Geodesic Dome
A Triangulation of a Platonic Solid or other Polyhedron to produce a closeapproximation to a Sphere. The 
th order geodesation operation replaces each polygon of the polyhedron by the projectiononto the Circumsphere of the order regular tessellation of that polygon. The above figure shows geodesations oforders 1 to 3 (from top to bottom) of the Tetrahedron, Cube, Octahedron, Dodecahedron, andIcosahedron (from left to right).
R. Buckminster Fuller designed the first geodesic dome (i.e., geodesation of a Hemisphere). Fuller's dome wasconstructed from an Icosahedron by adding Isosceles Triangles about eachVertex and slightly repositioning the Vertices. In such domes,neither the Vertices nor the centers of faces necessarily lie at exactly the same distancesfrom the center. However, these conditions are approximately satisfied.
In the geodesic domes discussed by Kniffen (1994), the sum of Vertex angles is chosen to be aconstant. Given a Platonic Solid, let 
be the number of Edges meeting at aVertex and be the number of Edges of the constituentPolygon. Call the angle of the old Vertex point 
and the angle of the newVertex point 
. Then
 |  |  | (1) |
 | |  | (2) |
 | |  | (3) |
Solving for
gives | (4) |
 | (5) |
 | (6) |
 | (7) |
| Solid |  |  |  | | | |  |
| Tetrahedron | 3 | 3 | 45° | 90° | 270° | ||
| Cube | 24 | 14 | 3 | 4 | 51 ° | 81 ° | 308 ° |
| Octahedron | 4 | 3 | 38 ° | 108 ° | 308 ° | ||
| Dodecahedron | 60 | 32 | 3 | 5 | 56 ° | 71 ° | 337 ° |
| Icosahedron | 5 | 3 | 33 ° | 118 ° | 337 ° |
References
Kenner, H. Geodesic Math and How to Use It. Berkeley, CA: University of California Press, 1976. Kniffen, D. ``Geodesic Domes for Amateur Astronomers.'' Sky and Telescope, pp. 90-94, Oct. 1994. Pappas, T. ``Geodesic Dome of Leonardo da Vinci.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 81, 1989.
- Probability
- Probability Axioms
- Probability Density Function
- Probability Distribution Function
- Probability Function
- Probability Inequality
- Probability Integral
- Probability Measure
- Probability Space
- Probable Error
- Probable Prime
- Problem
- Procedure
- Proclus' Axiom
- Procrustian Stretch
- Product
- Product Formula
- Product-Moment Coefficient of Correlation
- Product Neighborhood
- Product Rule
- Product Space
- Program
- Projection
- Projection Operator
- Projective Collineation