Before we talk about 4 dimensions let us recall some basic facts about 2 dimensions:
A planar polygon has the same number of vertices and edges.
This fact, which just asserts that the Euler characteristic of is zero, can be reformulated as: Polygons and their duals have the same number of vertices.
A linear algebra statement
The spaces of affine dependencies of the vertices for a polygon P and its dual P* have the same dimension.
I am not aware of a pairing or an isomorphism for demonstrating this equality. (See, however Tom Braden’s comment.)
Given a 4-dimensional polytope P define
Here, is the number of -faces of . is the number of chains of the form 0-face 2-face. In other words it is the sum over all 2-faces of P, of the number of their vertices. In 1987 I discovered the following:
Theorem [Mysterious Four-dimensional Duality]: Let P and P* be dual four dimensional polytopes then
Here is an example: Let be the four dimensional cube. In this case , , , , and since every 2-face has 4 vertices . . is the 4-dimensional cross-polytope with , , , , and since every 2-face has 3 vertices . . Viola!
The proof of (1) is quite a simple consequence of Euler’s theorem.
For the 120-cell and its dual the 600-cell .
Frameworks, rigidity and Alexandrov-Whiteley’s theorem.
A framework based on a -polytope is obtained from by adding diagonals which triangulate every 2-face of . Adding the diagonals leads to an infinitesimally rigid framework. This was proved by Alexandrov for 3 dimension and by Walter Whiteley in higher dimensions.
Theorem (Alexandrov-Whiteley): For , every famework based on is infinitesimmaly rigid.
It follows from the Alexandrov-Whiteley theorem that for a four-dimensional polytope , is the dimension of the space of stresses of every framework based on . (An affine stress is an assignment of weights to edges so that every vertex lies in “equilibrium”.) Whiteley’s theorem implies that for every 4-dimensional polytope .
4-dimensional stress-duality: Let P and P* be dual four dimensional polytopes. Then the space of affine stresses for a frameworks based on P and on P* have the same dimension.
Again, I am not aware of a pairing or an isomorphism that demonstrates this equality between dimensions. (See, however Tom Braden’s comment.)
Remark: Given a polytope , let be the number of edges in a framework based on . We can define for every -polytope, . Whiteley’s theorem implies that for every . (For by Euler’s theorem .)
Just as a polytope in dimension greater than 2 need not have the same number of vertices as its dual, it is also no longer true in dimensions greater than 4 that equals . However, it is true in every dimension that if and only if .
Let be a toric variety based on a rational 4-dimensional polytope . The dimension of the primitive part of the 4th intersection homology group of is equal to . Our duality theorem thus asserts that the primitive part of the 4th intersection homology group of has the same dimension as the primitive part of the 4th intersection homology group of .
Some references: Whiteley’s theorem (Whiteley, 1984); The 4-d duality relation (Kalai, 1987); A more general relation (Bayer and Klapper, 1991); An even more general relation (Stanley; 1992) Related theorems on combinatorics of polytopes (Braden, 2006); Connection to Mirror symmetry (Batyrev and Borisov, 1995); Connection to Koszul duality (Braden, 2007); Rigidity and polarity in 3-d (Whiteley 1987)