ScrapBook

Random Thoughts on Geometry
Home

Another picture of a mutated Buckyball:

It has the same nymber ov vertices, edges and faces as a Buckyball, and also the valence of each vertex is the same. In this cse, it also has the same number (2) of hexagons, but they are arranged differently, producing a long and thin polyhedron. It can be produced from Buckyball, using a so-called Stone-Wales transormation.

A picture of a mutated Buckyball:

It has the same nymber ov vertices, edges and faces as a Buckyball, and also the valence of each vertex is the same. But the polygon orders are difference: Instead of 20 hexagons we have 10 heptagons and 10 extra pentagons.

A picture of 30 dice:

I posted this on mathstodon:Mathstodon post

A picture for Azimuth:

A graph of the first bernoulilli numbers:

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:

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:

(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

Source file of BristorBrot (A 3D fractal)

Deformable Klein Quartic

Instructions for making one:
KleinInstructions.pdf
KleinConnectors.pdf
KleinLegs.pdf

Idea for Leeuwarden 2018 (cultural capital)

2 new animations of cycloids, now with the additional property that the centre cycloid is standing still.

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

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.

More cycloids:

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.

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:

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 of 3D space with dodecahedra

Pentagrams on cubically stacked dodecahedra

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+ay
dz/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.

$JuliaBrot type fractal$

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

Youtube clip of stirling engine

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.