Spherical Grid Construction:

To create a mostly hexagonal spherical grid, two basic starting polyhedra are considered.

Icosahedron:
20 triangular faces
perfect triangles
and Pentakis-Dodecahedron:
60 triangular faces
almost perfect triangles
A Pentakis-Dodecahedron is a Dodecahedron with each pentagonal face divided into five spherical triangles.



Each triangle is recursively subdivided into four smaller triangles using a 4-fold division until a desired resolution is achieved. This iterative process is best described in pictoral form:

Planar Triangles:

All sub-triangles have the same area.
Spherical Triangles:

All sub-triangles do not have the same area.

To create a mostly hexagonal tessellation, vertices are converted to faces at the end of the iterative process. (The tessellation will contain twelve pentagonal faces, the rest will be hexagons.)
Global Grid Construction Discrete Global Grid






Polyhedron Quantitative Comparison:

A comparison of the Icosahedron and the Pentakis-Dodecahedron for equal-area spherical tessellation:

N Icosahedron Pentakis-Dodecahedron
0
V = 12
F = 20
V = 20
F = 12
1
V = 42
F = 80
V = 32
F = 60
2
V = 162
F = 320
V = 122
F = 240
- - -
8 V = 655,362
F = 1,310,720
V = 491,522
F = 983,040


For comparison purposes a Dodecahedron is used as iteration 0 for a Pentakis-Dodecahedron
N is iteration number, V is number of vertices, F is number of triangular faces

Eight iterations were run and the following metrics were calculated to determine which base polyhedron is more suitable for an equal-area tessellation:


Vertice seperation distance is the distance between neighboring vertices. Deviation is the percent difference between the closest neighboring vertice pair and the farthest neighboring vertice pair across the entire sphere.


Estimated Vertice Area is the square of a vertex's average distance to its neighbors times PI. In more visual terms, estimated vertice area is the area of a circle that would best fit the vertex. Deviation is the percent difference in minimum to maximum estimated vertice area across the entire sphere.

Distribution of Vertice Area Estimations:
Icosahedron
Pentakis-Dodecahedron
Small Area Large Area


The Pentakis-Dodecahedron method was selected for the virtual planet model because area and seperation distance errors are minimized.