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# Geodesic Polyhedra

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

## 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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