last updated Fri, February 25, 2022

Geodesic Polyhedra

written by Rene K. Mueller, Copyright (c) 2007, last updated Fri, February 25, 2022

Introduction

After studying the geodesic dome (often direct associated to Buckminster Fuller but he wasn't the first), which essentially is derived from a Icosahedron, I thought to check the possibility to triangulate other platonic and archimedean solids. Based on the information I gathered from regular & semi-regular polyhedra I made this overview.

Geodesic Procedure


Triangle Divisions (Class 1/Alternate)

Normalizing to Sphere
The L0 is the original solid or face (n-sided), the L1 I created by centerpoint triangulation of the larger polygons (4, 5, 6, 8 or 10 sided polygons) until there are only triangles and normalized, the L2 is then the triangulated (class 1 or alternate method) & normalized version of L1.

Note: The common nV or nν notion introduced by Fuller looks alike, but those are derived from the original triangles, whereas L2 is derived from the already geodesized L1. I will later discuss the advantages and disadvantages of nV vs Ln in more details.

The Ln vertices are normalized (spherisized or spherical projected) to 1, so r = 1 or d = 2.

Triangulation Methods/Classes


Triangulation Methods
There are multiple ways to triangulate a triangle,

for sake of completeness.

Yet, for now I only focus on the class 1 / alternate and class 2 / triacon methods or classes. A more detailed overview of classes and methods is covered in Geodesic Math by Jay Salsburg.

Class 1 / Alternate Subdivision


Icosahedron 1V/L1

Icosahedron L2 (pre-normalized)

Icosahedron L2 or 2V

The class 1 or alternate subdivision provides a very even distribution of the triangles, alike the original triangle.

Class 2 / Triacon Subdivision


Icosahedron 1V/L1

Icosahedron L2T (pre-normalized)

Icosahedron L2T or 2V Triacon

The class 2 or triacon subdivision provides more options to cut the resulting sphere into a dome, yet, adds also one strut per triangle to an existing junction. E.g. a 5-way connector (with 5 triangles) triaconized results in 10-way connector, which in real-life poses a challange to implement, e.g. with a complex and rather large hub.

For triacon subdivision I add 'T' to the existing notion, e.g. L1T is a L1 with triacon subdivision.

As you notice, depending which kind of subdivision is used, the possible cut for a dome variant is completely different.

Where suitable, I also rendered the dome / hemisphere option. The overview here has become very comprehensive already within a short time since I started worked on it, and I plan to extend it further.

Please Note:

Geodesic Tetrahedron

Tetrahedron


Tetrahedron
I tried to "geodesize" using class 1 / alternate method the Tetrahedron into a sphere but you get a distorted sphere, no dome possible. And when the Tetrahedron solely is used as direct dome too many struts occur, L1 4 strut lengths and L2 already 12 strut lengths - not suitable. So, I won't even list all the details.

Geodesic Tetrahedron L2
Geodesic Tetrahedron L3
Surprisingly when I "triaconized" the Tetrahedron, it turned out to be quite more suitable:

Geodesic Tetrahedron L2T


Geodesic Tetrahedron L2T


Geodesic Tetrahedron Dome L2T


Geodesic Tetrahedron Dome L2T

Geodesic Tetrahedron Dome L2T

d= A= x 14, B= x 7,
It looks like a prepared (reduced to triangles using centerpoint triangulation) cube at the first sight.

Geodesic Tetrahedron L3T


Geodesic Tetrahedron L3T


Geodesic Tetrahedron Dome L3T


Geodesic Tetrahedron Dome L3T

Geodesic Tetrahedron Dome L3T

d= A= x 12, B= x 24, C= x 28, D= x 12, E= x 14, F= x 24,

Geodesic Truncated Tetrahedron

Truncated Tetrahedron


Truncated Tetrahedron

Geodesic Truncated Tetrahedron L1


Geodesic Truncated Tetrahedron L1

Geodesic Truncated Tetrahedron L2


Geodesic Truncated Tetrahedron L2
No base line, so likely not suitable for dome variant.

Geodesic Truncated Tetrahedron L2T


Geodesic Truncated Tetrahedron L2T


Geodesic Truncated Tetrahedron Dome L2T


Geodesic Truncated Tetrahedron Dome L2T

Geodesic Truncated Tetrahedron Dome L2T

d= A= x 7, B= x 24, C= x 14, D= x 19, E= x 24, F= x 7, G= x 24, H= x 14,
Using triacon subdivision, all of the sudden a base line appears - so I will explore the dome variant as well.

Geodesic Octahedron

Octahedron


Octahedron
Octahedron looks very suitable for a dome construction, straight hemisphere line.

The L1 is the same as the original aka L0, since there are just triangles, so jump direct on the L2.

Geodesic Octahedron L2


Geodesic Octahedron L2


Geodesic Octahedron Dome L2


Geodesic Octahedron Dome L2

Geodesic Octahedron Dome L2

d= A= x 16, B= x 12,

Geodesic Octahedron L3


Geodesic Octahedron L3


Geodesic Octahedron Dome L3


Geodesic Octahedron Dome L3

Geodesic Octahedron Dome L3

d= A= x 32, B= x 24, C= x 24, D= x 12, E= x 12,
The L3 has even a square skylight, but likely it needs to be composed by triangles so the rain flows down from the top. I likely build a L3 as a model to explore the details.

Geodesic Octahedron L2T


Geodesic Octahedron L2T


Geodesic Octahedron Dome L2T


Geodesic Octahedron Dome L2T

Geodesic Octahedron Dome L2T

d= A= x 12, B= x 16, C= x 12,

Geodesic Octahedron L3T


Geodesic Octahedron L3T


Geodesic Octahedron Dome L3T


Geodesic Octahedron Dome L3T

Geodesic Octahedron Dome L3T

d= A= x 24, B= x 24, C= x 24, D= x 24, E= x 24, F= x 32, G= x 24, H= x 24, I= x 24,

Geodesic Cube

Cube


Cube

Geodesic Cube L1


Geodesic Cube L1


Geodesic Cube Dome L1


Geodesic Cube Dome L1

Geodesic Cube Dome L1

d= A= x 14, B= x 7,

Geodesic Cube L2


Geodesic Cube L2


Geodesic Cube Dome L2


Geodesic Cube Dome L2

Geodesic Cube Dome L2

d= A= x 28, B= x 24, C= x 14, D= x 12,

Geodesic Cube L2T


Geodesic Cube L2T


Geodesic Cube Dome L2T


Geodesic Cube Dome L2T

Geodesic Cube Dome L2T

d= A= x 12, B= x 24, C= x 28, D= x 12, E= x 14, F= x 24,
The L2 looks good to explore this form as well, straight hemisphere line makes it a good 4/8 dome.

Geodesic Cuboctahedron

Cuboctahedron


Cuboctahedron

Geodesic Cuboctahedron L1


Geodesic Cuboctahedron L1


Geodesic Cuboctahedron Dome L1


Geodesic Cuboctahedron Dome L1

Geodesic Cuboctahedron Dome L1

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

Geodesic Cuboctahedron L2


Geodesic Cuboctahedron L2


Geodesic Cuboctahedron Dome L2


Geodesic Cuboctahedron Dome L2

Geodesic Cuboctahedron Dome L2

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:



Geodesic Cuboctahedron Dome L2 Model (46cm diameter)

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


Geodesic Cuboctahedron Dome L2T


Geodesic Cuboctahedron Dome L2T

Geodesic Cuboctahedron Dome L2T

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

Geodesic Truncated Octahedron

Truncated Octahedron


Truncated Octahedron

Geodesic Truncated Octahedron L1


Geodesic Truncated Octahedron L1


Geodesic Truncated Octahedron Dome L1


Geodesic Truncated Octahedron Dome L1

Geodesic Truncated Octahedron Dome L1

d= A= x 16, B= x 20, C= x 24,

Geodesic Truncated Octahedron L2


Geodesic Truncated Octahedron L2


Geodesic Truncated Octahedron Dome L2


Geodesic Truncated Octahedron Dome L2

Geodesic Truncated Octahedron Dome L2

d= A= x 32, B= x 24, C= x 40, D= x 12, E= x 24, F= x 48, G= x 48,

Geodesic Truncated Octahedron L2T


Geodesic Truncated Octahedron L2T


Geodesic Truncated Octahedron Dome L2T


Geodesic Truncated Octahedron Dome L2T

Geodesic Truncated Octahedron Dome L2T

d= A= x 12, B= x 24, C= x 48, D= x 24, E= x 12, F= x 32, G= x 40, H= x 24, I= x 48, J= x 48, K= x 24,

Geodesic Truncated Cube

Truncated Cube


Truncated Cube

Geodesic Truncated Cube L1


Geodesic Truncated Cube L1

Geodesic Truncated Cube L2


Geodesic Truncated Cube L2
I actually like this form, in particular the L1. The L2 doesn't provide a straight hemisphere line, unfortunately.

Geodesic Truncated Cube L2T


Geodesic Truncated Cube L2T


Geodesic Truncated Cube Dome L2T


Geodesic Truncated Cube Dome L2T

Geodesic Truncated Cube Dome L2T

d= A= x 14, B= x 48, C= x 26, D= x 38, E= x 14, F= x 48, G= x 48, H= x 26,

Geodesic Rhombicuboctahedron

Rhombicuboctahedron


Rhombicuboctahedron

Geodesic Rhombicuboctahedron L1


Geodesic Rhombicuboctahedron L1

Geodesic Rhombicuboctahedron L2


Geodesic Rhombicuboctahedron L2
Two straight hemispherical lines, at 1/3 or 2/3 height approximately.

Geodesic Rhombicuboctahedron L2T


Geodesic Rhombicuboctahedron L2T


Geodesic Rhombicuboctahedron Dome L2T


Geodesic Rhombicuboctahedron Dome L2T

Geodesic Rhombicuboctahedron Dome L2T

d= A= x 38, B= x 72, C= x 14, D= x 38, E= x 76, F= x 48, G= x 72, H= x 14,

Geodesic Truncated Cuboctahedron

Truncated Cuboctahedron


Truncated Cuboctahedron

Geodesic Truncated Cuboctahedron L1


Geodesic Truncated Cuboctahedron L1
The L1 looks very good, the Truncate Cuboctahedron has 8-, 6-, and 4-sided polygons, once centerpoint triangulated make up a beautiful shape. Unfortunately no straight hemisphere line to derive a dome from.

Geodesic Truncated Cuboctahedron L2


Geodesic Truncated Cuboctahedron L2

Geodesic Truncated Cuboctahedron L2T


Geodesic Truncated Cuboctahedron L2T

Geodesic Snub Cube

Snub Cube


Snub Cube

Geodesic Snub Cube L1


Geodesic Snub Cube L1

Geodesic Snub Cube L2


Geodesic Snub Cube L2

Geodesic Snub Cube L2T


Geodesic Snub Cube L2T
At the first sight it looks as if there is a horizontal base, yet, it's slightly distorted and unsuitable therefore for a dome consideration, unfortunately.

Geodesic Icosahedron

Icosahedron


Icosahedron

Let's skip the L1 as it's the same as L0 since the Icosahedron is composed already with triangles only.

Geodesic Icosahedron L2


Geodesic Icosahedron L2


Geodesic Icosahedron Dome L2


Geodesic Icosahedron Dome L2

Geodesic Icosahedron Dome L2

d= A= x 30, B= x 35,

Geodesic Icosahedron L3


Geodesic Icosahedron L3


Geodesic Icosahedron Dome L3


Geodesic Icosahedron Dome L3

Geodesic Icosahedron Dome L3

d= A= x 60, B= x 60, C= x 70, D= x 30, E= x 30,
When compared to other platonic and archimedean solids, it's apparent that it's the best source to triangulate from and get a firmly even distribution of vertices.

For more details look at my Geodesic Dome Notes, where I cover 2V, 3V and 4V in more details.

L2 4V Fuller
A 0.27590 A 0.25318
B 0.28547 C 0.29453
C 0.31287 D 0.31287
D 0.32124 B 0.29524
E 0.32492 E 0.32492
A 0.27590 F 0.29859
17% variance 28% variance
I noticed that the strut ratio at DesertDomes.com for the 4V (which is supposed to use Fuller's ratios), and mine differ. After a close checking, I realized the 4V is direct derived from the original Icosahedron, whereas I created the L2/2V, and created from that the L3.

Now, based on that result I keep the L-notion and describe this more thorough, the difference and maybe advantages of Ln vs nV:


Geodesic Icosahedron L2T


Geodesic Icosahedron L2T


Geodesic Icosahedron Dome L2T


Geodesic Icosahedron Dome L2T

Geodesic Icosahedron Dome L2T

d= A= x 32, B= x 32, C= x 32,

Geodesic Icosahedron L3T


Geodesic Icosahedron L3T


Geodesic Icosahedron Dome L3T


Geodesic Icosahedron Dome L3T

Geodesic Icosahedron Dome L3T

d= A= x 60, B= x 60, C= x 64, D= x 60, E= x 60, F= x 64, G= x 60, H= x 64, I= x 60,

Geodesic Dodecahedron

Dodecahedron


Dodecahedron

Geodesic Dodecahedron L1


Geodesic Dodecahedron L1

Geodesic Dodecahedron L2


Geodesic Dodecahedron L2
Looks very symmertrical, L1 & L2 - yet, no straight hemisphere line.

Geodesic Dodecahedron L2T


Geodesic Dodecahedron L2T


Geodesic Dodecahedron Dome L2T


Geodesic Dodecahedron Dome L2T

Geodesic Dodecahedron Dome L2T

d= A= x 32, B= x 60, C= x 64, D= x 32, E= x 32, F= x 60,

Geodesic Icosidodecahedron

Icosidodecahedron


Icosidodecahedron

Geodesic Icosidodecahedron L1


Geodesic Icosidodecahedron L1


Geodesic Icosidodecahedron Dome L1


Geodesic Icosidodecahedron Dome L1

Geodesic Icosidodecahedron Dome L1

d= A= x 30, B= x 35,
After triangulating it's the same as the Icosahedron 2V or L2. It's a bit of redundancy to list it anyway.

Geodesic Icosidodecahedron L2


Geodesic Icosidodecahedron L2


Geodesic Icosidodecahedron Dome L2


Geodesic Icosidodecahedron Dome L2

Geodesic Icosidodecahedron Dome L2

d= A= x 60, B= x 60, C= x 70, D= x 30, E= x 30,
The L2 is the same as a Icosahedron with 4V or L3.

Geodesic Icosidodecahedron L2T


Geodesic Icosidodecahedron L2T


Geodesic Icosidodecahedron Dome L2T


Geodesic Icosidodecahedron Dome L2T

Geodesic Icosidodecahedron Dome L2T

d= A= x 30, B= x 60, C= x 30, D= x 60, E= x 30, F= x 70, G= x 60, H= x 30,
Very beautiful construct, the same would have been achieved to triaconize a 2V or L2 Icosahedron (2V.2VT)

Geodesic Truncated Icosahedron

Truncated Icosahedron


Truncated Icosahedron

Geodesic Truncated Icosahedron L1


Geodesic Truncated Icosahedron L1


Geodesic Truncated Icosahedron Dome L1


Geodesic Truncated Icosahedron Dome L1

Geodesic Truncated Icosahedron Dome L1

d= A= x 30, B= x 40, C= x 50,
The L1 seems like having two straight hemisphere lines, at 3/7 and 4/7 approximately, looks like a Icosahedron 3V.

Geodesic Truncated Icosahedron L2


Geodesic Truncated Icosahedron L2


Geodesic Truncated Icosahedron Dome L2


Geodesic Truncated Icosahedron Dome L2

Geodesic Truncated Icosahedron Dome L2

d= A= x 60, B= x 60, C= x 90, D= x 30, E= x 65, F= x 120, G= x 130,

Geodesic Truncated Icosahedron L2T


Geodesic Truncated Icosahedron L2T


Geodesic Truncated Icosahedron Dome L2T


Geodesic Truncated Icosahedron Dome L2T

Geodesic Truncated Icosahedron Dome L2T

d= A= x 32, B= x 60, C= x 120, D= x 64, E= x 64, F= x 32, G= x 92, H= x 120, I= x 60, J= x 120, K= x 64,

Geodesic Truncated Dodecahedron

Truncated Dodecahedron


Truncated Dodecahedron

Geodesic Truncated Dodecahedron L1


Geodesic Truncated Dodecahedron L1

Geodesic Truncated Dodecahedron L2


Geodesic Truncated Dodecahedron L2
Beautiful shape, the L1 & L2, unfortunately no straight hemisphere line.

Geodesic Truncated Dodecahedron L2T


Geodesic Truncated Dodecahedron L2T

Geodesic Rhombicosidodecahedron

Rhombicosidodecahedron


Rhombicosidodecahedron

Geodesic Rhombicosidodecahedron L1


Geodesic Rhombicosidodecahedron L1

Geodesic Rhombicosidodecahedron L2


Geodesic Rhombicosidodecahedron L2

Geodesic Rhombicosidodecahedron L2T


Geodesic Rhombicosidodecahedron L2T

Geodesic Truncated Icosidodecahedron

Truncated Icosidodecahedron


Truncated Icosidodecahedron

Geodesic Truncated Icosidodecahedron L1


Geodesic Truncated Icosidodecahedron L1

Geodesic Truncated Icosidodecahedron L2


Geodesic Truncated Icosidodecahedron L2

Geodesic Truncated Icosidodecahedron L2T


Geodesic Truncated Icosidodecahedron L2T

Geodesic Snub Dodecahedron

Snub Dodecahedron


Snub Dodecahedron

Geodesic Snub Dodecahedron L1


Geodesic Snub Dodecahedron L1

Geodesic Snub Dodecahedron L2


Geodesic Snub Dodecahedron L2

Geodesic Snub Dodecahedron L2T


Geodesic Snub Dodecahedron L2T

Geodesic Pentakis Dodecahedron

It's not a platonic or archimedean polyhedra, but the dual of the archimedean truncated icosahedron:

Pentakis Dodecahedron


Pentakis Dodecahedron
In its original version there is no straight hemisphere line, and already two kinds of edge lengths.

Pentakis Dodecahedron vs Geodesic Dodecahedron L1

  A B Strut Variance
Pentakis Dodecahedron 1.000000 1.127322 12.7%
Geodesic Dodecahedron L1 1.000000 1.113583 11.4%
While I thought I knew this form, and going through previous forms I found it looks like the Geodesic Dodecahedron L1, and to my surprise the structure almost is the same, yet the ratio between both strut length is off a bit. The difference of B's is 1.21%, very small but still too significant - so the Pentakis Dodecahedron and the Geodesic Dodecahedron L1 (triangulated the pentagons and spherical projected) are structure-wise the same, but the strut ratios are a bit off.

Geodesic Pentakis Dodecahedron L2


Geodesic Pentakis Dodecahedron L2

Geodesic Pentakis Dodecahedron L2T


Geodesic Pentakis Dodecahedron L2T
The triacon version provides hemisphere lines for a dome version.

Epcot Spaceship Earth


Epcot Spaceship Earth on opening day (1982)
The pentakis dodecahedron is quite famous by its Epcot (Experimental Prototype Community Of Tomorrow) "Spaceship Earth" version at Disney World in Tampa, Florida (USA):


Epcot: Spaceship Earth (2006)
Epcot: Spaceship Earth - Pentakis Dodecahedron 8V (2006)
Epcot: Spaceship Earth Simplified Model (Geodesic Pentakis Dodecahedron 8V)
Epcot: Spaceship Earth Detailed Model (Geodesic Pentakis Dodecahedron 8V & Centerpoint Triangulated Faces)

As far I figured out from the photo, the geodesic sphere is 8V, whereas each 8V triangle is center point triangulated into 3 faces again, 60 x 8 x 8 x 3 = 11'520 faces finally. Yet, according Wikipedia: Spaceship Earth the sphere is composed by

60 x 16 x 4 x 3 = 11'520 faces. It would be interesting to get more details to see why the leveled subdivisions were chosen the way they were.

Epcot Construction of Spaceship Earth (courtesy by Von Johnson and Associates, Inc., 1981) (1 of 2)
Epcot Construction of Spaceship Earth (courtesy by Von Johnson and Associates, Inc., 1981) (2 of 2)

These construction photos confirm the description of the Wikipedia article.

Other Solids

From the Johnson Solid I tried

but didn't provide a straight hemisphere line at L1 or L2, and strut lengths were much higher than the platonic and archimedean solids.

As mentioned in the Geodesic Dome Notes I deepen some of the dome options for real-life applications.

Links