The Excel sheet that generates
these, can be downloaded here:
Canonicl thickening 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
spiraled patterns here
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:
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.