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There's a great deal of "stuff" in corners, side-bars, etc nowadays, and one of
the things that caught my was the following:
The Mathematics of the Honeycomb (1964)
https://www.youtube.com/watch?v=HphxVVJavIY&pp=ygUcdGhlIG1hdGhlbWF0aWNzIG9mIGhvbmV5Y29tYg%3D%3D
(worth the watch, IMO )
As usual, the presentation and derivation were interesting, but actually making
the geometry reveals that a few crucial bits of information get left out.
It was a little tricky, but I got that worked out using the dot product of the
two vectors, and the indispensable Pythorean theorem.
A good exercise, and something I think that Thomas Fester would enjoy.
Then I came across:
https://www.researchgate.net/publication/338237346_Mathematical_modeling_and_computer_simulation_of_the_three-dimension
al_pattern_formation_of_honeycombs
https://pmc.ncbi.nlm.nih.gov/articles/PMC3730681/
which provide support for the assertion that bees DON'T actually do anything
astounding with geometry and angles, but simply build hemispherically capped
cylinders, and the heated wax yields to the tension from gravity and adopts the
final honeycomb shape (rounded hexagons with trihedral pyramidal caps) as a
post-assembly deformation.
Wikipedia has a decent page on the honeycomb
https://en.wikipedia.org/wiki/Honeycomb
as well as other optimized space-filling polyhedra
https://en.wikipedia.org/wiki/Weaire%E2%80%93Phelan_structure
- BE
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