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Geodesic Polyhedra

written by Rene K. Mueller, Copyright (c) 2007, last updated Sun, October 28, 2012

Geodesic Icosahedron

Icosahedron


Icosahedron
  • Uniform Polyhedron: U22
  • Platonic Solid
  • Platonic Element: Water
  • Wythoff symbol: 5|2 3
  • Symmetry Group: icosahedral
  • Vertex Configuration: {3, 3, 3, 3, 3}
  • Dual: dodecahedron
  • V: s3 * 5/12 * (3 + √5)
  • A: s2 * 20 / 4 * √3
  • rinner: s / 12 * (3 * √3 + √15)
  • router: s / 4 * √(10 + 2 * √5)
  • Vertices/Connectors: 12 (5-way)
  • Faces: 20 (3-sided)
  • Edges/Struts:
    • A x 30: 1.05146

Let's skip the L1 as it's the same as L0 since the Icosahedron is composed already with triangles only.

Geodesic Icosahedron L2


Geodesic Icosahedron L2
  • Vertices/Connectors: 42
    • 12 x 5-way
    • 30 x 6-way
  • Faces: 80 (3-sided)
  • Edges/Struts:
    • A x 60: 0.54653
    • B x 60: 0.61803
    • total 120 struts (2 kinds)
    • strut variance 13.0%


Geodesic Icosahedron Dome L2


Geodesic Icosahedron Dome L2

Geodesic Icosahedron Dome L2
  • Vertices/Connectors: 26
    • 10 x 4-way
    • 6 x 5-way
    • 10 x 6-way
  • Faces: 40 (3-sided)

  • Icosahedron Dome L2 Construction Map
  • Edges/Struts:
    • A x 30: 0.54653
    • B x 35: 0.61803
    • total 65 struts (2 kinds)
    • strut variance 13.0%

d= A= x 30, B= x 35,

Geodesic Icosahedron L3


Geodesic Icosahedron L3
  • Vertices/Connectors: 162
    • 12 x 5-way
    • 150 x 6-way
  • Faces: 320 (3-sided)
  • Edges/Struts:
    • A x 120: 0.27590
    • B x 120: 0.28547
    • C x 120: 0.31287
    • D x 60: 0.32124
    • E x 60: 0.32492
    • total 480 struts (5 kinds)
    • strut variance 17.8%


Geodesic Icosahedron Dome L3


Geodesic Icosahedron Dome L3

Geodesic Icosahedron Dome L3
  • Vertices/Connectors: 91
    • 20 x 4-way
    • 6 x 5-way
    • 65 x 6-way
  • Faces: 160 (3-sided)

  • Icosahedron Dome L3 Construction Map
  • Edges/Struts:
    • A x 60: 0.27590
    • B x 60: 0.28547
    • C x 70: 0.31287
    • D x 30: 0.32124
    • E x 30: 0.32492
    • total 250 struts (5 kinds)
    • strut variance 17.8%

d= A= x 60, B= x 60, C= x 70, D= x 30, E= x 30,
When compared to other platonic and archimedean solids, it's apparent that it's the best source to triangulate from and get a firmly even distribution of vertices.

For more details look at my Geodesic Dome Notes, where I cover 2V, 3V and 4V in more details.

L2 4V Fuller
A 0.27590 A 0.25318
B 0.28547 C 0.29453
C 0.31287 D 0.31287
D 0.32124 B 0.29524
E 0.32492 E 0.32492
A 0.27590 F 0.29859
17% variance 28% variance
I noticed that the strut ratio at DesertDomes.com for the 4V (which is supposed to use Fuller's ratios), and mine differ. After a close checking, I realized the 4V is direct derived from the original Icosahedron, whereas I created the L2/2V, and created from that the L3.

Now, based on that result I keep the L-notion and describe this more thorough, the difference and maybe advantages of Ln vs nV:

  • Ln creates more alike struts - less variance, and seems less strut lengths
  • nV provides more fine tuned triangulation, e.g. 3V or 5V, whereas L1 = 1V, L2 = 2V, L3 ~ 4V, L4 ~ 8V etc. ('~' means similar)


Geodesic Icosahedron L2T


Geodesic Icosahedron L2T
  • Vertices/Connectors: 62
    • 30 x 4-way
    • 20 x 6-way
    • 12 x 10-way
  • Faces: 120 (3-sided)
  • Edges/Struts:
    • A x 60: 0.36284
    • B x 60: 0.54653
    • C x 60: 0.64085
    • total 180 struts (3 kinds)
    • strut variance 76.6%


Geodesic Icosahedron Dome L2T


Geodesic Icosahedron Dome L2T

Geodesic Icosahedron Dome L2T
  • Vertices/Connectors: 37
    • 4 x 3-way
    • 17 x 4-way
    • 12 x 6-way
    • 4 x 10-way
  • Faces: 60 (3-sided)

  • Icosahedron Dome L2T Construction Map
  • Edges/Struts:
    • A x 32: 0.36284
    • B x 32: 0.54653
    • C x 32: 0.64085
    • total 96 struts (3 kinds)
    • strut variance 76.6%

d= A= x 32, B= x 32, C= x 32,

Geodesic Icosahedron L3T


Geodesic Icosahedron L3T
  • Vertices/Connectors: 362
    • 180 x 4-way
    • 120 x 6-way
    • 30 x 8-way
    • 20 x 12-way
    • 12 x 20-way
  • Faces: 720 (3-sided)
  • Edges/Struts:
    • A x 120: 0.11762
    • B x 120: 0.15406
    • C x 120: 0.18218
    • D x 120: 0.19272
    • E x 120: 0.22385
    • F x 120: 0.27590
    • G x 120: 0.29886
    • H x 120: 0.32473
    • I x 120: 0.38444
    • total 1080 struts (9 kinds)
    • strut variance 225.4%


Geodesic Icosahedron Dome L3T


Geodesic Icosahedron Dome L3T

Geodesic Icosahedron Dome L3T
  • Vertices/Connectors: 193
    • 12 x 3-way
    • 84 x 4-way
    • 4 x 5-way
    • 60 x 6-way
    • 4 x 7-way
    • 13 x 8-way
    • 4 x 11-way
    • 8 x 12-way
    • 4 x 20-way
  • Faces: 360 (3-sided)

  • Icosahedron Dome L3T Construction Map
  • Edges/Struts:
    • A x 60: 0.11762
    • B x 60: 0.15406
    • C x 64: 0.18218
    • D x 60: 0.19272
    • E x 60: 0.22385
    • F x 64: 0.27590
    • G x 60: 0.29886
    • H x 64: 0.32473
    • I x 60: 0.38444
    • total 552 struts (9 kinds)
    • strut variance 225.4%

d= A= x 60, B= x 60, C= x 64, D= x 60, E= x 60, F= x 64, G= x 60, H= x 64, I= x 60,


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