Some New Equiprojective Polyhedra Nabila Rahman, Masud Hasan, and Saad Altaful Quader Department of Computer Science and Engineering, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh. Email: [email protected], [email protected], [email protected]

Abstract A convex polyhedron P is k-equiprojective if all of its orthogonal projections, i.e, shadows, except those parallel to the faces of P are k-gon for some fixed value of k. Since 1968, it is an open problem to construct all equiprojective polyhedra. In this paper we discover some new equiprojective polyhedra. Keywords: Polyhedra, orthogonal projection, algorithm, zonohedra, prism.

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I. INTRODUCTION A polyhedron is the region bounded by a finite number of plane polygons called faces such that (i) if two faces intersect, then it is only at a common edge or a vertex, (ii) every edge of every face is an edge of exactly one other face, and (iii) faces surrounding each vertex form a simple circuit [1]. A polyhedron is convex if a line segment connecting any two of its points is entirely inside of it, otherwise it is non-convex. In a different way, a convex polyhedron is the region bounded by the intersection of a finite number of half-spaces [2]. The problem of constructing new polyhedra based on certain mathematical properties have been extensively studied since antiquity in the fields of architecture, art, ornament, nature, cartography, and even in philosophy and literature. In this regard, regularity of faces, edges and vertices, and symmetry are some properties based on which construction of polyhedra has been extensively studied. For example, in a platonic solid all faces are the same regular convex polygon and all vertices are incident to the same number of faces. (See the book [3] and the web page [4] for some interesting discussions on the history of discovering new polyhedra. Also see the PhD thesis of Hasan [5] for other different applications of constructing polyhedra.) A convex polyhedron P is k-equiprojective if its shadow is a k-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 pgonal prism is p + 2-equiprojective), and a tetrahedron is not equiprojective. See Figure 1. In 1968, in a paper entitled, “Twenty problems on convex polyhedra,” Shepherd [6] defined equiprojective polyhedra, gave the examples above, and asked

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Fig. 1: (a) A cube is 6-equiprojective. (b) A triangular prism is 5-equiprojective. (c) A tetrahedron is not equiprojective.

for a method to construct all equiprojective polyhedra. Croft, Falconer, and Guy included this problem in their book “Unsolved Problems in Geometry” [7]. So far this open problem has been addressed very little. As the pioneering work, only recently Hasan and Lubiw [8] have given a characterization and linear time recognition algorithm for equiprojective polyhedra. Using that characterization, Biedl et al. [9] have proven that there is no 3- or 4equiprojective polyhedron and triangular prism is the only 5-equiprojective, and thus the minimum equiprojective, polyhedron. They also have discovered some new equiprojective polyhedra by cutting some well known polyhedra. In this paper, in continuation of exploring the whole class of equiprojective polyhedra, we discover some new equiprojective polyhedra. Our construction also uses the characterization given by Hasan and Lubiw [8] and our technique is to apply some cutting and gluing technique on some well known polyhedra. We will have two types of polyhedra. For one type we will play with four zonohedra. For the other type we will join two prisms similar to the 26th Johnson solid. For each of our new polyhedron we give the construction and justification of their equiprojectiv-

ity. Most of these polyhedra are non-trivial and to our knowledge none of them has been discovered before as equiprojective polyhedron. II. PRELIMINARIES For the rest of the paper by projection (shadow) we mean non-degenerate orthogonal projection (shadow) (i.e, faces not parallel to the projection direction) and by polyhedra we mean it to be convex. Consider a convex polyhedron P . For edge e in face f of P , we call (e, f ) an edge-face duple. Two edge-face duples (e, f ) and (e
, f
) are parallel if e is parallel to e
and f is parallel or equal to f
. Define the direction of duple (e, f ) to be a unit vector in the direction of edge e as encountered in clockwise traversal of the outside of face f . Two edge-face duples (e, f ) and (e
, f
) compensate each other if they are parallel and their directions are opposite (i.e. one is the negation of the other). In particular, this means that either f = f
and e and e
are parallel edges lying at opposite ends of f , or f and f
are distinct parallel faces and e and e
are parallel edges lying at the “same side” of f and f
, where by “same side” it means that a plane through e and e
will have f and f
in one half-space. An edgeface duple has at most two compensating duples. A face f is called self-compensating when its edgeface duples are compensated within themselves, e.g, the edges of f are in parallel pairs. See Figure 2. f2

e3

e4 e2

f3 e1

f1

Fig. 2: Examples of some compensating edge-face duples: (e1 , f1 ) is compensated by (e2 , f1 ) and by (e4 , f2 ) but not by (e3 , f2 ). Face f3 is selfcompensating.

will subsequently give two parallel edges in a parallel face. For more information on zonohedra, see the web pages [4, 12, 13]. A k-gonal prism consists of two similar copies of kgons with each pair of corresponding edges connected by a parallelogram. The characterization of equiprojective polyhedra given in [8] is based on partitioning the edge-face duples into compensating pairs. Any edge e of the shadow of P corresponds to some edge of P . As the projection direction changes, e may leave the shadow boundary. This only happens when a face f containing e in P becomes parallel to the projection direction. In order to preserve the size of the shadow, some other edge e
must join the shadow boundary. For these events to occur simultaneously, e
must be parallel to e and be an edge of f , or of a face parallel to f . This gives some idea that the condition for equiprojectivity involves a pairing-up of parallel edge-face duples of P . Theorem 1 [8] Polyhedron P is equiprojective iff its set of edge-face duples can be partitioned into compensating pairs. Some examples would reveal the power of the above theorem. A zonohedron is equiprojective since all of its faces are self-compensating. A prism is equiprojective since the corresponding edges in two opposite bases compensate each other and the remaining faces are self-compensating. On the contrary, a pyramid is not equiprojective since the edges in the trianguler faces do not have any compensating pairs (as the triangles do not have parallel faces). In this paper, we will work with four zonohedra and two prisms. The first three zonohedra are truncated octahedron (TO), truncated cuboctahedron (TC), and truncated icosidodecahedron (TI), and they better fall in the class of Archimedean solids. The other zonohedron is rhombic dodecahedron (RD). Each of these four zonohedra has all edges of equal length. Each vertex of a TO is incident on one square and two hexagons, that of a TC is incident on one square, one hexagon and one octagon, and that of a TI is incident on one square, one hexagon and one decagon. The RD has twelve rhombic faces with vertices incident on three or four of them. See Figure 3. III. THE NEW POLYHEDRA

Zonohedra are the convex polyhedra where every face consists of parallel pairs of edges [10]. (“Zonohedra” has been defined by several mathematicians and there are some confusions in their definitions, see [10] for the history). Zonohedra have the property that every face has a parallel face with corresponding edges parallel—each edge of a face has a parallel edge in the adjacent face which in turn has another parallel edge in the other face adjacent to the later face, and so on; so a pair of parallel edges in a face

A. Polyhedra from zonohedra We will get new equiprojective polyhedra by cutting zonohedra. We define a cut to be a plane that divides a polyhedron into two pieces. Our technique is motivated by removing a “zone” from a zonohedron, where removing a zone transforms a zonohedron into another one. We have two types of construction. The first one is similar for TO, TC and TI, and we only describe for TC. The second

Hexagonal cut

Octagonal cut

Truncated octahedron (TO)

Truncated cuboctahedron (TC)

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P1 =⇒

P P2

(b) Truncated icosidodecahedron (TI)

Rhombic dodecahedron (RD)

Fig. 3: Truncated octahedron (TO), truncated cuboctahedron (TC), truncated icosidodecahedron (TI), and rhombic dodecahedron (RD) (these figures are taken from Wikipedia: http://www.wikipedia.com).

one is more complicated and is for TO, RD and one of the polyhedra constructed from TC by the first construction. First construction Consider a TC P . (See Figure 4.) Consider a cut whose plane is perpendicular and along an edge of a square of P . This cut will pass through an similar edge in the opposite square. We call this cut an edge cut. Two types of edge cuts are possible, octagonal : an edge cut that does not cut any hexagon and hexagonal : an edge cut that cuts hexagons too. See Figure 4(a). Observe that for a TO and for a TI only one type of edge cut is possible. After applying an edge cut, octagonal or hexagonal, to P we get two new polyhedra P1 and P2 . P1 and P2 are equiprojective—the faces not affected by the cut are self-compensating, the completely new octagonal face, which we call the base, is self-compensating since its edges come from pairs of parallel faces, and the remaining faces come in similar parallel pairs and in each pair the corresponding edges compensate each other. We conclude the above results in the following theorem. Theorem 2 The polyhedra P1 and P2 created from a TC by the above procedure and the polyhedra that are similarly created from TO and TI are equiprojective.

Fig. 4: Construction of P1 and P2 by applying a (octagonal) edge cut to TC.

Second construction Here the initial part of our construction is similar for TO and RD. Consider a TO (resp. RD) P . Consider a cut whose plane is perpendicular and along a diagonal of a square (resp. parallelogram) of P . This cut will pass through a similar diagonal in the opposite square (resp. parallelogram). We call this cut a diagonal cut. A diagonal cut divides P into two equal pieces each of which is called P3 (resp. P4 ). By the argument similar to that for P1 and P2 , P3 and P4 are equiprojective. See Figure 5(a). Rest of the construction is similar for P3 , P4 , and P1 . Observe that the bases of P3 , P4 , and P1 are not regular (see [14] to get the mathematical measure for P4 ). We apply an adjusting cut to each of them which is parallel to and a little distance apart from the base such that the base becomes regular. A little exercise will show that such a cut always exists and will not vanish any edge from the adjacent faces. We take two copies of this modified polyhedron and join them by their bases such that similar faces are not adjacent. See Figure 5. The resulting polyhedron P6 , P7 and P8 , from P3 , P4 and P1 respectively, are equiprojective, since only two self-compensating bases are lost in each case. Moreover, the polyhedron P5 which is left alone from P4 due to an adjusting cut, is also equiprojective, since the two non-self compensating faces compensate each other. See Figure 5(b). Theorem 3 The polyhedra P3 , P4 , P5 , P6 , P7 , and P8 are equiprojective. B. Joining prisms Our construction here is the generalization of the

A diagonal cut

An adjusting cut

=⇒

=⇒

P3

P

P6

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=⇒

=⇒

P

P5

=⇒

P4

P7

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Fig. 6: Modified 26th Johnson solid by joining arbitrary prisms. =⇒

P1

(c)

P8

Fig. 5: Construction of new polyhedra by applying diagonal and adjusting cuts to a TO, a RD and P1 .

26th Johnson solid, which is also called as gyrobifastigium. Gyrobifastigium is the join of two identical triangular prisms and is shown in Figure 6(a). Note that gyrobifastigium is 6-equiprojective. The construction of gyrobifastigium can be generalized by generalizing the two prisms. These two prisms can be k1 - and k2 -gonal, for arbitrary values of k1 , k2 . The only criterion required is that the two faces by which the two prisms are joined should have solid angles at their edges such that after the joining the resulting polyhedron is convex. We call this generalized construction equiprojective bi-prism. Our construction is shown in Figure 6(b), where the two prisms joined are triangular and 4-gonal respectively. The justification of why an equiprojective bi-prism is equiprojective is easy to follow. Any prism is equiprojective and in the joining of two prisms we only lose two self-compensating faces. Moreover, observe that an equiprojective bi-prism P is (k1 + k2 )equiprojective, since the two prisms contribute exactly k1 and k2 edges, respectively, in the shadow of P. Theorem 4 Equiprojective bi-prisms are (k1 + k2 )equiprojective, where the two joining prisms are k1 and k2 -gonal respectively.

IV. CONCLUSION In this paper we discovered some new equiprojective polyhedra. We believe that the whole class of equiprojective polyhedra is very rich, and so the actual open problem is still open: construct all equiprojective polyhedra. We also believe that our technique of cutting and gluing existing polyhedra may help discovering such an algorithm. Any generalized algorithm for constructing even a subclass of equiprojective polyhedra would be interesting and challenging.

REFERENCES [1] H.S.M. Coxeter, Regular Polytopes, New York: Dover Publications Inc., 1973. [2] G.M. Ziegler, Lectures on Polytopes, Graduate Text in Mathematics, vol. 152, New York: Springer, 1995. [3] P.R. Cromwell, Polyhedra, Cambridge: Cambridge University Press, 1997. [4] G. Hart, Zonohedra, http://www.georgehart. com/virtual-polyhedra/zonohedra-info. html. [5] M. Hasan, Reconstruction and Visualization of Polyhedra Using Projections, PhD Thesis, School of Computer Science, University of Waterloo, Canada, 2005. [6] G.C. Shephard, Twenty problems on convex polyhedra II, Math. Gaz., vol. 52, 1968, 359– 367.

[7] H. Croft, K. Falconer, and R. Guy, Unsolved Problems in Geometry, Springer-Verlag, 1991. [8] M. Hasan and A. Lubiw, Equiprojective polyhedra, Proc. 15th Canadian Conference on Computational Geometry, Aug. 2003, pp. 47–50. [9] T. Biedl, M. Hasan, C.S. Kaplan, A. L´ opezOrtiz, and A. Lubiw, Smallest and Some New Equiprojective Polyhedra, Manuscript, 2006. [10] J.E. Taylor, Zonohedra and generalized zonohedra, American Mathematical Monthly, vol. 99, 1992, pp. 108–111. [11] M. Hasan and A. Lubiw, Equiprojective polyhedra, Manuscript (Update version of [8]), 2006.

[12] D. Eppstein, The Geometry Junkyard: Zonohedra, http://www.ics.uci.edu/~eppstein/ junkyard/zono.html. [13] D. Eppstein, Zonohedra and Zonotopes, Mathematica in Education and Research, vol. 5, 1996, pp. 15-21, http://www.ics.uci.edu/ ~eppstein/pubs/Epp-TR-95-53.pdf. [14] E.W. Weisstein, RhombicDodecahedron, Math World – A Wolfram Web Resource, http://mathworld.wolfram.com/ RhombicDodecahedron.html.