On Perfect 4-Polytopes

The concept of perfection of a polytope was introduced by S. A. Robertson. Intuitively speaking, a polytope P is perfect if and only if it cannot be deformed to a polytope of different shape without changing the action of its symmetry group G(P ) on its face-lattice F (P ). By Rostami’s conjecture, the perfect 4-polytopes form a particular set of Wythoffian polytopes. In the present paper first this known set is briefly surveyed. In the rest of the paper two new classes of perfect 4-polytopes are constructed and discussed, hence Rostami’s conjecture is disproved. It is emphasized that in contrast to an existing opinion in the literature, the classification of perfect 4-polytopes is not complete as yet. 1. Perfect polytopes First we briefly summarize some basic notions about polytopes and symmetry [2, 16, 21, 22]. A (convex) n-polytope P is an intersection of finitely many closed half-spaces in a Euclidean space, which is bounded and n-dimensional. A supporting hyperplane of an npolytope in E is an affine (n−1)-plane H such that H ∩P is non-empty and P lies in one of the closed half-spaces bounded by H. A proper face F of P is the non-empty intersection of P with a supporting hyperplane H. A proper face of dimension 0, 1, k and n−1 is called a vertex, edge, k-face and facet, respectively. The empty set ∅ and P itself are improper faces of dimension −1 and n, respectively. For an n-polytope P , let fi(P ) denote the number of i-faces of P . Then the n-tuple f(P ) = (f0(P ), f1(P ), . . . , fn−1(P )) is called the f-vector of P . The set F (P ) of all faces of a polytope P ordered by inclusion forms a lattice, the face-lattice of P . We say that the polytopes P and Q are combinatorially equivalent if and only if there is a lattice isomorphism λ : F (P )→ F (Q). On the other hand, in case F (P ) and F (Q) are anti-isomorphic, i.e. there is an order-reversing bijection between them, then P and Q are said to be dual to each other (or duals of each other). 0138-4821/93 $ 2.50 c ©2002 Heldermann Verlag 244 G. Gévay: On Perfect 4-Polytopes By a symmetry transformation of an n-polytope P we mean an isometry of E keeping P setwise fixed. The group G(P ) of all symmetry transformations of P is called the symmetry group of P . The action of G(P ) on P induces an action αP on F (P ), αP : G(P )×F (P )→ F (P ). Following Robertson [11, 21], we define an equivalence relation on the set of all n-polytopes. Definition 1.1. Two n-polytopes P and Q are said to be symmetry equivalent if and only if there exists an isometry φ of E and a face-lattice isomorphism λ : F (P )→ F (Q) such that for each g ∈ G(P ) and each A ∈ F (P ), λ ( g(A) ) = (φgφ−1) ( λ(A) ) . Each symmetry equivalence class is called a symmetry type. We are now ready to define perfection of polytopes. Definition 1.2. A polytope P is said to be perfect if and only if all polytopes symmetry equivalent to P are similar to P . (We note that here similarity is meant in the usual geometric sense.) We have two remarks here. First, the notions of symmetry equivalence and perfection have been elaborated by Robertson not only for the restricted class of polytopes, but for studying the much wider class of n-solids [11, 20, 21]. On the other hand, in the particular class of 3-polytopes the symmetry types are well known and applied for a long time. Namely, the types of “simple closed crystal forms” are just the symmetry types (in the sense just defined) belonging to the crystallographic point groups. These usually are listed in standard crystallographic textbooks (see e.g. [3, 18]) and have been determined even for the non-crystallographic point groups as well [12, 17]. Moreover, in [10, p. 144] a definition of the type of crystal forms appears which is consistent with Definition 1.1. There are some known constructions by which new perfect polytopes can be obtained from given perfect polytopes. These are the polar P ∗ of P , and the binary operations P Q and P Q. The polar of P is defined by P ∗ = {y ∈ E : ∀x ∈P, 〈x, y〉 ≤ 1}, where 0 is in the interior of P [2, 16]. The additional condition, i.e. that the origin is an interior point of P , guarantees that P ∗ is a polytope (actually, the dual of P ). For our purposes, however, it is more appropriate to choose a sharper form of this notion, as used by Robertson and his co-workers [11, 19, 20]. Namely, we require that the origin coincide with the centroid of P . (We note that, equivalently, it is also said that P and P ∗ are reciprocal of each other [1, 6].) In this case G(P ) and G(P ∗) are equal. P Q is called the rectangular product by Pólya [6, 21], or simply the product by Robertson et al. [11], and is defined as follows. Let P and Q an m-polytope in E and an n-polytope in E, respectively. Then let E× E be identified with E by mapping (x, y) ∈ E× E to z ∈ E, where zi = xi for i = 1, . . . ,m, and zm+j = yj for j = 1, . . . , n. Thus E and E are embedded as the images of E × 0 and 0 × E, as orthogonal complements. Then P Q is taken as the image of the Cartesian product P ×Q under this mapping. The co-product [11] P Q is defined using the same identification of E× E with E as above. Then P Q equals the convex hull conv ( P ×{d} ∪ {c}×Q ) , where c and d denotes the centroid of P and Q, respectively. G. Gévay: On Perfect 4-Polytopes 245 An n-polytope which is not isometric to a product or co-product of polytopes of dimension less than n is called prime. In fact, either of the first two conditions is sufficient, since we have the equality (P Q)∗ = P ∗ Q∗ for any polytopes P and Q with common centroid [11]. Now it is known [19] that P ∗ is perfect if P perfect. On the other hand, P Q and P Q are perfect only if P and Q are isometric and perfect. Perfection is a kind of generalization of the notion of regularity. In fact, every regular polytope is perfect [21]. Recall that by one of the usual definition an n-polytope P is regular in case for all k, 0 ≤ k ≤ n− 1, the symmetry group G(P ) of P is transitive on the k-faces of P . In other words, for all k, 0 ≤ k ≤ n− 1, there is a single orbit of k-faces of P under the action of G(P ). Thus, in the study of perfect polytopes it is useful to introduce the following notion [19, 21]. Definition 1.3. The orbit vector of an n-polytope P is θ(P ) = ( θ0, . . . , θn−1 ) , where θi is the number of orbits of i-faces of P , for each i = 0, . . . , n− 1, under the action of G(P ). Non-regular perfect polytopes appear first in dimension 3 (perfect 2-polytopes coincide with the regular polygons). These are the cuboctahedron, the icosidodecahedron, and their polars, which are the rhombic dodecahedron and the rhombic triacontahedron, respectively [21]. Naturally arises the following Problem 1.4. Characterize the orbit vectors of perfect n-polytopes.