|
|
COMMENTS
|
Polytetrahedra (abbreviated polytets or n-tets) are a 3-dimensional generalization of polyiamonds.
Polytetrahedra are composed of identically-sized regular tetrahedra in Euclidean 3-space. More tetrahedra can be placed in proximity in Hyperbolic space, but that is beyond the scope of this paper. Herein we make use of the dihedral angle of the regular tetrahedron (the angle between two faces) as being arccos(1/3) radians, or (180/pi) arccos(1/3) ~ 70.5287794 degrees. For n running from 0 through 4, we enumerate different classes of polytetrahedra, where tetrahedra are variously connected in a "floppy" edge-to-edge (e2e) manner, 2, 3, 4, or 5 sharing an edge; or in a rigid face-to-face (f2f) manner; or in a combination of the two ("semifloppy"). Note that a "floppy" polytetrahedron with only e2e connections can actually be mechanically rigid.
|
|
|
REFERENCES
|
Andrew I. Campbell, Valerie J. Anderson, Jeroen S. van Duijneveldt and Paul Bartlett, "Dynamical Arrest in Attractive Colloids: The Effect of Long-Range Repulsion", Phys. Rev. Lett. 94, 208301 (2005).
Jonathan Vos Post, Polytetrahedra, preprint, Draft 4.0, approx. 6750 words, 15 pages, available as Word file by email upon request.
J. F. Sadoc, "Boerdijk-Coxeter helix and biological helices", Eur. Phys. J. B 12, 309-318.
|
|
|
EXAMPLE
|
a(0) = 1 because there is only one kind of n-tet with zero tetrahedra, namely the null set.
a(1) = 1 because there is only one kind of n-tet with one tetrahedron, namely the regular tetrahedron itself.
a(2) = 2 because there are two ways to establish vertex-to-vertex connections between two regular tetrahedra, neglecting a pair of tetrahedra which only touch at a single vertex (beyond the scope of this paper). First, we may join the two tetrahedra face-to-face (f2f) to get the Triangular Dipyramid. The triangular (or trigonal) dipyramid is one of the 8 convex deltahedra and Johnson solid J12. It has 5 vertices (2 tips and a girdle of three around the joined triangular face), 9 edges and degree sequence (3, 3, 3, 3, 3, 3). This is a rigid hexahedron, with 6 vertices, 11 edges and 8 faces. It is one of the 7 convex hexahedra. Second, we may join the two tetrahedra edge-to-edge (f2f) to get the "floppy 2-tet." It is floppy because there is no constraint on the angle that the two tetrahedra may make about the "hinge" between them, until reaching the dihedral angle arccosine (1/3) ~ 70.53 degrees, upon which it has folded into a triangular dipyramid. The floppy 2-tet has 6 vertices, 11 edges and 8 faces.
a(3) = 7 because the unique rigid 3-tet is called a "boat." The boat is a concave irregular octahedron which, since all faces are identical equilateral triangles, is a deltahedron. It has 6 vertices: two tips (prow and stern), the two extrema of the concave hinge and the two extrema of the convex "keel." It has 12 edges: 3 each adjacent to the stern and bow, the unique concave edge, 4 connecting the concave edge to the keel and one keel edge. We also have 2 purely floppy 3-tets and 2 semifloppy 3-tets, as described below.
(a) triangular dipyramid with e2e tet along one of the 3 triangular girdle edges (semifloppy, as it has 1 f2f and 1 e2e connection);
(b) triangular dipyramid with e2e tet along one of the 6 edges adjacent to a tip (semifloppy, as it has 1 f2f and 1 e2e connection);
(c) floppy 2-tet with 3rd tet added e2e so that the three tets' centroids are coplanar and can form a straight line when the hinges are both at zero degrees;
(d) floppy 2-tet with 3rd tet added e2e so that, when both hinges are both at zero degrees, the three tets' centroids are coplanar, but the lines connecting one pair of the tets is perpendicular to the line connecting the other tets' centroids. Comparing (c) with (d), we may look at the shadows of the edges on a plane, i.e. the projection. A single tetrahedron may be oriented so that its projection onto a plane is a square with crossing diagonals. Similarly, the projection of (c) onto a plane parallel to the plane of the 3 centroids is three squares end-to-end, i.e. a straight triomino, with each square containing crossed diagonals. The projection of (d) onto a plane parallel to the plane of the 3 centroids is three squares in an L shape, i.e. a L-triomino, with each square containing crossed diagonals. In the hydrocarbon world, we analogize these two to n-propane and iso-propane. We can generalize this to an infinite class of purely floppy polytetrahedra whose planar projections are polyominoes. The unique 1-tet projects to a monomino; the unique floppy 2-tet projects to a domino. Since there are five free tetrominoes there are at least five fully floppy 4-tets (a sixth from the square teromiono with one e2e connection broken). Since there are 12 free pentominoes there are at least 12 fully floppy 5-tets. Since there are 35 free hexominoes there are at least 35 fully floppy 5-tets. And so on, where the published enumerations of polyominoes immediately define a partial set of polytetrahedra: 108 floppy 7-tets, 369 floppy 8-tets (which do not include the floppy 8-tet ring which has no 2-D equivalent) and so forth. As Andrew Carmichael Post pointed out, if we bend (c) above into a ring and close the ring with an additional e2e connection, we have a mechanically rigid "floppy" 3-tet with a partially (3/4) surrounded tetrahedral hole. We can describe this also as a central tetrahedron with 3 external tetrahedra joined f2f with 3 of the 4 faces and then the central tetrahedron removed. We have the triple-edged 3-tet, in which 3 tets share an edge. This is possible because 3 x 70.53 degrees = 211.59 degrees, which is sufficiently smaller than 360 degrees that the floppiness allows for 148.41 degrees to be distributed between the tets. The planar projection of this onto a plane parallel to the plane of the 3 tets's centroids, when bent to equiangular, is three identical triangles meeting at a vertex, akin to the radiation warning logo.
|
