Study on methodology of geodesic breakdown systems- Methods 1 and 2 as explained in Geodesic Math by Kenner, pages 64-68 etc.
Method 1 – polyhedral edges divided equally and projected to the great circle. Resulting chords are unequal, but triangle faces of this dome are more nearer to equilateral (less variation). More variation of strut lengths.
Method 2 – the circle corresponding to the polyhedra edge is divided equally, which defines the chords. Resulting triangles are less equal, while there are less variations in strut lengths. Pentagon areas in icosahedral domes and octa (square) areas in octahedral domes have shallower (flatter) vertex angles. This is probably critical only in domes with high (6+) frequencies. All my domes up to this point (January 2002) have been Method 1 types. This page gives some graphical explanation. This page has more on dome terminology.
Class 1 Method 2, 4 frequency dome:
- As can be seen here, the Method 2 dome has triangle sizes more varied, and pentagons are larger. Only 4 strut lengths, 5 triangle face types.
- Method 1 dome has more uniform triangle sizes. 6 strut lengths, 5 triangle face types.
Quote from Kenner: “In general, to minimize component inventory, we use the lowest frequency we can. How long a component we can tolerate is a function of its slenderness ratio length divided by diameter. Fuller has suggested a 24/1 as a slenderness ratio for wood, 30/1 for metal or pre stressed concrete.
“Finally, very high frequencies bring each vertex so near flatness as to be dangerous unless virtual thickness is imparted to the shell by trussing (Chapter 16). ‘Very high’ for the icosa, means anything over 12v with careful workmanship, or over, say, 8v when tolerances are less than exacting.”