One thing that comes up a lot in homology is the idea of simplices and simplicial complexes. They’re interesting in their own right, and they’re one more thing that we can talk about
that will help make understanding the homology and the homological chain complexes easier when we get to them.
A simplex is a member of an interesting family of filled geometric figures. Basically, a simplex is an N-dimensional analogue of a triangle. So a 1-simplex is a line-segment; a 2-simplex is a triangle; a three simplex is a tetrahedron; a four-simplex is a pentachoron. (That cool image to the right is a projection of a rotating pentachoron from wikipedia.) If the lengths of the sides of the simplex are equal, it’s called a regular simplex.
There are some neat things about simplices. For any N≥2, an N-simplex is a figure with N+1 faces, each of which is an N-1 simplex. So a tetrahedron – a 3-simplex – has four faces, each of which is a triangle; a pentachoron (a 4-simplex) has 5 tetrahedral faces. Also, for any N, an N-simplex is the convex hull of N+1 linearly independent points embedded in ℝN.
A simplicial complex is where simplices start meeting up with topology. A simplicial complex is a topological space formed from a set of simplices. Basically, a topological T space is a simplicial complex K if/f it can be decomposed into a collection of simplices where:
- For every simplex S in K, every face of S is also in K.
- Every intersection between 2 simplices is a face of both of the intersecting simplices.
There’s one annoying part of that second requirement, which is that you always consider intersections using the lowest-dimension simplices that can include the intersection as a face. So you can have two tetrahedrons intersecting along an edge – they’re still a simplicial complex, because the intersection is a 1-simplex, so you consider it using the 2-simplices in K – and the line segment is a face of all of the triangles that meet at that edge.
A simplicial complex where the largest dimension of any simplex in the complex is N is called a simplicial N-complex. It’s called a pure N-complex if every simplex of dimension <N is a face of an N-simplex in the complex.
Every N-simplex is homeomorphic to an N-ball in ℝN; and for N≤3, a manifold which is a subspace of ℝN is homeomorphic to
a pure simplicial complex. The simplicial complex is sometimes called a triangulation of the space. Many of the early results about manifolds in topology were done using triangulations of
manifolds; the division of the manifold into simplexes was a major tool used to make topological proofs tractable.
See also Pentatope
http://mathworld.wolfram.com/Pentatope.html
Tetrahedral Number, Pentatope Number.
====================
http://www.research.att.com/~njas/sequences/A119602
A119602 Number of nonisomorphic polytetrahedra with n identical regular tetrahedra connected face-to-face and/or edge-to-edge (chiral shapes counted twice).
1, 1, 2, 7, 39
n a(n)
0 1
1 1
2 2
3 7
4 39
OFFSET 0,3
COMMENT
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.
LINKS
Eric W. Weisstein et al., Tetrahedron.
http://mathworld.wolfram.com/Tetrahedron.html
Wikipedia, Polyiamond.
http://en.wikipedia.org/wiki/Polyiamond
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 tetromino 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.
CROSSREFS
Cf. A000577, A000105.
KEYWORD
nonn,uned,obsc
AUTHOR
Jonathan Vos Post, Jun 02 2006
====================
Um, looks like you need to close a superscript tag, before “is homeomorphic to a pure simplicial complex.”
Thanks Blake. Fixed now.
And part of each of those proofs was a justification that your choice of triangulation didn’t change your answer. Many of those arguments (in low dimensions) are streamlined by the fact that there’s a finite list of “moves” so that any two triangulations of a manifold are related by a finite sequence of those moves. If the result is left invariant under each move, it doesn’t depend on the choice of triangulation.
So why care about that? Because those “moves” on triangulations form the morphisms of a category! Your invariant goes from being a function on manifold to a functor on the category of triangulated manifolds. And studying the equations your functor has to satisfy leads to a lot of interesting mathematical structures, like the Zamolodchikov tetrahedron equation.
http://arxiv.org/pdf/physics/0603068
From: Phil Fraundorf
Date (v1): Thu, 9 Mar 2006 19:48:43 GMT (50kb)
Date (revised v2): Tue, 6 Mar 2007 22:51:43 GMT (195kb)
A simplex model for layered niche networks
Authors: P. Fraundorf
Comments: 10 pages, 9 figures, 16 references,
cf. this http URL
http://newton.umsl.edu/philf//correlatedstates.html
Subj-class: Physics and Society; Other
The standing crop of correlations in metazoan communities may be assessed by an inventory of niche structures focused inward and outward from the physical boundaries of skin (self), gene-pool (family), and meme-pool (culture). We consider tracking the progression from three and four correlation layers in many animal communities, to five of six layers for the shared adaptation of most humans, with an attention-slice model that maps the niche-layer focus of individuals onto the 6-variable space of a 5-simplex. The measure puts questions about the effect, on culture and species, of policy and natural events into a common context, and may help explore the impact of electronically-mediated codes on community health.
John: I didn’t know that. What is the list of moves called?
It’s probably worth mentioning that not all manifolds have a triangulation.
I’ve been computing the hypervolumes of various 4-dimensional polytopes in Euclidean space, and in hyperbolic space of constant negative curvature. My work is slow, as visualization comes only in flashes, after much effort. But the results trickle in, a polytope at a time. In several cases, I’ve found things that I simply couldn’t locate in the literature, although it seems that they must have been known, as they deal with shapes known for a century.
It is conventional to “pentatopalize” the polytope, dividing it into 4-D simplices, and simply adding up the hypervolumes of each in the simplicial complex. The pentatope’s hypervolume in Euclidean space is the Cayley-Menger determinant. In the cases I’ve considered, symmetries allow me to side-step the computationally intense pentatopalization. No software package available that I know does this in general. Experts that I ask suggest that I write such software, and they’ll cheer from the sidelines. My last 6-month software contract was at $82.50/hour; I really can’t afford to write “spec” software, with a family to support.
In curved space, the problem is more subtle, but the theory is well-established.
In Einstein-Minkowski space, there are unsolved elementary problems.
I’d also like to know even a tiny fraction as much as John Armstrong on the subject that he mentions.