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%
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%
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)
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.
