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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 Cuboctahedron

 Cuboctahedron
• Uniform Polyhedron: U7
• Archimedean Solid: A1
• Vertices: 12
• Edges: 24
• Faces: 14
• Wythoff symbol: 2|3 4
• Symmetry Group: octahedral
• Vertex Configuration: {3, 4, 3, 4}
• Dual: rhombic dodecahedron
• V: s3 * 5/3 * √2
• A: s2 * (6 + 2 * √3)
• rinner: s * 3/4
• router: s

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

Preparing the Cuboctahedron

Cuboctahedron L0/0V (original)
Cuboctahedron L1/1V flat (only triangles & flat)
Cuboctahedron L1/1V (only triangles & normalized)
The 6 squares are centerpoint triangulated for further processing.

1V/L1 Cuboctahedron Dome

 Geodesic 1V Cuboctahedron Dome (front view)
 Geodesic 1V Cuboctahedron Dome (bird view)
• vertices/connectors: 12
• 9 x 4-way
• 3 x 6-way
•  1V Cuboctahedron Dome Construction Map
• edges/struts & bending angles (αstrut):
• A x 12: 0.76537 (22.50°)
• B x 15: 1.00000 (30.00°)
• total 27 struts (2 kinds)1)
• strut variance 30.7%
• faces: 16 (3-sided)
• A-A-B x 12 (49.21°, 49.21°, 81.57°)
• B-B-B x 4 (60.00°, 60.00°, 60.00°)
• 2 kinds of faces2)
• height: 0.817 or 40.83% 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 Cuboctahedron Dome Calculator

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

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

• Alath = x 12,
• Blath = x 15,
• h =

Cut optimization:
• normalize Alath+Blath = 1.0, therefore Alath+Blath =
• cuts:
• 12 x (Alath+Blath):
• 3 x (Blath):
• total 15 laths to cut

2V/L2 Cuboctahedron Dome

 Geodesic 2V Cuboctahedron Dome (front view)
 Geodesic 2V Cuboctahedron Dome (bird view)
• vertices/connectors: 39
• 15 x 4-way
• 24 x 6-way
•  2V Cuboctahedron Dome Construction Map
• edges/struts & bending angles (αstrut):
• A x 24: 0.39018 (11.25°)
• B x 24: 0.42291 (12.21°)
• C x 30: 0.51764 (15.00°)
• D x 12: 0.54120 (15.70°)
• E x 12: 0.57735 (16.78°)
• total 102 struts (5 kinds)1)
• strut variance 48.0%
• faces: 64 (3-sided)
• A-A-D x 12 (46.09°, 46.09°, 87.81°)
• A-B-C x 24 (47.72°, 53.31°, 78.96°)
• B-B-D x 12 (50.22°, 50.22°, 79.56°)
• C-C-E x 12 (56.10°, 56.10°, 67.80°)
• E-E-E x 4 (60.00°, 60.00°, 60.00°)
• 5 kinds of faces2)
• height: 0.943 or 47.15% 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 Cuboctahedron Dome Calculator

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

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

• Alath = x 24,
• Blath = x 24,
• Clath = x 30,
• Dlath = x 12,
• Elath = x 12,
• h =

Cut optimization:
• normalize Alath+Elath = 1.0, therefore Alath+Elath =
• cuts:
• 12 x (Alath+Elath):
• 12 x (Alath+Dlath):
• 24 x (Blath+Clath):
• 6 x (Clath):
• total 54 laths to cut

There might be a better optimization available, but that's what I came up at the first sight.

3V Cuboctahedron Dome

 Geodesic 3V Cuboctahedron Dome (front view)
 Geodesic 3V Cuboctahedron Dome (bird view)
• vertices/connectors: 82
• 21 x 4-way
• 61 x 6-way
•  3V Cuboctahedron Dome Construction Map
• edges/struts & bending angles (αstrut):
• A x 24: 0.25480 (7.32°)
• B x 12: 0.27355 (7.86°)
• C x 24: 0.27468 (7.89°)
• D x 24: 0.27569 (7.92°)
• E x 24: 0.29082 (8.36°)
• F x 30: 0.33193 (9.55°)
• G x 12: 0.35741 (10.29°)
• H x 24: 0.36346 (10.47°)
• I x 27: 0.37796 (10.89°)
• J x 24: 0.38517 (11.10°)
• total 225 struts (10 kinds)1)
• strut variance 51.2%
• faces: 144 (3-sided)
• A-A-G x 12 (45.47°, 45.47°, 89.07°)
• A-D-F x 24 (48.51°, 54.14°, 77.35°)
• B-C-H x 24 (48.34°, 48.60°, 83.05°)
• C-C-G x 12 (49.42°, 49.42°, 81.16°)
• D-E-H x 24 (48.28°, 51.94°, 79.78°)
• E-E-I x 12 (49.46°, 49.46°, 81.07°)
• F-F-I x 12 (55.29°, 55.29°, 69.42°)
• I-J-J x 24 (58.77°, 60.62°, 60.62°)
• 8 kinds of faces2)
• 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

3V Cuboctahedron Dome Calculator

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

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

• Alath = x 24,
• Blath = x 12,
• Clath = x 24,
• Dlath = x 24,
• Elath = x 24,
• Flath = x 30,
• Glath = x 12,
• Hlath = x 24,
• Ilath = x 27,
• Jlath = x 24,
• h =

4V Cuboctahedron Dome

 Geodesic 4V Cuboctahedron Dome (front view)
 Geodesic 4V Cuboctahedron Dome (bird view)
• vertices/connectors: 141
• 27 x 4-way
• 114 x 6-way
•  4V Cuboctahedron Dome Construction Map
• edges/struts & bending angles (αstrut):
• A x 24: 0.18820 (5.40°)
• B x 24: 0.20036 (5.75°)
• C x 24: 0.20240 (5.81°)
• D x 24: 0.20386 (5.85°)
• E x 24: 0.20571 (5.90°)
• F x 24: 0.21338 (6.12°)
• G x 24: 0.21629 (6.21°)
• H x 24: 0.21961 (6.30°)
• I x 30: 0.24197 (6.95°)
• J x 24: 0.26458 (7.60°)
• K x 12: 0.26497 (7.61°)
• L x 24: 0.27316 (7.85°)
• M x 24: 0.27685 (7.96°)
• N x 12: 0.27735 (7.97°)
• O x 30: 0.28011 (8.05°)
• P x 12: 0.28580 (8.22°)
• Q x 24: 0.29180 (8.39°)
• R x 12: 0.30151 (8.67°)
• total 396 struts (18 kinds)1)
• strut variance 60.2%
• faces: 256 (3-sided)
• A-A-K x 12 (45.25°, 45.25°, 89.50°)
• A-C-I x 24 (49.12°, 54.41°, 76.47°)
• B-B-K x 12 (48.61°, 48.61°, 82.78°)
• B-D-L x 24 (46.95°, 48.03°, 85.02°)
• C-F-J x 24 (48.66°, 52.34°, 78.99°)
• D-E-J x 24 (49.46°, 50.06°, 80.48°)
• E-G-L x 24 (47.98°, 51.37°, 80.65°)
• F-H-O x 24 (48.73°, 50.67°, 80.60°)
• G-G-P x 12 (48.65°, 48.65°, 82.70°)
• H-H-P x 12 (49.40°, 49.40°, 81.19°)
• I-I-N x 12 (55.04°, 55.04°, 69.91°)
• M-M-N x 12 (59.94°, 59.94°, 60.12°)
• M-O-Q x 24 (57.85°, 58.95°, 63.19°)
• Q-Q-R x 12 (58.89°, 58.89°, 62.21°)
• R-R-R x 4 (60.00°, 60.00°, 60.00°)
• 15 kinds of faces2)
• height: 0.985 or 49.24% 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 Cuboctahedron Dome Calculator

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

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

• Alath = x 24,
• Blath = x 24,
• Clath = x 24,
• Dlath = x 24,
• Elath = x 24,
• Flath = x 24,
• Glath = x 24,
• Hlath = x 24,
• Ilath = x 30,
• Jlath = x 24,
• Klath = x 12,
• Llath = x 24,
• Mlath = x 24,
• Nlath = x 12,
• Olath = x 30,
• Plath = x 12,
• Qlath = x 24,
• Rlath = x 12,
• h =

L3 Cuboctahedron Dome

 Geodesic L3 Cuboctahedron Dome (front view)
 Geodesic L3 Cuboctahedron Dome (bird view)
• vertices/connectors: 141
• 27 x 4-way
• 114 x 6-way
•  L3 Cuboctahedron Dome Construction Map
• edges/struts & bending angles (αstrut):
• A x 48: 0.19603 (5.62°)
• B x 24: 0.19991 (5.74°)
• C x 24: 0.20044 (5.75°)
• D x 48: 0.21266 (6.10°)
• E x 24: 0.21671 (6.22°)
• F x 24: 0.21746 (6.24°)
• G x 60: 0.26105 (7.50°)
• H x 24: 0.26433 (7.59°)
• I x 24: 0.26899 (7.73°)
• J x 24: 0.27316 (7.85°)
• K x 12: 0.27590 (7.93°)
• L x 12: 0.27686 (7.96°)
• M x 24: 0.29180 (8.39°)
• N x 12: 0.29886 (8.59°)
• O x 12: 0.30151 (8.67°)
• total 396 struts (15 kinds)1)
• strut variance 53.8%
• faces: 256 (3-sided)
• A-A-K x 12 (45.27°, 45.27°, 89.47°)
• A-B-J x 24 (45.78°, 46.96°, 87.27°)
• A-D-H x 24 (47.00°, 52.53°, 80.47°)
• A-E-G x 24 (47.32°, 54.37°, 78.32°)
• B-B-K x 12 (46.36°, 46.36°, 87.28°)
• C-D-G x 24 (48.74°, 52.92°, 78.34°)
• C-E-H x 24 (48.00°, 53.47°, 78.53°)
• D-D-L x 12 (49.39°, 49.39°, 81.22°)
• D-F-J x 24 (49.80°, 51.36°, 78.84°)
• F-F-L x 12 (50.46°, 50.46°, 79.07°)
• G-G-N x 12 (55.08°, 55.08°, 69.83°)
• G-I-M x 24 (55.32°, 57.91°, 66.78°)
• I-I-N x 12 (56.25°, 56.25°, 67.50°)
• M-M-O x 12 (58.89°, 58.89°, 62.21°)
• O-O-O x 4 (60.00°, 60.00°, 60.00°)
• 15 kinds of faces2)
• height: 0.985 or 49.24% 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 Cuboctahedron Dome Calculator

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

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

• Alath = x 48,
• Blath = x 24,
• Clath = x 24,
• Dlath = x 48,
• Elath = x 24,
• Flath = x 24,
• Glath = x 60,
• Hlath = x 24,
• Ilath = x 24,
• Jlath = x 24,
• Klath = x 12,
• Llath = x 12,
• Mlath = x 24,
• Nlath = x 12,
• Olath = x 12,
• 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