Escher's Stars

In M. C. Escher's Stars are the following polyhedra: a compound of three octahedra, a compound of two cubes with common 3-fold axis, a stella octangula, a compound of a cube and an octahedron, a rhombic dodecahedron, a cuboctahedron, a rhombicuboctahedron, a square trapezohedron, a trapezoidal icositetrahedron, a triakis octahedron, and all five Platonic solids.

A polyhedron is a three-dimensional object, or a closed portion of space, bounded on all sides by polygons (plane surfaces) and whose edges are shared by exactly two polygons. "Polyhedron" comes from the Greek poly for "many" and -hedron meaning "base," "seat," or "face."


Every polyhedron in three-dimensional space consists of (two-dimensional) faces, (one-dimensional) edges, and (zero-dimensional) vertices. Sometimes the term "polyhedron" is used to apply to figures in more than three dimensions; however, analogs of polyhedra in the fourth dimension or higher are also referred to as polytopes.


Polyhedra, like polygons, may be convex or non-convex. If a line that connects any two points on the surface of a polyhedron is completely inside or on the polyhedron, the figure is convex. Otherwise, it is non-convex or concave.


A polyhedron is regular if all of its faces are exactly the same size and shape and if the same number of faces meet at each vertex. There are only five regular convex polyhedra – the Platonic solids. A further four regular polyhedra, the so-called Kepler-Poinsot solids, exist that are non-convex. However, the term "regular polyhedra" is sometimes used to describe only the Platonic solids.


A convex polyhedron is said to be semiregular if its faces have a similar arrangement of non-intersecting regular plane convex polygons of two or more different types about each vertex. These solids, of which there are 13 different kinds, are commonly called the Archimedean solids. A dual of a polyhedron is another polyhedron in which faces and vertices occupy complementary locations. The duals of the Archimedean solids are known as the Catalan solids. A quasiregular polyhedron is the solid region interior to two dual regular polyhedra; only two exist: the cuboctahedron and the icoidodecahedron. There are also infinite families of prisms and antiprisms, which, like the Archimedean solids, are considered to be semiregular if all their faces are regular polygons. In total there are 92 convex polyhedra with regular polygonal faces (and not necessary equivalent vertices); these are the Johnson solids.


A self-intersecting polyhedron is a polyhedron with faces that cross other faces.


Euler's formula for polyhedra, the oldest known formula in topology, relates the number of faces of a polyhedron, the number of edges, and the number of vertices.


Also, if A denotes the number of angles of a polyhedron, then:


A = 2E, A ≥ 3F, V ≤ 2/3 E, F ≤ 2/3 E, F ≤ 2V - 4


where E is the number of edges, F is the number of faces, and V is the number of vertices.


The oldest known examples of human-made polyhedra were found on the islands of northeastern Scotland and date back to Neolithic times, between 2000 and 3000 BC. These stone figures are about two inches in diameter and many are carved into rounded forms of regular polyhedra. Examples including cubical, tetrahedral, octahedral, and dodecahedral forms, one which is the dual of the pentagonal prism, are on display in the Museum of Scotland and in Oxford's Ashmolean Museum.



1. Cromwell, Peter R. Polyhedra. Cambridge: Cambridge University Press, 1997.