## Two dimensions

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 $S^1$ 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.)

## Four dimensions

Given a 4-dimensional polytope P define $\gamma (P) =f_1(P) + f_{02}(P)-3f_2(P)-4f_0+10.$

Here, $f_i(P)$ is the number of $i$-faces of $P$. $f_{02}(P)$ is the number of chains of the form 0-face $\subset$ 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

### (1)   γ(P)=γ(P*)

Here is an example: Let $P$ be the four dimensional cube. In this case $f_0=16$, $f_1=32$, $f_2=24$, $f_3=8$, and since every 2-face has 4 vertices $f_{02}=96$. $\gamma (P)=32+96-72-80+10=2$. $P^*$ is the 4-dimensional cross-polytope with $f_0=8$, $f_1=24$, $f_2=32$, $f_3=16$, and since every 2-face has 3 vertices $f_{02}=96$ $\gamma (P^*)=24+96-96-40+10=2$. Viola!

The proof of (1) is quite a simple consequence of Euler’s theorem. For the 120-cell and its dual the 600-cell $\gamma=250$.

## Frameworks, rigidity and Alexandrov-Whiteley’s theorem.

A framework based on a $d$-polytope $P$ is obtained from $P$ by adding diagonals which triangulate every 2-face of $P$. Adding the diagonals leads to an infinitesimally rigid framework. This was proved by Alexandrov for 3 dimension and by Walter Whiteley in higher dimensions. Walter Whiteley

Theorem (Alexandrov-Whiteley): For $d\ge 3$, every famework based on $P$ is infinitesimmaly rigid.

It follows from the Alexandrov-Whiteley theorem that for a four-dimensional polytope $P$, $\gamma (P)$ is the dimension of the space of stresses of every  framework based on $P$. (An affine stress is an assignment of weights to edges so that every vertex lies in “equilibrium”.) Whiteley’s theorem implies that $\gamma (P) \ge 0$ for every 4-dimensional polytope $P$.

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 $d-$polytope $P$,  let $f_1^+(P)$ be the number of edges in a framework based on $P$. We can define for every $d$-polytope, $\gamma (P)=f_1^+(P)-df_0(P)+{{d+1} \choose {2}}$. Whiteley’s theorem implies that $\gamma (P) \ge 0$ for every $P$. (For $d=3$ by Euler’s theorem $\gamma =0$.)

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 $\gamma(P)$ equals $\gamma (P^*)$. However, it is true in every dimension that $\gamma (P)=0$ if and only if $\gamma (P^*)=0$.

## Toric varieties

Let $T(P)$ be a toric variety based on a rational 4-dimensional polytope $P$. The dimension of the primitive part of the 4th intersection homology group of $T(P)$ is equal to $\gamma (P)$.  Our duality theorem thus asserts that the primitive part of the 4th intersection homology group of $T(P)$ has the same dimension as the primitive part of the 4th intersection homology group of $T(P^*)$.

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)

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### 5 Responses to A Mysterious Duality Relation for 4-dimensional Polytopes.

1. Tom Braden says:

Hi Gil — I have a few comments. In the last paragraph you want the primitive intersection (co)homology of T(P), since T(P) may not be smooth. Also rather than working with primitive IH of a projective toric variety you can just use the IH of the affine toric variety T(cP) defined by an unbounded cone over P.

In both the 2d and 4d cases the vector spaces for dual polytopes (or more precisely dual cones) are naturally dually paired, after choosing an orientation of the ambient vector space. I pointed this out after Theorem 4.3 in my paper “Remarks on combinatorial IH of fans“. For polygons it’s not hard to write down the pairing explicitly, but I don’t think I was ever able to write a formula for the pairing for 4-polytopes.

• Gil Kalai says:

Dear Tom, many thanks for the remark. I meant IH, of course, but I was not aware of your pairing. I will update the post. (I also added the link to your paper in your comment.)

2. Frédéric Chapoton says:

Your formula seems to be wrong.. For example the sum of (32, 96, -72, -80, 10) is -14.

• Frédéric Chapoton says:

It seems that it should use $-4 f_0$ instead.

And there is a typo, voilà !

GK: Corrected! Thanks!