In discrete geometry, a polytope is projectively unique (or projectively stable) if it has a unique convex realization up to projective transformations. The study of projectively unique polytopes was initiated by Geoffrey C. Shephard, Micha Perles, Peter McMullen and Branko Grünbaum. Later significant contributions are also due to Karim Adiprasito and Günter M. Ziegler.
Introduction Two convex polytope P , Q ⊂ R n {\displaystyle P,Q\subset {\mathbb {R}}^{n}} are combinatorially equivalent if they have the same number of faces with the same incidence relations between them. Formally this means that P {\displaystyle P} and Q {\displaystyle Q} have isomorphic face lattices, or equivalently, isomorphic underlying abstract polytopes. One also says that both P {\displaystyle P} and Q {\displaystyle Q} are realizations of the face lattice (or abstract polytope).
Given a convex polytope P ⊂ R n {\displaystyle P\subset {\mathbb {R}}^{n}} many combinatorially equivalent polytopes can be generated through transformation of the ambient space, such as isometries or affine transformations. The largest class of transformations of Euclidean space that preserves coplanarity, and hence maps polytopes onto combinatorially equivalent polytopes, is the class of projective transformations. A polytope P {\displaystyle P} is projectively unique if all other realizations are obtained through projective transformations; or in other words, if P {\displaystyle P} has a unique convex realization up to projective transformations. For example, simplices, such as triangles and tetrahedra, are projectively unique. In contrast, the cube is not projectively unique, as it has realizations that are not projectively equivalent to the standard cube (see the figure).
Known projectively unique polytopes Projective transformations on n {\displaystyle n} -dimensional Euclidean space can map any n + 2 {\displaystyle n+2} points onto any other n + 2 {\displaystyle n+2} points. Hence, any n {\displaystyle n} -dimensional polytope with at most n + 2 {\displaystyle n+2} vertices is projectively unique. Being projectively unique is moreover closed under polar duality. Since duality swaps vertices and facets, any n {\displaystyle n} -dimensional polytope with at most n + 2 {\displaystyle n+2} facets is projectively unique as well. This shows that, for example, the product of two simplices is projectively unique (such as the (3,3)-duoprism). In dimension up to three, these are the only polytopes that are projectively unique, but in dimensions n ≥ 4 {\displaystyle n\geq 4} there are examples with more than n + 2 {\displaystyle n+2} vertices and facets.
Dimension two The following is the complete list of projectively unique polytopes in dimension two:
triangle quadrangle
Dimension three A 3-polytope is projectively unique if and only if it has at most nine edges. The following is therefore the complete list of projectively unique polytopes in dimension three:
tetrahedron square pyramid triangular prism triangular bipyramid
Dimension four The following list of projectively unique 4-polytopes was compiled by Geoffrey C. Shephard (but never formally published by himself) and is conjectured to be complete:
* self-dual
Operations McMullen showed that projectively unique polytopes are closed under the following operations:
The join P ⋆ Q {\displaystyle P\star Q} of two polytopes is projectively unique if and only if the P {\displaystyle P} and Q {\displaystyle Q} are projectively unique. The vertex sum ( P , v ) ⊕ ( Q , w ) {\displaystyle (P,v)\oplus (Q,w)} of two projectively unique polytopes is projectively unique. If P {\displaystyle P} is projectively unique and v {\displaystyle v} is a vertex of P {\displaystyle P} so that P {\displaystyle P} is not a vertex sum with distinguished vertex v {\displaystyle v} , then the subdirect sum ( P , v ) ⊕ Δ 1 {\displaystyle (P,v)\oplus \Delta _{1}} is projectively unique (this is also known as splitting the vertex v {\displaystyle v} ). These operations are sufficient to generate Shephard's list of projectively unique 4-polytopes. But not all projectively unique polytopes can be generated using these operations, for example, Perles' non-rational polytope (see below).
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