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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. Links Geodesic Dome , Mathworld AntiPrism.com suite of programs for the generation, manipulation, and visualisation of point distributions and polyhedra (End of Article) Page << Prev | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | Next >> Content: Page 1: Introduction, Geodesic Procedure, Triangulation Methods/Classes, Class 1 / Alternate Subdivision ... Page 2: Geodesic Truncated Tetrahedron Page 3: Geodesic Octahedron Page 4: Geodesic Cube Page 5: Geodesic Cuboctahedron Page 6: Geodesic Truncated Octahedron Page 7: Geodesic Truncated Cube Page 8: Geodesic Rhombicuboctahedron Page 9: Geodesic Truncated Cuboctahedron Page 10: Geodesic Snub Cube Page 11: Geodesic Icosahedron Page 12: Geodesic Dodecahedron Page 13: Geodesic Icosidodecahedron Page 14: Geodesic Truncated Icosahedron Page 15: Geodesic Truncated Dodecahedron Page 16: Geodesic Rhombicosidodecahedron Page 17: Geodesic Truncated Icosidodecahedron Page 18: Geodesic Snub Dodecahedron Page 19: Geodesic Pentakis Dodecahedron, Pentakis Dodecahedron vs Geodesic Dodecahedron L1, Epcot Spaceship Earth ... Home   ·   About   ·   Tipi   ·   Yurt   ·   Dome   ·   Features   ·   Gallery Print   ·   Bookmark Creative Commons (CC) BY-SA-NC 2005-2017, developed, designed and written by René K. 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