In mathematics, a

convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

is defined to be

k

-equiprojective if every

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it we ...

of the polygon onto a plane, in a direction not parallel to a face of the polyhedron, forms a

k

-gon. For example, a

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

is 6-equiprojective: every projection not parallel to a face forms a

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is de ...

, More generally, every prism over a convex

k

(k+2)

-equiprojective.

Zonohedra In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

are also equiprojective. Hasan and his colleagues later found more equiprojective polyhedra by truncating equally the tetrahedron and three other

Johnson solid In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...

s. shows there is an

O(n \log n)

time algorithm to determine whether a given polyhedron is equiprojective.

References

{{reflist, refs= {{cite arXiv , last = Buffière , first = Thèophile , year = 2023 , title = Many equiprojective polytopes , class = math.MG , eprint = 2307.11366 {{cite book , last1 = Croft , first1 = Hallard , last2 = Falconer , first2 = Kenneth , last3 = Guy , first3 = Richard , year = 1991 , title = Unsolved Problems in Geometry: Unsolved Problems in Intuitive Mathematics , url = https://books.google.com/books?id=rdDTBwAAQBAJ&pg=PA60 , pages = 60 , doi = 10.1007/978-1-4612-0963-8 , isbn = 978-1-4612-0963-8 {{cite arXiv , title = Some New Equiprojective Polyhedra∗ , first1 = Masud , last1 = Hasan , first2 = Mohammad Monoar , last2 = Hossain , first3 = Alejandro , last3 = Lopez-Ortiz , first4 = Sabrina , last4 = Nusrat , first5 = Saad Altaful , last5 = Quader , first6 = Nabila , last6 = Rahman , year = 2010 , class = cs.CG , eprint = 1009.2252 {{cite journal , journal = Computational Geometry , volume = 40 , issue = 2 , year = 2008 , pages = 148–155 , title = Equiprojective polyhedra , last1 = Hasan , first1 = Masud Hasan , last2 = Lubiw , first2 = Anna , doi = 10.1016/j.comgeo.2007.05.002 {{cite journal , last = Shephard , first = G. C. , title = Twenty Problems on Convex Polyhedra: Part I , journal =

The Mathematical Gazette ''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journ ...

, volume = 52 , issue = 380 , year = 1968 , pages = 136–147 , doi = 10.2307/3612678 , jstor = 3612678 See Problem IX. Polyhedra