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# Geodesic Dome Notes & Calculator

written by Rene K. Mueller, Copyright (c) 2005, 2006, 2007, last updated Tue, January 29, 2019

## The Rhombicuboctahedron

 Rhombicuboctahedron
• Uniform Polyhedron: U10
• aka Small Rhombicuboctahedron
• Archimedean Solid: A6
• Vertices: 24
• Edges: 48
• Faces: 26
• Wythoff symbol: 3 4|2
• Symmetry Group: octahedral
• Vertex Configuration: {4, 3, 4, 4}
• Dual: deltoidal icositetrahedron
• V: s3 / 3 * (12 + 10 * √2)
• A: s2 * (18 + 2 * √3)
• rinner: s / 17 * (6 + √2) * √(5 + 2 * √2)
• router: s / 2 * √(5 + 2 * √2)

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

### Preparing the Rhombicuboctahedron

Rhombicuboctahedron L0/0V (original)
Rhombicuboctahedron L1/1V flat (only triangles & flat)
Rhombicuboctahedron L1/1V (only triangles & normalized)
The squares are triangulated for further processing.

### 1V/L1 3/8 Rhombicuboctahedron Dome

 Geodesic 1V 3/8 Rhombicuboctahedron Dome (front view)
 Geodesic 1V 3/8 Rhombicuboctahedron Dome (bird view)
• vertices/connectors: 17
• 13 x 4-way
• 4 x 7-way
•  1V 3/8 Rhombicuboctahedron Dome Construction Map
• edges/struts & bending angles (αstrut):
• A x 20: 0.52372 (15.18°)
• B x 20: 0.71481 (20.94°)
• total 40 struts (2 kinds)1)
• strut variance 36.5%
• faces: 24 (3-sided)
• A-A-B x 20 (46.96°, 46.96°, 86.07°)
• B-B-B x 4 (60.00°, 60.00°, 60.00°)
• 2 kinds of faces2)
• height: 0.643 or 32.14% 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

#### 1V 3/8 Rhombicuboctahedron Dome Calculator

 Geodesic 1V 3/8 Rhombicuboctahedron Dome (Human is 170cm/5'7")

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

• Alath = x 20,
• Blath = x 20,
• h =

### 1V/L1 5/8 Rhombicuboctahedron Dome

 Geodesic 1V 5/8 Rhombicuboctahedron Dome (front view)
 Geodesic 1V 5/8 Rhombicuboctahedron Dome (bird view)
• vertices/connectors: 33
• 13 x 4-way
• 8 x 5-way
• 12 x 7-way
•  1V 5/8 Rhombicuboctahedron Dome Construction Map
• edges/struts & bending angles (αstrut):
• A x 52: 0.52372 (15.18°)
• B x 36: 0.71481 (20.94°)
• total 88 struts (2 kinds)1)
• strut variance 36.5%
• faces: 56 (3-sided)
• A-A-B x 52 (46.96°, 46.96°, 86.07°)
• B-B-B x 4 (60.00°, 60.00°, 60.00°)
• 2 kinds of faces2)
• height: 1.358 or 67.88% 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

#### 1V 5/8 Rhombicuboctahedron Dome Calculator

 Geodesic 1V 5/8 Rhombicuboctahedron Dome (Human is 170cm/5'7")

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

• Alath = x 52,
• Blath = x 36,
• h =

This is a nice variant, with square walls, optionally instead of square triangulation either replaced by one strut (B * √2) or leave it out at all.

The vertical struts of the vertical wall make it look very artistic, yet also edgy, but given it's a 1V it's understandable. With only 88 struts you get a 5/8 sphere.

At d = 6m the height is apprx. 4m and a 2nd level can be done at 2m height, just at the level of existing connectors.

### 2V/L2 3/8 Rhombicuboctahedron Dome

 Geodesic 2V 3/8 Rhombicuboctahedron Dome (front view)
 Geodesic 2V 3/8 Rhombicuboctahedron Dome (bird view)
• vertices/connectors: 57
• 21 x 4-way
• 32 x 6-way
• 4 x 7-way
•  2V 3/8 Rhombicuboctahedron Dome Construction Map
• edges/struts & bending angles (αstrut):
• A x 40: 0.26418 (7.59°)
• B x 40: 0.27386 (7.87°)
• C x 40: 0.36346 (10.47°)
• D x 20: 0.37033 (10.67°)
• E x 12: 0.38268 (11.03°)
• total 152 struts (5 kinds)1)
• strut variance 44.9%
• faces: 96 (3-sided)
• A-A-D x 20 (45.51°, 45.51°, 88.98°)
• A-B-C x 40 (46.39°, 48.64°, 84.97°)
• B-B-D x 20 (47.47°, 47.47°, 85.06°)
• C-C-E x 12 (58.24°, 58.24°, 63.53°)
• E-E-E x 4 (60.00°, 60.00°, 60.00°)
• 5 kinds of faces2)
• height: 0.643 or 32.14% 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 3/8 Rhombicuboctahedron Dome Calculator

 Geodesic 2V 3/8 Rhombicuboctahedron Dome (Human is 170cm/5'7")

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

• Alath = x 40,
• Blath = x 40,
• Clath = x 40,
• Dlath = x 20,
• Elath = x 12,
• h =

### 2V/L2 5/8 Rhombicuboctahedron Dome

 Geodesic 2V 5/8 Rhombicuboctahedron Dome (front view)
 Geodesic 2V 5/8 Rhombicuboctahedron Dome (bird view)
• vertices/connectors: 121
• 21 x 4-way
• 8 x 5-way
• 80 x 6-way
• 12 x 7-way
•  2V 5/8 Rhombicuboctahedron Dome Construction Map
• edges/struts & bending angles (αstrut):
• A x 104: 0.26418 (7.59°)
• B x 104: 0.27386 (7.87°)
• C x 72: 0.36346 (10.47°)
• D x 52: 0.37033 (10.67°)
• E x 12: 0.38268 (11.03°)
• total 344 struts (5 kinds)1)
• strut variance 44.9%
• faces: 224 (3-sided)
• A-A-D x 52 (45.51°, 45.51°, 88.98°)
• A-B-C x 104 (46.39°, 48.64°, 84.97°)
• B-B-D x 52 (47.47°, 47.47°, 85.06°)
• C-C-E x 12 (58.24°, 58.24°, 63.53°)
• E-E-E x 4 (60.00°, 60.00°, 60.00°)
• 5 kinds of faces2)
• height: 1.383 or 69.14% 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 5/8 Rhombicuboctahedron Dome Calculator

 Geodesic 2V 5/8 Rhombicuboctahedron Dome (Human is 170cm/5'7")

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

• Alath = x 104,
• Blath = x 104,
• Clath = x 72,
• Dlath = x 52,
• Elath = x 12,
• h =

This is also a nice variant, alike the 1V 5/8 one. Multiple squares in a spherical setting, a nice crossover of rectangular to circular/spherical space.

With 344 struts still considerable for temporary buildings.

### 3V 3/8 Rhombicuboctahedron Dome

 Geodesic 3V 3/8 Rhombicuboctahedron Dome (front view)
 Geodesic 3V 3/8 Rhombicuboctahedron Dome (bird view)
• vertices/connectors: 121
• 29 x 4-way
• 88 x 6-way
• 4 x 7-way
•  3V 3/8 Rhombicuboctahedron Dome Construction Map
• edges/struts & bending angles (αstrut):
• A x 40: 0.17453 (5.01°)
• B x 20: 0.18015 (5.17°)
• C x 40: 0.18031 (5.17°)
• D x 40: 0.18048 (5.18°)
• E x 40: 0.18562 (5.33°)
• F x 40: 0.23805 (6.84°)
• G x 20: 0.24588 (7.06°)
• H x 40: 0.24779 (7.12°)
• I x 32: 0.25307 (7.27°)
• J x 24: 0.25516 (7.33°)
• total 336 struts (10 kinds)1)
• strut variance 46.2%
• faces: 216 (3-sided)
• A-A-G x 20 (45.20°, 45.20°, 89.59°)
• A-D-F x 40 (46.82°, 48.96°, 84.22°)
• B-C-H x 40 (46.56°, 46.59°, 86.85°)
• C-C-G x 20 (47.01°, 47.01°, 85.99°)
• D-E-H x 40 (46.54°, 48.28°, 85.19°)
• E-E-I x 20 (47.01°, 47.01°, 85.98°)
• F-F-I x 12 (57.89°, 57.89°, 64.21°)
• I-J-J x 24 (59.46°, 60.27°, 60.27°)
• 8 kinds of faces2)
• height: 0.643 or 32.14% 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 3/8 Rhombicuboctahedron Dome Calculator

 Geodesic 3V 3/8 Rhombicuboctahedron Dome (Human is 170cm/5'7")

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

• Alath = x 40,
• Blath = x 20,
• Clath = x 40,
• Dlath = x 40,
• Elath = x 40,
• Flath = x 40,
• Glath = x 20,
• Hlath = x 40,
• Ilath = x 32,
• Jlath = x 24,
• h =

### 3V 5/8 Rhombicuboctahedron Dome

 Geodesic 3V 5/8 Rhombicuboctahedron Dome (front view)
 Geodesic 3V 5/8 Rhombicuboctahedron Dome (bird view)
• vertices/connectors: 265
• 29 x 4-way
• 8 x 5-way
• 216 x 6-way
• 12 x 7-way
•  3V 5/8 Rhombicuboctahedron Dome Construction Map
• edges/struts & bending angles (αstrut):
• A x 104: 0.17453 (5.01°)
• B x 52: 0.18015 (5.17°)
• C x 104: 0.18031 (5.17°)
• D x 104: 0.18048 (5.18°)
• E x 104: 0.18562 (5.33°)
• F x 72: 0.23805 (6.84°)
• G x 52: 0.24588 (7.06°)
• H x 104: 0.24779 (7.12°)
• I x 48: 0.25307 (7.27°)
• J x 24: 0.25516 (7.33°)
• total 768 struts (10 kinds)1)
• strut variance 46.2%
• faces: 504 (3-sided)
• A-A-G x 52 (45.20°, 45.20°, 89.59°)
• A-D-F x 104 (46.82°, 48.96°, 84.22°)
• B-C-H x 104 (46.56°, 46.59°, 86.85°)
• C-C-G x 52 (47.01°, 47.01°, 85.99°)
• D-E-H x 104 (46.54°, 48.28°, 85.19°)
• E-E-I x 52 (47.01°, 47.01°, 85.98°)
• F-F-I x 12 (57.89°, 57.89°, 64.21°)
• I-J-J x 24 (59.46°, 60.27°, 60.27°)
• 8 kinds of faces2)
• height: 1.380 or 68.99% 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 5/8 Rhombicuboctahedron Dome Calculator

 Geodesic 3V 5/8 Rhombicuboctahedron Dome (Human is 170cm/5'7")

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

• Alath = x 104,
• Blath = x 52,
• Clath = x 104,
• Dlath = x 104,
• Elath = x 104,
• Flath = x 72,
• Glath = x 52,
• Hlath = x 104,
• Ilath = x 48,
• Jlath = x 24,
• h =

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Creative Commons (CC) BY, SA, NC 2005-2017, developed, designed and written by René K. Müller,
Graphics & illustrations made with Inkscape, Tgif, Gimp, PovRay, GD.pm