General analytical solution: angle between surfaces of a pyramid/polyhedron
A step further: Polygons & Polyhedra, and Giant Molecular Compounds
The faces of regular polyhedra comprise regular polygons, arranged in regular pyramid-shaped groupings. The angles (Theta) between any two adjacent surfaces of the polyhedron are identical. "F-V Inverse" means FACE-VERTEX Inverse, i.e., the shape you would have if you exchanged all the vertices for faces centred nearest the vertex of the previous shape, and all the faces for vertices located at the centres of the faces of the previous shape.
| Polyhedron: | Faces: | Vertices per Face: | Faces per Vertex | Vertices: | F-V Inverse: |
|---|
| Tetrahedron | 4 | 3 | 3 | 4 | Tetrahedron |
| Cube | 6 | 4 | 3 | 8 | Octahedron |
| Octahedron | 8 | 3 | 4 | 6 | Cube |
| Dodecahedron | 12 | 5 | 3 | 20 | Icosahedron |
| Icosahedron | 20 | 3 | 5 | 12 | Dodecahedron |
- Regular Polyhedron Space-Filling Theorem (informally, but not rigorously, proven):
- Every regular polyhedron for which 360° is a round (or 'integer') multiple of the angle Theta between two adjacent faces, can fill space as an infinite tessellation; (and therefore represents the structure of a feasible giant molecular compound) - the converse also being true, that every regular polyhedron for which 360° is not an integer multiple of Theta may not fill space.
| Polyhedron: | Description of Regular Pyramid Formations: (Theta representing the internal angle between surfaces of each regular polyhedron) |
|---|
| Tetrahedron | n=3, Alpha=60°, Theta = cos-1(1/3) = 70.529° (to 5 s.f.) |
| Cube | n=3, Alpha=90°, Theta is obviously 90°
Represents the cubic and face-centred cubic formations in giant molecular compounds |
| Octahedron | n=4, Alpha=60°, Theta = 109.471° (to 5 s.f.) |
| Dodecahedron | n=3, Alpha=108° (the internal vertex angle of a regular pentagon)
Theta = 116.565° (to 5 s.f.)
i.e. structures similar to C20, C60, may not be infinitely tesselated in the structure their shape encourages on a smaller scale. An alternative informal counter-proof is possible by observing the unfavourable properties of five-fold symmetry... |
| Icosahedron | n=5, Alpha=60°, Theta = 138.190° (to 5 s.f.) |
This work proved useful in Chemistry, in disproving that an old idea I had for a super-hard giant molecular compound to replace industrial diamonds. While the proposed structure could be built and demonstrated on the small scale, and 'proven' in a small-scale experiment, the giant molecular structure I had envisaged would never grow beyond a certain (small) size, because of a small flaw in one of its internal angles, that I could only discern mathematically. (Once the mathematics disproved the idea, it was easier to see evidence of the problem in the original experiment.)
A relatively simple calculation prevented me from wasting time pursuing an impossible project.
Lessons Learned:
- Don't extrapolate from a small-scale experiment (like a small 'MolyMod' molecular structure model, for instance) without applying some serious mathematical scrutiny. The results from small-scale experiments do not always apply to large-scale situations.
- A small amount of work overcoming mathematical (or design) headaches can save you from a large amount of work overcoming experimental (or build) headaches. (Any good design technology teacher will tell you this.)
|
