11 Choosing a Polyhedron
1If our requirements are sufficiently modest, we need not start from a polyhedron at all. Standing at the center of the system we propose to design, we can surround ourselves with as many points as we like, obtained by writing coordinates by hunch, or even at random. We can join each point to its two nearest neighbors to obtain a triangulated network, and use Equation 9.2 to calculate the lengths of these connections. This gives us a table of chord factors. If we then choose a radius and let the chord factors determine the lengths of members, we can build a spherical environment which meets two important criteria. (1) It is spherical; every vertex point is at the same distance from the center. This fact is guaranteed by Equation 9.2, which treats and as spherical coordinates. (2) It is omnitriangulated, hence self-sustaining. We ensured this by our method of joining the points.
2 But it has grave deficiencies. (1) It is unlikely, except by chance, to possess any symmetries whatever, save that of the spherical envelope (a constant radius). This means that every single member length is probably unique, which is wildly inefficient. It also means that numerous short members are quite likely to be clustered in zones where they make no structural sense, and that elsewhere long unassisted spans are likely to bound large triangular openings. It means that if we want a dome instead of a sphere, no truncation plane exists, nor any logical place to go about inserting one. And it means that the structure looks as unpleasingly random as the design process was.
3 Moreover (2), it is very unlikely indeed that any sequence of vertices will be aligned along a great circle. Any two of them will be, because an equator can be made to pass through any two points on a globe. But that is not what we want to achieve. The geodesic dome, as we have seen, is supported by tension, and the tension networks are most economical when their strands run for considerable distances without changing direction. This means aligning sequences of vertices along great-circle arcs.
4 Hence the polyhedra, which we described in Chapter 8 as devices for finding symmetry systems.
5 There are only three such systems to be found: tetrahedral, octahedral, icosahedral. All other regular or semi-regular polyhedra present variations on these themes; thus the icosidodecahedron we encountered in Chapter 3 has icosahedral symmetry, and the cuboctahedron has octahedral. It is possible to execute a geodesic breakdown on any polyhedron, by triangulating its non-triangular faces before commencing, and a few polyhedra have special advantages. Thus A. J. Pugh has pointed out that since truncated octahedra cluster naturally, they make a natural starting point for domical clusters (Polyhedra, University of California Press, 1976). To isolate first principles, though, we shall concentrate on the three prime symmetry systems, tetrahedral, octahedral, icosahedra.
6 We commence by setting the tetrahedron aside, for use; only in a few very special situations we shall look at much later. Its overwhelming disadvantage is inefficiency. Having only four face triangles, it uses up so much of the sphere’s surface before repeating anything that to obtain an average chord factor of about 0.33 we must go all the way to subdivision (Class I). We then confront twenty different member lengths, varying over a range of . (By contrast, a icosa yields a slightly tighter mesh—average chord factor about 0.30—using only six different lengths which vary over a range of .) A Class II attack on the tetrahedron gives more attractive results and underlies most of its usefulness. It remains a special-case polyhedron merely.
7 For most purposes, then, we have the icosahedron and the octahedron to choose from.
8 Useful characteristics of the octahedron include:
- a natural equator at all frequencies;
- vertical slicing line at all frequencies, permitting half a dome to be joined to a rectangular structure;
- simple generation of non-spherical contours;
- provision for square openings: either one at the zenith or, by a simple rotation, three surrounding the zenith partway down;
- relatively simple computations.
9 Useful characteristics of the icosahedron include:
- lower frequency for comparable average chord factor (a icosa yields sphere subdivisions comparable to a octa);
- angles uniformly averaged among nearly equilateral triangles (octa has right-angled triangles at its principal vertices);
- less variation in the lengths of members for comparable average chord factor ( for icosa, for octa);
- curvature more uniformly distributed (principal octa vertices are nearly flat; hence they are potential weak areas);
- generally superior appearance.
10 Some of the icosa’s advantages disappear once we contemplate non-spherical contours. Since member lengths of elliptical or similar structures must vary continuously from the 0° zenith down to a 90° equator, the icosa loses most of the economy it derives from its generally lower frequency of subdivision. In addition, it requires us to subdivide two different triangles differently en route to the equator, while the octa comprehends this span in a single triangular face.
11 A minor disadvantage of icosas—lack of a natural equator—disappears at even frequencies, though only with Class I breakdowns, The octa can always be rotated to yield an equator, whether the subdivision is Class I or Class II.
12 As for the relative ease of computation afforded by the octahedron, this advantage is peripheral; computations, however troublesome, need only be performed once.
13 There seems to be no doubt that a wooden dome is easier to build from octahedral geometries; plywood panels, for instance, fit more easily at square than at pentagonal apices.
14 In Shelter (Bolinas, Ca., Shelter Publications: 1973) Bill Woods of Dyna Domes, whose user-built wooden structures use octahedral geometries and are reportedly trouble-free, is quoted as saying flatly that the icosahedron is ‘‘no good.’’ No reasons are given, but Woods presumably had in mind the octa’s adaptability to hammer-and-nail technology. Doors and windows, for instance, are easier to fit when squares, not pentagons, are available at the principal vertices.
11.1 Other Criteria
15We shall see in Chapter 14 that rotating an octahedron until the center of a face is uppermost places three square windows in symmetrical array around the zenith and about 50° down from it. Should we want this trio of skylights, only the octahedron will provide it, and only in a Class I breakdown.
16 Similarly, the Class II tetrahedron (Method 1), truncated on one of the great circles it makes available, affords at ground level four pairs of large, virtually square openings, each with a single diagonal brace, removal of which leaves a natural door (try an model). The tetrahedron’s drawbacks, notably its extreme variation in member length, might well be worth coping with should we want a structure with large doors in pairs at the four compass points.
17 As these examples indicate, some of the inherent difficulties of designing useful geodesic structures can be minimized by choosing the right polyhedron and turning it right way up. Truncation planes will often seem uneven, but in Chapter 14 we shall learn how easy it is to deal with that.
18 So there may well be practical reasons, deriving from a structure’s intended use, for selecting one polyhedron rather than another.
19 Or there may be aesthetic reasons. A geodesic framing system is normally visible, and we should give some thought to what it looks like. Near-equality of member lengths makes icosahedral systems especially graceful, and functions of the celebrated Golden Section pervade the proportions of the icosa itself and no doubt help account for its visual appeal. At the other extreme, a designer in quest of bold effects might do worse than investigate the Class II Method 3 tetra, a system of elongated diamonds with incised dihedral angles (again, try an model).