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:
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
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:
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+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.
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
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.
