In geometry, a composite polyhedron is a convex polyhedron that produces two convex, regular-faced polyhedra when sliced by a plane.[dubious–discuss] Repeated slicing of this type until it cannot produce more such polyhedra again is called the elementary polyhedron or non-composite polyhedron.
Definition and examples
The regular icosahedron is a composite, because of the construction by attaching two pentagonal pyramids onto the bases of a pentagonal prism. Slicing three pyramids of a regular icosahedron produces a tridiminished icosahedron, which cannot produce more convex, regular-faced polyhedra again.
A convex polyhedron with regular faces is said to be composite if there exists a plane through a cycle of its edges that is not a face. Slicing the polyhedron on this plane produces two convex-regular-faced polyhedra, having together the same faces as the original polyhedron, along with two new faces on the plane of the slice.[1] Repeated slicing of a polyhedron that cannot produce more convex, regular-faced polyhedra again is called the elementary polyhedron or non-composite polyhedron. One can alternatively define a composite polyhedron as the result of attaching two or more non-composite polyhedra.[2][3]
The regular-faced elementary polyhedra can be enumerated from the convex regular-faced polyhedra. Zalgaller (1967) expressed interest in enumerating the elementary polyhedra whose faces are either regular polygons or the sums of regular polygons, providing twenty-eight examples. These are called Zalgaller solids.[6][4]Ivanov (1971) and Pryakhin (1973) provide six more examples, respectively the five Ivanov solids and one Pryakhin solid.[7][8][4].
