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# Geodesic Polyhedra

written by Rene K. Mueller, Copyright (c) 2007, last updated Fri, February 25, 2022

## Geodesic Pentakis Dodecahedron

It's not a platonic or archimedean polyhedra, but the dual of the archimedean truncated icosahedron:

### Pentakis Dodecahedron

 Pentakis Dodecahedron
• Dual: truncated icosahedron
• s1: 1/19 * (18*√5-9)
• s2: 3/2 * (√5-1)
• V: s13 * 5/3 * √(1/2 * (421+63*√5))
• A: s12 * 5/36 * (41+25*√5)
• Vertices/Connectors: 32
• 12 x 5-way
• 20 x 6-way
• Faces: 60 (3-sided)
• Edges/Struts:
• A x 60: 0.67765
• B x 30: 0.76393
• total 90 struts (2 kinds)
• strut variance 12.7%
In its original version there is no straight hemisphere line, and already two kinds of edge lengths.

#### Pentakis Dodecahedron vs Geodesic Dodecahedron L1

 A B Strut Variance Pentakis Dodecahedron 1.000000 1.127322 12.7% Geodesic Dodecahedron L1 1.000000 1.113583 11.4%
While I thought I knew this form, and going through previous forms I found it looks like the Geodesic Dodecahedron L1, and to my surprise the structure almost is the same, yet the ratio between both strut length is off a bit. The difference of B's is 1.21%, very small but still too significant - so the Pentakis Dodecahedron and the Geodesic Dodecahedron L1 (triangulated the pentagons and spherical projected) are structure-wise the same, but the strut ratios are a bit off.

### Geodesic Pentakis Dodecahedron L2

 Geodesic Pentakis Dodecahedron L2
• Vertices/Connectors: 122
• 12 x 5-way
• 110 x 6-way
• Faces: 240 (3-sided)
• Edges/Struts:
• A x 60: 0.32036
• B x 60: 0.32910
• C x 120: 0.33867
• D x 60: 0.36284
• E x 60: 0.38161
• total 360 struts (5 kinds)
• strut variance 19.4%

### Geodesic Pentakis Dodecahedron L2T

 Geodesic Pentakis Dodecahedron L2T
• Vertices/Connectors: 182
• 90 x 4-way
• 60 x 6-way
• 12 x 10-way
• 20 x 12-way
• Faces: 360 (3-sided)
• Edges/Struts:
• A x 60: 0.18738
• B x 120: 0.21357
• C x 60: 0.32036
• D x 60: 0.32910
• E x 60: 0.36284
• F x 60: 0.36388
• G x 120: 0.40553
• total 540 struts (7 kinds)
• strut variance 117.1%
The triacon version provides hemisphere lines for a dome version.

### Epcot Spaceship Earth

 Epcot Spaceship Earth on opening day (1982)
The pentakis dodecahedron is quite famous by its Epcot (Experimental Prototype Community Of Tomorrow) "Spaceship Earth" version at Disney World in Tampa, Florida (USA):
• 50 m / 165 feet diameter
• 11'520 faces (60 x 16 x 4 x 3 faces)
• 2 years and 2 months building time
• 7'040'000 kg total weight

Epcot: Spaceship Earth (2006)
Epcot: Spaceship Earth - Pentakis Dodecahedron 8V (2006)
Epcot: Spaceship Earth Simplified Model (Geodesic Pentakis Dodecahedron 8V)
Epcot: Spaceship Earth Detailed Model (Geodesic Pentakis Dodecahedron 8V & Centerpoint Triangulated Faces)

As far I figured out from the photo, the geodesic sphere is 8V, whereas each 8V triangle is center point triangulated into 3 faces again, 60 x 8 x 8 x 3 = 11'520 faces finally. Yet, according Wikipedia: Spaceship Earth the sphere is composed by

• pentakis dodecohedron 60 faces, which each is
• subdivided 16 equilateral triangles, which each is
• subdivided in 4 triangles, which each is
• subdivided in 3 isoceles triangles

60 x 16 x 4 x 3 = 11'520 faces. It would be interesting to get more details to see why the leveled subdivisions were chosen the way they were.

Epcot Construction of Spaceship Earth (courtesy by Von Johnson and Associates, Inc., 1981) (1 of 2)
Epcot Construction of Spaceship Earth (courtesy by Von Johnson and Associates, Inc., 1981) (2 of 2)

These construction photos confirm the description of the Wikipedia article.

## Other Solids

From the Johnson Solid I tried
• Gyroelongated Triangular Bicupola
• Gyroelongated Square Bicupola

but didn't provide a straight hemisphere line at L1 or L2, and strut lengths were much higher than the platonic and archimedean solids.

As mentioned in the Geodesic Dome Notes I deepen some of the dome options for real-life applications.

• Geodesic Dome , Mathworld
• AntiPrism.com suite of programs for the generation, manipulation, and visualisation of point distributions and polyhedra

(End of Article)

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