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:
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.
Link to openscad file for generating ruled
Source file of BristorBrot (A 3D
Animation of 'Eating
3D animation with
Instructions for making one:
Idea for Leeuwarden 2018 (cultural capital)
2 new animations of cycloids, now with the additional property that the
centre cycloid is standing still.
"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
Roling Epicycloid, for
The Excel sheet that generates
these, can be downloaded
here: Henry Segerman posed an
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
Sufficiently elongated ellipses (eggs) can be arranged in a pentagonal
What is the least eccentric ellipse that can do this?
Can you construct a quasicristal with ellipses?
Sunflower seeds look a bit similar...
Nice pictures of tiling and
Regular dodecahedra almost pack space. (In a suitable neighbourhood of
black hole, they would form a perfect packing, due to the curvature of
A "Stewart Toroid" I discovered years ago, based on dodecahedra and
Note that icosa-dodecahedra nicely fit in the holes, forming a quasi
to this, here are some
funky structures you can build with rhombicosidodecahedra:
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
Below are 2 pictures on how the packing works.
Animation of the lattice D5.
One of the simplest chaotic
systems is the Rossler
on 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
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.
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:
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)
clip of stirling engine
with Farey sequence mod 6 double cover
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.