Polyhedron Quantitative Comparison:
A comparison of the Icosahedron and the Pentakis-Dodecahedron for equal-area spherical tessellation:
N
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Icosahedron
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Pentakis-Dodecahedron
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0
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V = 12
F = 20
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V = 20
F = 12
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1
|
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V = 42
F = 80
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|
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V = 32
F = 60
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2
|
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V = 162
F = 320
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|
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V = 122
F = 240
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-
|
-
|
-
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8
|
V = 655,362
F = 1,310,720
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V = 491,522
F = 983,040
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For comparison purposes a Dodecahedron is used as iteration 0 for a Pentakis-Dodecahedron
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N is iteration number, V is number of vertices, F is number of triangular faces
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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.
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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.
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Distribution of Vertice Area Estimations:
Icosahedron
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Pentakis-Dodecahedron
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Small Area
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Large Area
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The Pentakis-Dodecahedron method was selected for the virtual planet model because area and seperation distance
errors are minimized.
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