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Geodesic Dome Notes & Calculator

written by Rene K. Mueller, Copyright (c) 2005, 2006, 2007, last updated Sun, September 22, 2013

The Icosahedron


Icosahedron
  • Uniform Polyhedron: U22
  • Platonic Solid
  • Platonic Element: Water
  • Vertices: 12
  • Edges: 30
  • Faces: 20
  • 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)


s = , V = , A = , rinner = , router = , ravg =

1V/L1 2/3 Icosahedron Dome


Geodesic 1V 2/3 Icosahedron Dome (front view)

Geodesic 1V 2/3 Icosahedron Dome (bird view)
  • vertices/connectors: 11
    • 5 x 4-way
    • 6 x 5-way

  • 1V 2/3 Icosahedron Dome Construction Map
  • edges/struts & bending angles (αstrut):
    • A x 25: 1.05146 (31.72°)
  • faces: 15 (3-sided)
    • A-A-A x 15 (60.00°, 60.00°, 60.00°)
  • diameter: 2.000, radius: 1.000
  • height: 1.447 or 72.36% of diameter


1V 2/3 Icosahedron Dome Calculator


Geodesic 1V 2/3 Icosahedron Dome (Human is 170cm/5'7")

d = , lhole =
(Edit numbers & hit TAB to recalculate)

  • Alath = x 25,
  • h =


2V/L2 Icosahedron Dome


Geodesic 2V Icosahedron Dome (front view)

Geodesic 2V Icosahedron Dome (bird view)
  • vertices/connectors: 26
    • 10 x 4-way
    • 6 x 5-way
    • 10 x 6-way

  • 2V Icosahedron Dome Construction Map
  • edges/struts & bending angles (αstrut):
    • A x 30: 0.54653 (15.86°)
    • B x 35: 0.61803 (18.00°)
    • total 65 struts (2 kinds)1)
    • strut variance 13.1%
  • faces: 40 (3-sided)
    • A-A-B x 30 (55.57°, 55.57°, 68.86°)
    • B-B-B x 10 (60.00°, 60.00°, 60.00°)
    • 2 kinds of faces2)
  • diameter: 2.000, radius: 1.000
  • height: 1.000 or 50.00% of diameter

1) strut lengths sorted by 1/10'000th or +/-0.00005 exact
2) clock wise (cw) and counter clock wise (ccw) orientation neglected

2V Icosahedron Dome Calculator


Geodesic 2V Icosahedron Dome (Human is 170cm/5'7")

d = , lhole =
(Edit numbers & hit TAB to recalculate)

  • Alath = x 30,
  • Blath = x 35,
  • h =

Cut optimization:
  • normalize Alath+Blath = 1.0, therefore Alath+Blath =
  • cuts:
    • 30 x (Alath+Blath):
    • 5 x (Blath):
    • total 35 laths to cut


3V 4/9 Icosahedron Dome

In some literature this variant is also called "3V 3/8 Dome".

Geodesic 3V 4/9 Icosahedron Dome (front view)

Geodesic 3V 4/9 Icosahedron Dome (bird view)
  • vertices/connectors: 46
    • 15 x 4-way
    • 6 x 5-way
    • 25 x 6-way

  • 3V 4/9 Icosahedron Dome Construction Map
  • edges/struts & bending angles (αstrut):
    • A x 30: 0.34862 (10.04°)
    • B x 40: 0.40355 (11.64°)
    • C x 50: 0.41241 (11.90°)
    • total 120 struts (3 kinds)1)
    • strut variance 18.3%
  • faces: 75 (3-sided)
    • A-A-B x 30 (54.63°, 54.63°, 70.74°)
    • B-C-C x 45 (58.59°, 60.70°, 60.70°)
    • 2 kinds of faces2)
  • diameter: 2.000, radius: 1.000
  • height: 0.828 or 41.42% of diameter

1) strut lengths sorted by 1/10'000th or +/-0.00005 exact
2) clock wise (cw) and counter clock wise (ccw) orientation neglected

3V 4/9 Icosahedron Dome Calculator


Geodesic 3V 4/9 Icosahedron Dome (Human is 170cm/5'7")

d = , lhole =
(Edit numbers & hit TAB to recalculate)

  • Alath = x 30,
  • Blath = x 40,
  • Clath = x 50,
  • h =

Cut optimization:
  • normalize Blath+Clath = 1.0, therefore Blath+Clath =
  • cuts:
    • 40 x (Blath+Clath):
    • 10 x (Alath+Clath):
    • 10 x (Alath+Alath):
    • total 60 laths to cut

As you see the bottom isn't entirely even, for heavy construction you have to level them.

3V 5/9 Icosahedron Dome

In some literature this variant is also called "3V 5/8 Dome".

Geodesic 3V 5/9 Icosahedron Dome (front view)

Geodesic 3V 5/9 Icosahedron Dome (bird view)
  • vertices/connectors: 61
    • 15 x 4-way
    • 6 x 5-way
    • 40 x 6-way

  • 3V 5/9 Icosahedron Dome Construction Map
  • edges/struts & bending angles (αstrut):
    • A x 30: 0.34862 (10.04°)
    • B x 55: 0.40355 (11.64°)
    • C x 80: 0.41241 (11.90°)
    • total 165 struts (3 kinds)1)
    • strut variance 18.3%
  • faces: 105 (3-sided)
    • A-A-B x 30 (54.63°, 54.63°, 70.74°)
    • B-C-C x 75 (58.59°, 60.70°, 60.70°)
    • 2 kinds of faces2)
  • diameter: 2.000, radius: 1.000
  • height: 1.188 or 59.38% of diameter

1) strut lengths sorted by 1/10'000th or +/-0.00005 exact
2) clock wise (cw) and counter clock wise (ccw) orientation neglected

3V 5/9 Icosahedron Dome Calculator


Geodesic 3V 5/9 Icosahedron Dome (Human is 170cm/5'7")

d = , lhole =
(Edit numbers & hit TAB to recalculate)

  • Alath = x 30,
  • Blath = x 55,
  • Clath = x 80,
  • h =

Cut optimization:
  • normalize Blath+Clath = 1.0, therefore Blath+Clath =
  • cuts:
    • 55 x (Blath+Clath):
    • 25 x (Alath+Clath):
    • 3 x (Alath+Alath):
    • total 83 laths to cut

As you see here as well that the bottom isn't entirely even, for heavy construction you have to level them.

4V Icosahedron Dome


Geodesic 4V Icosahedron Dome (front view)

Geodesic 4V Icosahedron Dome (bird view)
  • vertices/connectors: 91
    • 20 x 4-way
    • 6 x 5-way
    • 65 x 6-way

  • 4V Icosahedron Dome Construction Map
  • edges/struts & bending angles (αstrut):
    • A x 30: 0.25318 (7.27°)
    • B x 60: 0.29453 (8.47°)
    • C x 30: 0.29524 (8.49°)
    • D x 30: 0.29859 (8.59°)
    • E x 70: 0.31287 (9.00°)
    • F x 30: 0.32492 (9.35°)
    • total 250 struts (6 kinds)1)
    • strut variance 28.3%
  • faces: 160 (3-sided)
    • A-A-C x 30 (54.34°, 54.34°, 71.32°)
    • B-B-C x 30 (59.92°, 59.92°, 60.16°)
    • B-D-E x 60 (57.52°, 58.80°, 63.68°)
    • E-E-F x 30 (58.72°, 58.72°, 62.55°)
    • F-F-F x 10 (60.00°, 60.00°, 60.00°)
    • 5 kinds of faces2)
  • diameter: 2.000, radius: 1.000
  • height: 1.000 or 50.00% of diameter

1) strut lengths sorted by 1/10'000th or +/-0.00005 exact
2) clock wise (cw) and counter clock wise (ccw) orientation neglected

4V Icosahedron Dome Calculator


Geodesic 4V Icosahedron Dome (Human is 170cm/5'7")

d = , lhole =
(Edit numbers & hit TAB to recalculate)

  • Alath = x 30,
  • Blath = x 60,
  • Clath = x 30,
  • Dlath = x 30,
  • Elath = x 70,
  • Flath = x 30,
  • h =

Cut optimization:
  • normalize Elath+Elath = 1.0, therefore Elath+Elath =
  • cuts:
    • 60 x (Blath+Elath):
    • 5 x (Elath+Elath):
    • 30 x (Alath+Flath):
    • 30 x (Clath+Dlath):
    • total 125 laths to cut

The 4V looks very attractive, yet, I would recommend using the L3 version instead with less strut lengths and less strut variance.

L3 Icosahedron Dome


Geodesic L3 Icosahedron Dome (front view)

Geodesic L3 Icosahedron Dome (bird view)
  • vertices/connectors: 91
    • 20 x 4-way
    • 6 x 5-way
    • 65 x 6-way

  • L3 Icosahedron Dome Construction Map
  • edges/struts & bending angles (αstrut):
    • A x 60: 0.27590 (7.93°)
    • B x 60: 0.28547 (8.21°)
    • C x 70: 0.31287 (9.00°)
    • D x 30: 0.32124 (9.24°)
    • E x 30: 0.32492 (9.35°)
    • total 250 struts (5 kinds)1)
    • strut variance 17.8%
  • faces: 160 (3-sided)
    • A-A-D x 30 (54.40°, 54.40°, 71.20°)
    • A-B-C x 60 (54.68°, 57.60°, 67.72°)
    • B-B-D x 30 (55.77°, 55.77°, 68.46°)
    • C-C-E x 30 (58.72°, 58.72°, 62.55°)
    • E-E-E x 10 (60.00°, 60.00°, 60.00°)
    • 5 kinds of faces2)
  • diameter: 2.000, radius: 1.000
  • height: 1.000 or 50.00% of diameter

1) strut lengths sorted by 1/10'000th or +/-0.00005 exact
2) clock wise (cw) and counter clock wise (ccw) orientation neglected

L3 Icosahedron Dome Calculator


Geodesic L3 Icosahedron Dome (Human is 170cm/5'7")

d = , lhole =
(Edit numbers & hit TAB to recalculate)

  • Alath = x 60,
  • Blath = x 60,
  • Clath = x 70,
  • Dlath = x 30,
  • Elath = x 30,
  • h =

Cut optimization:
  • normalize Clath+Clath = 1.0, therefore Clath+Clath =
  • cuts:
    • 60 x (Blath+Clath):
    • 5 x (Clath+Clath):
    • 30 x (Alath+Dlath):
    • 30 x (Alath+Elath):
    • total 125 laths to cut

The L3 might be a better choice for a 250 strut dome than the 4V variant: the L3 provides less strut variance 17.8% (5 kinds of struts) instead of 28.3% (6 kinds of struts).


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