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Geodesic Dome Notes & Calculator

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

Overview of Variants

Since the possibilities are so vast I have selected a couple of platonic and archimedean solids - as I previously studied in Geodesic Polyhedra - which seem reasonable for dome constructions, and created variants with up 1000 struts, and max 30 different strut lengths. So, these are the results:

Icosahedron-based Geodesic Domes

1V 2/3
25 struts (1 kind)
65 struts (2 kinds)
3V 4/9
120 struts (3 kinds)
3V 5/9
165 struts (3 kinds)
250 struts (6 kinds)
250 struts (5 kinds)
5V 7/15
350 struts (9 kinds)
5V 8/15
425 struts (9 kinds)
555 struts (9 kinds)
555 struts (10 kinds)
7V 10/21
700 struts (15 kinds)
7V 11/21
805 struts (15 kinds)
980 struts (19 kinds)
980 struts (14 kinds)

Octahedron-based Geodesic Domes

8 struts (1 kind)
28 struts (2 kinds)
60 struts (3 kinds)
104 struts (6 kinds)
104 struts (5 kinds)
L3 3/8
60 struts (5 kinds)
L3 5/8
144 struts (5 kinds)
160 struts (9 kinds)
228 struts (9 kinds)
228 struts (10 kinds)
228 struts (7 kinds)
308 struts (16 kinds)
400 struts (20 kinds)
400 struts (15 kinds)
L4 7/16
308 struts (15 kinds)
L4 9/16
488 struts (15 kinds)
504 struts (18 kinds)
504 struts (15 kinds)
620 struts (30 kinds)
620 struts (28 kinds)
620 struts (24 kinds)

Cube-based Geodesic Domes

21 struts (2 kinds)
78 struts (4 kinds)
171 struts (10 kinds)
300 struts (14 kinds)
300 struts (11 kinds)
465 struts (21 kinds)
666 struts (29 kinds)
666 struts (27 kinds)
666 struts (22 kinds)
903 struts (42 kinds)

Cuboctahedron-based Geodesic Domes

27 struts (2 kinds)
102 struts (5 kinds)
225 struts (10 kinds)
396 struts (18 kinds)
396 struts (15 kinds)
615 struts (28 kinds)
882 struts (36 kinds)
882 struts (34 kinds)
882 struts (28 kinds)

Truncated Octahedron-based Geodesic Domes

60 struts (4 kinds)
228 struts (7 kinds)
504 struts (15 kinds)
888 struts (25 kinds)
888 struts (19 kinds)

Rhombicuboctahedron-based Geodesic Domes

1V 3/8
40 struts (2 kinds)
1V 5/8
88 struts (2 kinds)
2V 3/8
152 struts (5 kinds)
2V 5/8
344 struts (5 kinds)
3V 3/8
336 struts (10 kinds)
3V 5/8
768 struts (10 kinds)

Numerical Overview

In order to assist to overview the options further, consider to study following table carefully, it's worth it.
  • Types: strut types or amount of different strut lengths, the lesser the better
  • Variance: strut variance, difference between longest and shortest strut, the lesser the better (more even triangles)
  • The L-variant is better than the comperable V-variant, in sense of strut variance and amount of different struts.

The list is sorted by the amount of struts, so you can choose how complex the dome should become.

Octahedron 1V54810%
Cube 1V101221225.6%
Icosahedron 1V 2/311152510%
Cuboctahedron 1V121627230.7%
Octahedron 2V131628230.7%
Rhombicuboctahedron 1V 3/8172440236.5%
Octahedron 3V253660346.1%
Octahedron L3 3/8253660548.0%
Truncated Octahedron 1V253660446.1%
Icosahedron 2V264065213.1%
Cube 2V314878437.4%
Rhombicuboctahedron 1V 5/8335688236.5%
Cuboctahedron 2V3964102548.0%
Octahedron 4V4164104680.2%
Octahedron L34164104548.0%
Icosahedron 3V 4/94675120318.3%
Octahedron L3 5/85392144548.0%
Rhombicuboctahedron 2V 3/85796152544.9%
Octahedron 5V61100160992.9%
Icosahedron 3V 5/961105165318.3%
Cube 3V641081711050.1%
Cuboctahedron 3V821442251051.2%
Octahedron 2V.3V851442281051.2%
Octahedron 3V.2V85144228753.5%
Octahedron 6V85144228995.4%
Truncated Octahedron 2V85144228753.5%
Icosahedron 4V91160250628.3%
Icosahedron L391160250517.8%
Cube 4V1091923001456.9%
Cube L31091923001141.5%
Octahedron 7V11319630816107.9%
Octahedron L4 7/161131963081553.8%
Rhombicuboctahedron 3V 3/81212163361046.2%
Rhombicuboctahedron 2V 5/8121224344544.9%
Icosahedron 5V 7/15126225350932.1%
Cuboctahedron 4V1412563961860.2%
Cuboctahedron L31412563961553.8%
Octahedron 8V14525640020112.6%
Octahedron L41452564001553.8%
Icosahedron 5V 8/15151275425932.1%
Cube 5V1663004652158.9%
Octahedron L4 9/161733164881553.8%
Octahedron 3V.3V1813245041555.1%
Octahedron 9V18132450418113.2%
Truncated Octahedron 3V1813245041555.1%
Icosahedron 2V.3V1963605551018.9%
Icosahedron 6V196360555933.2%
Cuboctahedron 5V2164006152862.8%
Octahedron 10V22140062030119.3%
Octahedron 2V.5V2214006202862.8%
Octahedron 5V.2V2214006202498.1%
Cube 2V.3V2354326662744.4%
Cube 3V.2V2354326662252.3%
Cube 6V2354326662963.1%
Icosahedron 7V 10/212464557001536.5%
Rhombicuboctahedron 3V 5/82655047681046.2%
Icosahedron 7V 11/212815258051536.5%
Cuboctahedron 2V.3V3075768823454.6%
Cuboctahedron 3V.2V3075768822853.6%
Cuboctahedron 6V3075768823663.1%
Truncated Octahedron 4V3135768882558.0%
Truncated Octahedron L33135768881955.6%
Cube 7V3165889034264.0%
Icosahedron 8V3416409801937.7%
Icosahedron L43416409801419.0%

Note: all strut lengths have been sorted by 1/10'000th or +/-0.00005 exact

Amount of Struts vs. Strut Variance

Amount of Struts vs Strut Variance

Note: Click on the graphic, and click "full scale" or scroll down, and press "Print" link at the bottom of the page.

In the graphic you actually see why the Icosahedron looks best, it has the least strut variance, which is an indication of more or less similiar triangles through the entire structure. And as realized before, the Ln variants additionally provide better results than the nV counterparts. The Octahedron variants, on the other hand, have high strut variances as seen above.

I personally like the Cuboctahedron variants, but they have more strut variance than the Icosahedron, in other words, at the end it's your personal choice and favours which lead you to choose a polyhedral geodesic dome variant, the list above and this graphic may just assist you in your overall considerations.

Amount of Struts vs. Strut Types

Amount of Struts vs Strut Types

Note: Click on the graphic, and click "full scale" or scroll down, and press "Print" link at the bottom of the page.

In this graphic above the Icosahedron does quite well, with low amount of strut lengths, and again there also Ln provides better results than the nV variants. Interesting also the Rhombicuboctahedron comes with even lower amount of strut lengths than the Icosahedron.

The amount of different strut lengths has a direct impact on the construction, the lesser the amount of different strut lengths the better - as you have less cut optimization to calculate and therefore less waste to expect. So it's certainly an aspect to value when choosing a variant to build. As mentioned already, the struts are sorted in 1/10'000th and you may notice on the details of the variants on the following pages, that you can sum together near the same length struts in case you compose a smaller dome, e.g. < 8m or so. For large scale domes it may be crucial to remain as precise as you can and I leave it up to you to treat certain struts of alike length as the same or not.

How to Use the Notes

I recommend you choose a dome variant you like to construct ...

  • look what base appeals to you most, as a first step disregard any other kind of argument
  • check the amount of different strut types, amount of struts itself - and become aware of the overall overhead (e.g. whether building an edgy 1V or 2V, or a smoother 4V or higher with far more struts)
  • how temporary shall the dome become? e.g. a 4V Icosahedron Dome skeleton with 250 struts I raise as a single person in 4-5 hours
  • the higher the subdivision the more exact you need to work, little difference will distort and weaken the construction

... and once you decided which variant(s) you consider more closely, then ...

  • print out that particular page of the dome variant with your settings - as the notes are subject of change, and may later not be available or available with a different notion
  • make a 10:1 or 20:1 sized model (e.g. 6m full scale dome, create a 0.60m model), it gives you a sense of the overhead, amount of struts, connectors etc
  • think about the kind of strut and its connectors you gonna use, e.g. metal, wood, plastic etc and make tests with the actual material, create a 5-way or 6-way connector construction and test its stability

Strut Options & Notion

For each dome variant there is a small calculator, to calculate diameter to the different struts, additionally the lhole can be entered.

Notion of a dome strut (A vs Alath) & lhole

Wooden Strut with Flat Connector

Timber based strut, metallic plate as connector, rather cheap with average labour.

Wooden Strut with Pipe Hub

Pipe Hub Closeup (courtesy by Michal Wielgus)
Timber based strut, with a steel pipe as connector, moderately cheap with average labour, prefereable for more lasting constructions.

Michal Wielgus who sent me the photo used this hub for a 3V 5/9 icosahedron based geodesic dome, he calculated the holes in the pipe for the 5- and 6-way simply by dpipe π / 5 or 6, which was exact enough he said. For the 4-way connectors on the base require the same calculation as the 6-way, and leaving the two bottom directed connectors empty.

To be more precise each hub would require the angles according, A, B, C, etc ...

Following photos were kindly shared by Haan from Korea , who made a 2V icosahedron based geodesic dome, with 7m diameter, 11cm diameter steel pipe:

Haan's Dome (2v icosa, 7m diameter): 10cm thick wooden struts
2008/04/18 10:02
Haan's Dome (2v icosa, 7m diameter): drilling tool & anchor bolt
2008/04/18 13:53
Haan's Dome (2v icosa, 7m diameter): hub detail with steel pipe
2008/04/18 09:53
Haan's Dome (2v icosa, 7m diameter): hub detail with steel pipe
2008/04/18 09:58
Haan's Dome (2v icosa, 7m diameter): complete setup
2008/04/18 10:07

Pipe/Tube based Strut

Steel/aluminium/etc tube or pipe, and ends squeezed, also rather cheap with little labour. This options is very popular using conduits. Desert Domes: Conduit Domes Tips has some useful information on this option.

Bamboo Strut

Bamboo (different diameters) with soft-pipe as connector, very cheap but increased labour, and only suitable for small domes < 4-5m diameter and lower sub-division frequencies (e.g. 2V, 3V), e.g. for play domes.

Cut pointing ends
2007/02/14 12:47
4V Geodesic Dome (6.33m) (Closeup 1)
2007/03/16 16:53
Connector (Closeup)
2007/03/17 11:39
For temporary setups it's obvious to use struts which built-in connectors, e.g. endings which operate as connector. , german bamboo expert, has special connectors for more stable and large domes, with a large dome construction example.

Bending the Strut-Endings

Strut Angle αstrut
The angle αstrut can be calculated with a bit of trigonometry. α shall be the angle between the tangent (green in the illustration) and the strut:

  • h = √(r2 - lstrut2/4)
  • h = r * cos(γ/2)
  • γ = 2 acos(h/r) = 2 acos(√(r2 - lstrut2/4) / r)
  • α = 90-&beta
  • β = (180-γ)/2
  • α = 90 - (180-γ)/2 = γ/2
  • α = αstrut = acos(√(r2 - lstrut2/4) / r)

finally α is also the angle to bend the endings αstrut. The following pages with all the variant details provides you for each strut the corresponding αstrut as well.

Detailed Calculation of the Faces or Hub/Strut Angles

Hub/Strut Angles
In case you require the exact angles for the faces, e.g. for a pipe hub or for composing individual faces, according the SSS Theorem , known all 3 sides of a triangle searching the angles, we can calculate all angles:
  • α = acos((B2+C2-A2)/(2 B C))
  • β = acos((A2+C2-B2)/(2 A C))
  • γ = acos((A2+B2-C2)/(2 A B))

For now, all the faces are sorted and the angles are given for all variants. I may add later a list of hubs and their angles, but you can sort them your based on the construction maps.

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