On fixed size projection of simplicial polyhedra
Introduction
A (3D) convex polyhedron P is the bounded intersection of a finite number of half-spaces. The closed surface of P is made up of planar convex polygons, called faces, the faces meet at line segments, called edges, and the edges meet at endpoints, called vertices.
The problem of constructing new polyhedra based on certain mathematical properties has been extensively studied since antiquity in the fields of architecture, art, cartography, philosophy and literature. (See [2], [4] for some interesting discussions on the history of discovering new polyhedra.) Regularity of faces, edges and vertices, and symmetry are some popular criteria for constructing new polyhedra. For example, in a Platonic solid (Platonic solids are the most primitive convex polyhedra), each vertex is incident to the same number of identical regular faces [2], [4].
A convex polyhedron P is k-equiprojective (similarly, -biprojective) if its shadow (i.e., the orthogonal projection) is a k-gon (similarly, - or -gon) in every direction, except directions parallel to faces of P. A cube is 6-equiprojective, a triangular prism is 5-equiprojective (in fact, any n-gonal prism is -equiprojective [7]), a tetrahedron is -biprojective, an octahedron is -biprojective, and a rectangular pyramid is -biprojective. See Fig. 1.
In 1968, in a paper [9] entitled “Twenty problems on convex polyhedra,” Shephard defined equiprojective polyhedra, gave the examples in Fig. 1(a, b), and asked for a method to construct all equiprojective polyhedra. Later, Croft, Falconer, and Guy included this problem in their book “Unsolved Problems in Geometry” (Problem B10) [1]. Also recently, this problem has been added to “The Open Problem Project” (Problem 76), which maintains a list of open problems in computational geometry and related fields and is edited by Demaine, Mitchell, and OʼRourke [3].
So far, this problem has not been studied extensively. Hasan and Lubiw [7] gave a characterization and an -time recognition algorithm for equiprojective polyhedra. Their characterization shows that the class of equiprojective polyhedra is rich. For example, zonohedra, which is an infinite class of convex polyhedra, are equiprojective [7].
Based on this characterization, Rahman et al. [8] and Hasan et al. [6] (these two papers are combined in [5]) discovered some non-trivial equiprojective polyhedra by cutting and gluing some polyhedra. Hasan et al. [5], [6] also proved that no 3- or 4-equiprojective polyhedra exist, and that only triangular prisms can be 5-equiprojective. However, the original problem of constructing all equiprojective polyhedra remains open.
In this paper, we address the problem in another direction, that is, we try to find which classes of polyhedra are not equiprojective. We show that simplicial polyhedra cannot be equiprojective, where a simplicial polyhedron is a convex polyhedron with only triangular faces.
Then, we extend the idea of equiprojectivity to biprojectivity, which is a natural extension. While the class of equiprojective polyhedra is rich, it is also interesting to find what other classes of polyhedra have their shadow size fixed to more than one integer. We show that simplicial polyhedra having all faces in parallel pairs do not have their shadow size fixed to two consecutive integers, i.e., they are not -biprojective, for any k.
Section snippets
Preliminaries
Let P be a convex polyhedron with m edges. For an edge e in face f of P, we call an edge-side. Two edge-sides and are parallel if e is parallel to and f is parallel or equal to . Since in a convex polyhedron a face can have at most one parallel face, an edge-side has at most three parallel edge-sides, one in the same face and the other two in the parallel face. The direction of edge-side is the unit vector in the direction of edge e as encountered in a clockwise
Simplicial polyhedra
In this section, we first show that simplicial polyhedra cannot be equiprojective. Then we show that simplicial polyhedra whose faces are in parallel pairs are not -biprojective. Both of our results are based on the characterization mentioned in Theorem 1.
Theorem 2 Simplicial polyhedra are not equiprojective.
Proof Let P be a simplicial polyhedron. For contradiction assume that P is equiprojective. We shall first prove that all faces of P are in compensating pairs. Consider a face f of P and an edge e of
Conclusion
In this paper, we have shown that simplicial polyhedra are not equiprojective. Then, we have extended the idea of equiprojectivity to biprojectivity and have shown that simplicial polyhedra having all faces in parallel pairs are not -biprojective.
We believe that there may not exist many biprojective polyhedra. This is because, once a convex polyhedron gets more and more faces, it becomes unlikely to have its projection size fixed to two integers.
The actual problem of finding all
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