Random Thoughts on Geometry

Some pictures for a post on John Baez's weblog Azimuth:


Some pictures on the Kuramoto Sivashinsky equation:
Here is an electric circuit equivalent:
Electric circuit equivalent of Kuramoto Sivashinsky
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:
Kuramota Sivashinsky stripes

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:
Regular stripes of Kuramoto


(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

Source file of BristorBrot (A 3D fractal)
Spidron folding animation

Spidron animation

Deformable Klein Quartic
Klein Quartic Inside out
Instructions for making one:

Hinge polyhedron

Animation perspectagram leeuwarden 2018
Idea for Leeuwarden 2018 (cultural capital)

2 new animations of cycloids, now with the additional property that the centre cycloid is standing still.
Rolling hypocycloidsRolling EpicycloidsMore rolling hypocycloids

A "Perspectagram", or anagram implemented by viewing from 2 perspectives.

Mechanagram Voldemort
This "Mechanagram" illustrates a general method for a mechanical realisation of an arbitrary anagram. (For the Harry Potter fans!)

A "Mechanagram", inspired by an idea by Ikeda Yosuke

Roling Epicycloid, for this discusson.
Roling epicycloid

More cycloids:
Roling cycloids
Roling cycloids
Roling cycloidsRoling cycloids

5-hypocycloid rilling in 4-epicycloid


The Excel sheet that generates these, can be downloaded here:

Canonicl thickening
Henry Segerman posed an interesting question on Google plus .
If you have a network with vertices connected by lines, how do you thicken the lines (so you can 3D print them), such that around the vertices, things work out in a nice way.
A method is 'canoncal' if it does not depend on arbitrary choices.
One way to do it, is to trace all lines with circles. The envelope gives the thickened geometry. This will work in any dimension.
Canonical thickening of infinitely thin network
After you have created the envelope, you can proceed to mesh it.  The black thick dots are points you would want to be mesh vertices.

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:

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


JuliaBrot fractal.
JuliaBrot type fractal
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
        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


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