A geodesic sphere and its dual.
The mathematical object "chord" of the "geodesic sphere" corresponds to the structural "strut" of the physical "geodesic dome". The general definition of a chord is a (straight) line segment joining two points on a curve. For simple geodesic domes we recognize the associated curve to be the surface of a sphere. Here is how chords of geodesic spheres are generated. We first choose an underlying polyhedron with equal triangle faces. The regular icosahedron is most popular. The sphere we use is specifically the "circumscribing sphere" that contains the points (vertices) of the underlying polyhedron. The desired frequency of the subsequent geodesic sphere or dome is the number of parts or segments into which a side (edge) of the underlying polyhedral triangle is subdivided. The frequency has historically been denoted by the Greek letter "ν" (nu). By connecting like points along the subdivided sides we produce a natural triangular grid of segments inside each underlying triangle face. Each segment of the grid is then projected as a "chord" onto the surface of the circumscribing sphere. The technical definition of a chord factor is the ratio of the chord length to the radius of the circumscribing sphere. It is therefore convenient to think of the circumscribing sphere as scaled to radius = 1 in which "chord factors" are the same as "chord lengths" (decimal numbers less than one).
For geodesic spheres a well-known formula for calculating any "chord factor" is
chord factor = 2 Sin (θ / 2) where θ is the corresponding angle of arc for the given chord, that is, the "central angle" spanned by the chord with respect to the center of the circumscribing sphere. Determining the central angle usually requires some non-trivial spherical geometry.
In Geodesic Math and How to Use It Hugh Kenner writes, "Tables of chord factors, containing as they do the essential design information for spherical systems, were for many years guarded like military secrets. As late as 1966, some 3ν icosa figures from Popular Science Monthly were all anyone outside the circle of Fuller licensees had to go on." (page 57, 1976 edition). Other tables became available with publication of Lloyd Kahn's Domebook 1 (1970) and Domebook 2 (1971). With advent of personal computers, the mathematics became more accessible. Rick Bono's Dome software, outputs a script that can be used with the POV-ray raytrace to produce 3D pictures of domes. Domes based on the frameworks of different underlying polyhedra along with various methods for subdividing them will produce quite different results. Mathematical formulas developed by Peter W. Messer for calculating chord factors and dihedral angles for the general geodesic sphere appear in the Appendix of the 1999 Dover edition of Spherical Models by Magnus J. Wenninger.
