platonic_sphere

Builds class-1 geodesic polyhedra (based on Platonic solids).

The main purpose of this function is to generate points on the surface of a unit sphere that have an optimized uniformity of distribution. A secondary purpose is to facilitate exploration of geodesic polyhedra.

The functional form is:

[polyhedron,properties] = platonic_sphere(symmetry,freq,style)

where inputs are:

symmetry: 'icosahedral', 'octahedral' or 'tetrahedral'

freq: subdivision frequency (natural number)

style: subdivision style ('planar' or 'spherical

The most uniform spherical distribution is produced using icosahedral symmetry. If no input parameters are given, default is icosahedral symmetry with a subdivision frequency of 3 (each face divided as a tetractys) and a planar subdivision style.

Example:

[polyhedron,properties] = platonic_sphere('icosahedral',3)

where output is:

- polyhedron: triangulation object

- properties: structure with fields describing properties of the polyhedron

practicalities:

- visualize: trisurf(polyhedron); axis equal

- make an stl file: stlwrite(polyhedron,'my_psphere')

- make a movie: psphere_movie('octahedral',25)

AUTHOR: David Monteverde

RELEASED: 2024-11-10

UPDATED: 2024-11-29

REV: 1.3

NOTES

1) Surface density of the "platonic sphere" (number of vertices of the subdivided polyhedron) increases quadratically with subdivision frequency:

nvp = (nfb/2)*freq^2 + (neb-3*nfb/2)*freq +( nvb-neb+nfb)

where,

nvp : number of vertices of the subdivided polyhedron

freq : subdivision frequency

nvb : number of vertices of the basis polyhedron

neb : number of edges of the basis polyhedron

nfb : number of faces of the basis polyhedron (platonic solid)

Parenthetically, the above expression is derived from:

nvp = nvb + neb*(freq-1) + nfb*Trinum(freq-2)

where Trinum(n) is the triangular number of n.

For instance, the surface densities for a platonic sphere with icosahedral basis are as follows:

freq: 1 2 3 4 5 6 7 8 9 10 ...

vertices: 12 42 92 162 252 362 492 642 812 1002 ...

2) Additional useful information:

https://en.wikipedia.org/wiki/Geodesic_polyhedron

https://en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra

Pugh, Antony. Polyhedra: a visual approach. University of California Press, 1976