Equiprojective polyhedra

In mathematics, a convex polyhedron is defined to be ${\displaystyle k}$-equiprojective if every orthogonal projection of the polygon onto a plane, in a direction not parallel to a face of the polyhedron, forms a ${\displaystyle k}$-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 ${\displaystyle k}$ is ${\displaystyle (k+2)}$-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 ${\displaystyle O(n\log n)}$ time algorithm to determine whether a given polyhedron is equiprojective. ^[5]

References

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.