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

Updates

Wed, June 6, 2007: Added Pentakis Dodecahedron, with its famous 8V version of Epcot's "Spaceship Earth" sphere.

Wed, April 4, 2007: Added triacon variants and where suitable also the triacon dome variant as well.

Sun, March 25, 2007: Redefined and clarified L0, L1, L2 and so forth, and better explanation of Ln vs nV.

Mon, March 12, 2007: More details for each variant, incl. calculator for dome options.

Thu, March 1, 2007: Included dome variants when suitable, including strut maps.

Sun, February 25, 2007: First version with overview of platonic & archimedean geodesic forms, for now only L1 and L2 as I call them.

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,

the class 1 / alternate is the most prominant (creating 4 triangles),

the class 2 / triacon (creating 6 triangles) is also well known, and I added

the centerpoint (creating 3 triangles) and

the slashing (creating 2 triangles, 3 possible ways)

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:

I may change the notion from L1/2 to another more general approach later again; I likely stick with it as there is a clear distinction between nV and Ln notion

You can calculate for each dome variant the strut lengths, edit the yellow field and hit ENTER to calculate the struts

For more detailed variants, e.g. fine triangulation and more options see my Geodesic Dome Notes

Geodesic Tetrahedron

Tetrahedron

Tetrahedron

Uniform Polyhedron: U1

Platonic Solid

Platonic Element: Fire

Wythoff symbol: 3|2 3

Symmetry Group: {3, 3, 3}

Vertex Configuration: tetrahedral

Dual: tetrahedron

V: s^{3} / 12 * √2

A: s^{2} * √3

r_{inner}: s / 12 * √6

r_{outer}: s / 4 * √6

h: s / 3 * √6

Vertices/Connectors: 4 (3-way)

Faces: 4 (3-sided)

Edges/Struts:

A x 6: 1.63299

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

Vertices/Connectors: 14

6 x 4-way

8 x 6-way

Faces: 24 (3-sided)

Edges/Struts:

A x 24: 0.91940

B x 12: 1.15470

total 36 struts (2 kinds)

strut variance 25.7%

Geodesic Tetrahedron Dome L2T

Geodesic Tetrahedron Dome L2T

Geodesic Tetrahedron Dome L2T

Vertices/Connectors: 10

2 x 3-way

6 x 4-way

2 x 6-way

Faces: 12 (3-sided)

Tetrahedron Dome L2T Construction Map

Edges/Struts:

A x 14: 0.91940

B x 7: 1.15470

total 21 struts (2 kinds)

strut variance 25.7%

It looks like a prepared (reduced to triangles using centerpoint triangulation) cube at the first sight.