Polyhedron sliced by a plane into other polyhedrons
In geometry, a composite polyhedron is a convex polyhedron that produces other polyhedra when sliced by a plane. Examples can be found in Johnson solids.
Definition and examples
A convex polyhedron 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 polyhedra, having together the same faces as the original polyhedron, along with two new faces on the plane of the slice. Repeated slicing of this type decomposes any polyhedron into non-composite polyhedra or elementary polyhedra.[1][2] Some examples of non-composite polyhedra are the prisms, antiprisms, and seventeen of the Johnson solids.[1][3]Zalgaller (1967) showed that there are twenty-eight non-composite polyhedra, which are called Zalgaller solids.[4][5]Ivanov (1971) and Pyrakhin (1973) discovered five and one more polyhedra, having total of six, which are called Ivanov solids and Pyrakhin solid, respectively.[6][7][5] Among the regular polyhedra, the regular octahedron and regular icosahedron are composite.[5]
Any composite polyhedron can be constructed by attaching two or more non-composite polyhedra. Alternatively, it can be defined as a convex polyhedron that can be separated into two or more non-composite polyhedra.[1] Examples can be found in a polyhedron that is constructed by attaching the regular base of pyramids onto another polyhedron. This process is known as augmentation, although its general meaning is constructed by attaching pyramids, cupola, and rotundas.[8][9] Some Johnson solids are examples of that construction, and they have other constructions as in elongation (a polyhedron constructed by attaching those onto the bases of a prism), and gyroelongation (a polyhedron constructed by attaching those onto the bases of an antiprism).[5][9][10]
^Ivanov, B. A. (1971). "Polyhedra with faces that are composed of regular polygons". Ukrain. Geom. Sb (in Russian) (10): 20–34. MR0301634.
^Pyrakhin, Yu. A. (1973). "Convex polyhedra with regular faces". Ukrain. Geom. Sb. (in Russian) (14): 83–88. MR0338930.
^Huybers, P. (2002). "The Morphology of Building Structures". In Sloot, Peter M.A.; Tan, C.J. Kenneth; Dongaraa, Jack J.; Hoekstra, Alfons G. (eds.). Computational Science — ICCS 2002: International Conference Amsterdam, The Netherlands, April 21–24, 2002 Proceedings, Part III. Lecture Notes in Computer Science. Vol. 2331. p. 89. doi:10.1007/3-540-47789-6. ISBN978-3-540-43594-5.
