A structural property of convex 3-polytopes (original) (raw)
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Convex 3-polytopes with exactly two types of edges
Discrete Mathematics, 1990
We consider convex 3-polytopes with exactly two types of edges. The questions of the existence of such 3-polytopes are solved. The cardinalities of all classes are determined. 0012-365X/90/$03.50 0 1990 -Elsevier Science Publishers B.V. (North-Holland)
On the simplicial 3-polytopes with only two types of edges
Discrete Mathematics, 1984
For some families of graphs of simplicial 3-polytopes with two types of edges structural properties are described, for other ones their cardinality is determined. Griinbaum and Motzkin [3], Griinbaum and Zaks [4], and Malkevitch [6] investigated the structural properties of trivalent planar graphs with at most two types of faces. It seems that the knowledge of the structure of such graphs is useful for other reasons as well (cf., e.g. Griinbaum [1, 2], Jucovi~ [5], Owens [7], Zaks [8]). The dual problem may be formulated as follows: Characterize simplicial planar graphs with at most two types of vertices. (A planar graph is simplicial if all its faces are triangles.)
Monatshefte f�r Mathematik, 1990
We call a convex subset N of a convex d-polytope P c E d a k-nucleus of P if N meets every k-face of P, where 0 < k < d. We note that P has disjoint k-nuclei if and only if there exists a hyperplane in E d which bisects the (relative) interior of every k-face of P, and that this is possible only if/~-/~< k ~< d-1.
Extremal properties for dissections of convex 3-polytopes
A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3-polytope and analyze extremal size triangulations for specific non-simplicial polytopes: prisms, antiprisms, Archimedean solids, and combinatorial d-cubes.
Construction theorems for polytopes
Israel Journal of Mathematics, 1984
Certain construction theorems are represented, which facilitate an inductive combinatorial construction of polytopes. That is, applying the constructions to a d-polytope with n vertices, given combinatorially, one gets many combinatorial d-polytopes-and polytopes only-with n + I vertices. The constructions are strong enough to yield from the 4-simplex all the 1330 4-polytopes with up to 8 vertices.
Arbitrarily large neighborly families of congruent symmetric convex 3-polytopes With Scott Kim
We construct, for any positive integer n, a family of n congruent convex polyhedra in IR 3 , such that every pair intersects in a common facet. Our polyhedra are Voronoi regions of evenly distributed points on the helix (t, cos t, sin t). The largest previously published example of such a family contains only eight polytopes. With a simple modification, we can ensure that each polyhedron in the family has a point, a line, and a plane of symmetry. We also generalize our construction to higher dimensions and introduce a new family of cyclic polytopes.
Determination of the element numbers of the regular polytopes
Geometriae Dedicata, 2012
Let be a set of n-dimensional polytopes. A set of n-dimensional polytopes is said to be an element set for if each polytope in is the union of a finite number of polytopes in identified along (n − 1)-dimensional faces. The element number of the set of polyhedra, denoted by e( ), is the minimum cardinality of the element sets for , where the minimum is taken over all possible element sets ∈ E( ). It is proved in Theorem 1 that the element number of the convex regular 4-dimensional polytopes is 4, and in Theorem 2 that the element numbers of the convex regular n-dimensional polytopes is 3 for n ≥ 5. The results in this paper together with our previous papers determine completely the element numbers of the convex regular n-dimensional polytopes for all n ≥ 2.