame n=12Frame n=13Frame n=17
Frame n=37Frame n=17

Timeslice2012

Pythagoras tree with A4 paper
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
Animation of insects role in world food