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Define a quadtope P as a polytope which is the convex hull of a set Q of convex quadrilaterals, any pair of which intersect in at most one vertex, and are in maximally general position, or, a k-fold pyramid over such a convex hull. (So, a pair of quadrilaterals from Q with a common vertex subtends a 4-dimensional subpolytope of P, and a pair of disjoint quadrilaterals from Q subtends a five-dimensional subpolytope of P.)

A pure quadtope is a quadtope each of whose vertices is a vertex of an element of Q.

I assert that the number of combinatorial types of pure quadtopes for dimensions 1 through 4 are 0, 1, 0, 1, respectively. How many combinatorial types of pure quadtopes of dimension 5 are there?

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I think there are at least 6 or maybe 7. –  Dan Moore Dec 10 '10 at 22:15

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This partial answer will list 9 combinatorial types of 5-dimensional pure quadtopes. In the following, ‘quadrilateral’ is shortened to ‘quad’.

Types 2 – 5 are each the convex hull of three quads, with pairwise intersections at a vertex. Thus, the quads form a loop. For a given quad, say Q1, there are two ways it can connect to the other two quads; the common vertices can be adjacent on Q1 (described as crenellated), or they can be diagonally opposite on Q1 (described as diamondback). The term crenellated is suggested by the outline of the same-colored squares on the first two rows of a chessboard. There are four resulting combinatorial types, denoted ccc, dcd, cdc, and ddd, corresponding to whether the arrangement of quads 1, 2, and 3 (not necessarily in order) is crenellated or diamondback.

A specific example of each type 2 – 5 contains vertices V2 through V8 – the vertices of a 4-dimensional pure quadtope subtended by parallelograms Q1 (V3, 6, 5, 7) and Q2 (V2, 4, 3, 8), plus V1, affine to it. Each type 2 – 5 is completed by completing a parallelogram Q3 with V9, containing V1 and vertices from each of Q1 and Q2. Vertices V1 through V8 are as follows:

V1: (1,3,6,4,8) V2: (2,3,6,4,8) V3: (2,4,7,5,10) V4: (2,4,8,5,10) V5: (2,4,8,5,11) V6: (2,4,8,6,11) V7: (2,4,9,6,12) V8: (2,5,9,6,12)

For brevity in listing polytope elements, the 4-dimensional pure quadtope type will be denoted P([2]), a k-fold pyramid over a quad as P([1],k), and an n-simplex as sn. P([2]) has 8 facets – four P([1],1) and four s3, with f-vector (7,17,18,8). P([1],2) has 6 facets – two P([1],1) and four s3, with f-vector (6,13,13,6).

  1. The free join of two quads is like-faceted – its 8 facets are all P([1],2), with f-vector (8,24,32,24,8).
  2. ccc; V9 = V1 + V5 – V2. This polytope has 10 facets – the cyclic polytope C(4,6), three P([2]), three P([1],2) and three s4, with f-vector (9,30,46,33,10).
  3. dcd; V9 = V7 + V8 – V1. This polytope has 13 facets – one P([2]), eight P([1],2) and four s4, with f-vector (9,30,48,38,13).
  4. cdc; V9 = V1 + V7 – V8. This polytope has 13 facets – two P([2]), five P([1],2) and six s4, with f-vector (9,30,48,38,13).
  5. ddd; V9 = V7 + V4 – V1. This like-faceted polytope has 12 facets of type P([1],2), with f-vector (9,30,47,36,12).

Starting with one of the types 2 – 5, you can build another 5-dimensional combinatorial type by selecting one free vertex each from Q1 – Q3, adding a vertex V10 by completing another parallelogram Q4. So, each quad intersects at three different vertices with the other three quads. On any quad, there are two pairs of adjacent vertices intersecting with other quads, and one pair of opposite vertices intersecting with other quads. So, in describing the arrangement of quads in each of the four quad triplets (as above), ‘c’ must appear 8 times and ‘d’ 4 times. (The quad triplets subtend proper 5-dimensional subpolytopes). There are therefore at least four more combinatorial types:

Type 6. Quad triplets cdc, cdc, cdc, cdc. Beginning with type cdc, V10 = V2 + V6 – V1. This polytope has 17 facets – the direct sum s3 ⊕ s1, two P([2]), eight P([1],2), and six s4, with f-vector (10,36,62,51,17).

Type 7. Quad triplets ddd, cdc, ccc, ccc. Beginning with type cdc, V10 = V2 + V9 – V5. This polytope has 12 facets. Two of the polychoron facets have 9 ridges each: two P([1],1), six s3, and a common ridge which is a triangular bipyramid. The remaining facets are three P([2]), five P([1],2), and two s4. The f-vector is (10,36,57,41,12).

Type 8. Quad triplets dcd, cdc, cdc, ccc. Beginning with type dcd, V10 = V1 + V5 – V2. This polytope has 17 facets – the direct sum s1 ⊕ s1 ⊕ s2, two P([2]), six P([1],2), and eight s4, with f-vector (10,37,64,52,17). Interestingly, although Q3 is a (set) element of Q, Q3 is not a (polytope) element of P – it’s a non-element within the facet of type s1 ⊕ s1 ⊕ s2.

Type 9. Quad triplets dcd, dcd, ccc, ccc. Beginning with type dcd, V10 = V1+ V2 – V5. This polytope has a fifth quad element Q5, a parallelogram with vertices V4, V6, V9, and V10. Q5 can be included in set Q with the quadtope definition still met. This polytope has 12 facets - five P([2]), five P([1],2), and two s4, with f-vector (10,35,55,40,12). This polytope has a kind of rotational symmetry to it, with automorphism group the dihedral group D5, although with a seemingly unavoidable crimp at V10. The quads in the P([2]) elements are two apart (mod 5).

From types 6 – 8, a fifth quad can be added, so there are more 5-dimensional pure quadtope types. (I'll accept this answer for now & complete the list later.) The figure below contains diagrams for types 2 – 9, showing the quad arrangements and vertices. In the diagrams, arcs are used to join vertices to indicate that they are actually the same vertex.

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