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Geodesic Polyhedra

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: s3 / 12 * √2
  • A: s2 * √3
  • rinner: s / 12 * √6
  • router: 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%

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
  • Vertices/Connectors: 74
    • 36 x 4-way
    • 24 x 6-way
    • 6 x 8-way
    • 8 x 12-way
  • Faces: 144 (3-sided)
  • Edges/Struts:
    • A x 24: 0.29239
    • B x 48: 0.35693
    • C x 48: 0.47313
    • D x 24: 0.48701
    • E x 24: 0.60581
    • F x 48: 0.66092
    • total 216 struts (6 kinds)
    • strut variance 126.4%


Geodesic Tetrahedron Dome L3T


Geodesic Tetrahedron Dome L3T

Geodesic Tetrahedron Dome L3T
  • Vertices/Connectors: 43
    • 6 x 3-way
    • 15 x 4-way
    • 2 x 5-way
    • 12 x 6-way
    • 4 x 7-way
    • 2 x 8-way
    • 2 x 12-way
  • Faces: 72 (3-sided)

  • Tetrahedron Dome L3T Construction Map
  • Edges/Struts:
    • A x 12: 0.29239
    • B x 24: 0.35693
    • C x 28: 0.47313
    • D x 12: 0.48701
    • E x 14: 0.60581
    • F x 24: 0.66092
    • total 114 struts (6 kinds)
    • strut variance 126.4%

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


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