Equiprojective polyhedra

Convex polyhedron property

In mathematics, a convex polyhedron is defined to be  k {\displaystyle k} -equiprojective if every orthogonal projection of the polygon onto a plane, in a direction not parallel to a face of the polyhedron, forms a  k {\displaystyle k} -gon. For example, a cube is 6-equiprojective: every projection not parallel to a face forms a hexagon, More generally, every prism over a convex  k {\displaystyle k} is  ( k + 2 ) {\displaystyle (k+2)} -equiprojective.[1][2] Zonohedra are also equiprojective.[3] Hasan and his colleagues later found more equiprojective polyhedra by truncating equally the tetrahedron and three other Johnson solids.[4]

Hasan & Lubiw (2008) shows there is an  O ( n log ⁡ n ) {\displaystyle O(n\log n)} time algorithm to determine whether a given polyhedron is equiprojective.[5]

## References

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1. ^ Shephard, G. C. (1968). "Twenty Problems on Convex Polyhedra: Part I". The Mathematical Gazette. 52 (380): 136–147\. doi:10.2307/3612678. JSTOR 3612678. See Problem IX.
2. ^ Croft, Hallard; Falconer, Kenneth; Guy, Richard (1991). Unsolved Problems in Geometry: Unsolved Problems in Intuitive Mathematics. p. 60\. doi:10.1007/978-1-4612-0963-8. ISBN 978-1-4612-0963-8.
3. ^ Buffière, Thèophile (2023). "Many equiprojective polytopes". arXiv:2307.11366 [math.MG].
4. ^ Hasan, Masud; Hossain, Mohammad Monoar; Lopez-Ortiz, Alejandro; Nusrat, Sabrina; Quader, Saad Altaful; Rahman, Nabila (2010). "Some New Equiprojective Polyhedra∗". arXiv:1009.2252 [cs.CG].
5. ^ Hasan, Masud Hasan; Lubiw, Anna (2008). "Equiprojective polyhedra". Computational Geometry. 40 (2): 148–155\. doi:10.1016/j.comgeo.2007.05.002.