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Geodesic Polyhedra

written by Rene K. Mueller, Copyright (c) 2007, last updated Sun, October 28, 2012

Geodesic Cuboctahedron

Cuboctahedron


Cuboctahedron
  • Uniform Polyhedron: U7
  • Archimedean Solid: A1
  • 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
  • Vertices/Connectors: 12 (4-way)
  • Faces: 14
    • 8 x 3-sided
    • 6 x 4-sided
  • Edges/Struts:
    • A x 24: 1.00000

Geodesic Cuboctahedron L1


Geodesic Cuboctahedron L1
  • Vertices/Connectors: 18
    • 6 x 4-way
    • 12 x 6-way
  • Faces: 32 (3-sided)
  • Edges/Struts:
    • A x 24: 0.76537
    • B x 24: 1.00000
    • total 48 struts (2 kinds)
    • strut variance 30.7%


Geodesic Cuboctahedron Dome L1


Geodesic Cuboctahedron Dome L1

Geodesic Cuboctahedron Dome L1
  • Vertices/Connectors: 12
    • 9 x 4-way
    • 3 x 6-way
  • Faces: 16 (3-sided)

  • Cuboctahedron Dome L1 Construction Map
  • Edges/Struts:
    • A x 12: 0.76537
    • B x 15: 1.00000
    • total 27 struts (2 kinds)
    • strut variance 30.7%

d= A= x 12, B= x 15,

Geodesic Cuboctahedron L2


Geodesic Cuboctahedron L2
  • Vertices/Connectors: 66
    • 6 x 4-way
    • 60 x 6-way
  • Faces: 128 (3-sided)
  • Edges/Struts:
    • A x 48: 0.39018
    • B x 48: 0.42291
    • C x 48: 0.51764
    • D x 24: 0.54120
    • E x 24: 0.57735
    • total 192 struts (5 kinds)
    • strut variance 47.9%


Geodesic Cuboctahedron Dome L2


Geodesic Cuboctahedron Dome L2

Geodesic Cuboctahedron Dome L2
  • Vertices/Connectors: 39
    • 15 x 4-way
    • 24 x 6-way
  • Faces: 64 (3-sided)

  • Cuboctahedron Dome L2 Construction Map
  • Edges/Struts:
    • A x 24: 0.39018
    • B x 24: 0.42291
    • C x 30: 0.51764
    • D x 12: 0.54120
    • E x 12: 0.57735
    • total 102 struts (5 kinds)
    • strut variance 47.9%

d= A= x 24, B= x 24, C= x 30, D= x 12, E= x 12,
With a straight hemisphere line it is suitable, yet, the top for a skylight is a triangle, which means it's horizontal and water (rain) cannot flow down, so likely a slight tilt is required. It provides three square windows on the side.


Geodesic Cuboctahedron Dome L2 Model (Closeup)
I made following model with 4mm thick and 40cm long bamboo, and 6mm/4mm clear PVC pipe as connector. A quick check how to optimize the struts:

  • 12 x (A + E) = 1.0
  • 12 x (A + D)
  • 24 x (B + C)
  • 6 x C
  • total 54 laths



Geodesic Cuboctahedron Dome L2 Model (46cm diameter)

  • (A+E) = 1.0, therefore
    • A = 0.403, B = 0.436, C = 0.534, D = 0.559, E = 0.596
  • (Alath+Elath) = ~20cm (using 27 x 40cm bamboo equal 54 x 20cm bamboo)
    • dhole (distance strut to junction hole) = 0.75cm, e.g. A = Alath + 2 * dhole; therefore Alath = 7.5cm, Blath = 8.3cm, Clath = 10.6cm, Dlath = 11.1cm, Elath = 11.9cm

I first calculated A+E (instead of Alath+Elath) which made the struts 0.5cm longer A and B, for C, D and E the error was neglectable, but the resulting dome looked strange, until I realized those 0.75cm do really matter at that scale of the model. So I recommend considering the distance the connector itself requires as well, even you are doing a model at 20cm scale or so, it matters.

Geodesic Cuboctahedron L2T


Geodesic Cuboctahedron L2T
  • Vertices/Connectors: 98
    • 48 x 4-way
    • 32 x 6-way
    • 6 x 8-way
    • 12 x 12-way
  • Faces: 192 (3-sided)
  • Edges/Struts:
    • A x 24: 0.22232
    • B x 48: 0.29289
    • C x 24: 0.33820
    • D x 72: 0.39018
    • E x 48: 0.51764
    • F x 48: 0.55745
    • G x 24: 0.60581
    • total 288 struts (7 kinds)
    • strut variance 173.0%


Geodesic Cuboctahedron Dome L2T


Geodesic Cuboctahedron Dome L2T

Geodesic Cuboctahedron Dome L2T
  • Vertices/Connectors: 57
    • 4 x 3-way
    • 30 x 4-way
    • 2 x 5-way
    • 12 x 6-way
    • 2 x 7-way
    • 2 x 8-way
    • 5 x 12-way
  • Faces: 96 (3-sided)

  • Cuboctahedron Dome L2T Construction Map
  • Edges/Struts:
    • A x 14: 0.22232
    • B x 24: 0.29289
    • C x 14: 0.33820
    • D x 38: 0.39018
    • E x 24: 0.51764
    • F x 24: 0.55745
    • G x 14: 0.60581
    • total 152 struts (7 kinds)
    • strut variance 173.0%

d= A= x 14, B= x 24, C= x 14, D= x 38, E= x 24, F= x 24, G= x 14,


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