Random Thoughts on Geometry
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Egg packing
Sufficiently elongated ellipses (eggs) can be arranged in a pentagonal packing.
What is the least eccentric ellipse that can do this?
Can you construct a quasicrital with ellipses?
Sunflower seeds look a bit similar...
Nice pictures of tiling and spiraled patterns here
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 frunky structures you van 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:
The faces can be formed from intersecting pentagons.
This shape, toghether 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
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
