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# Polyhedrons

A polyhedron is a closed, three-dimensional solid bounded entirely by at least four polygons, no two of which are in the same plane. Polygons are flat, two-dimensional figures (planes) bounded by straight sides. A square and a triangle are two examples of polygons.

The number of sides of each polygon is the major feature distinguishing polyhedrons from one another. Some common polygons are the triangle (with three sides), the quadrilateral (with four sides), the pentagon (with five sides), the hexagon (with six sides), the heptagon (with seven sides), and the octagon (with eight sides).

A regular polygon, like the square, is one that contains equal interior angles and equal side lengths. A polygon is considered irregular if its interior angles are not equal or if the lengths of its sides are not equal.

Each of the polygons of a polyhedron is called a face. A straight side that intersects two faces is called an edge. A point where three or more edges meet is called a vertex. The illustration below indicates these features for a cube, which is a well-known polyhedron comprised of six square faces.

The relationship between the number of vertices (v ), faces (f ), and edges (e ) is given by the equation v + f e = 2. For example, the cube has 8 vertices, 6 faces, and 12 edges, which gives 8 + 6 12 = 2. The value of v + f e for a polyhedron is called the Euler characteristic of the polyhedron's surface, named after the Swiss mathematician Leonhard Euler (17071783). Using the Euler characteristic and knowing two of the three variables, one can calculate the third variable.

## Platonic and Archimedean Solids

There are many groupings of polyhedrons classified by certain characteristicstoo many to discuss here. One common group is known as the Platonic solids, so-called because its five members appeared in the writings of Greek philosopher Plato. The Platonic solids are within the larger grouping known as regular polyhedrons, in which the polygons of each are regular and congruent (that is, all polygons are identical in size and shape and all edges are identical in length), and are characterized by the same number of polygons meeting at each vertex.

The illustration below depicts the five Platonic solids (from left to right): tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

The tetrahedron consists of four triangular faces, and is represented as {3, 3}, in which the first 3 indicates that each face consists of three sides and the second 3 indicates that three faces meet at each vertex. The cube, sometimes called a hexahedron, has six square faces, and is represented as {4, 3}. The octahedron contains eight equilateral triangles, and is constructed by placing two identical square-based pyramids base to base. The octahedron is represented as {3, 4}. The dodecahedron consists of five sides to each face, and three pentagons meeting at each of the polyhedron' twenty vertices. It is represented by {5, 3}. The icosahedron is made by placing five equilateral triangles around each vertex. It contains congruent equilateral triangles for its twenty faces and twelve vertices, and is described as {3, 5}.

Archimedean Solids. Another common group of polyhedrons is the Archimedean solids, in which two or more different types of polygons appear. Each face is a regular polygon, and around every vertex the same polygons appear in the same sequence. For example, a truncated dodecahedron is made of the pentagon-pentagon-triangle sequence.

## Nets

A polyhedron can be "opened up" along some of its edges until its surface is spread out like a rug. The resulting map, similar to a dressmaker's pattern, is called a net. A net contains all faces of a polyhedron, some of them separated by angular gaps. Because a net is a flat pattern that can then be folded along the edges and taped together to regenerate the polyhedron of origin, a net therefore enables the easy construction of basic polyhedrons out of paper. The construction of polyhedron models can help make concepts in geometry easier to learn.

William Arthur Atkins with

Philip Edward Koth

## Bibliography

Henderson, Kenneth B. Modern Geometry: Its Structure and Function. St. Louis: Webster Division McGraw-Hill Book Company, 1962.

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## polyhedron

polyhedron (pŏl´ēhē´drən), closed solid bounded by plane faces; each face of a polyhedron is a polygon. A cube is a polyhedron bounded by six polygons (in this case squares) meeting at right angles. Although regular polygons are possible for any number of sides, there are only five possible regular polyhedrons, having congruent faces, each a regular polygon and meeting at equal angles. The five regular polyhedrons are also known as the Platonic solids, although they were known to the Greeks before the time of Plato. They are the tetrahedron, bounded by four equilateral triangles; the hexahedron, or cube, bounded by six squares; the octahedron, bounded by eight equilateral triangles; the dodecahedron, bounded by twelve regular pentagons; and the icosahedron, bounded by twenty equilateral triangles. The 18th-century Swiss mathematician Leonhard Euler showed that for any simple polyhedron, i.e., a polyhedron containing no holes, the sum of the number of vertices V and the number of faces F is equal to the number of edges E plus 2, or V+F=E+2.

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## polyhedron

pol·y·he·dron / ˌpäliˈhēdrən/ • n. (pl. -he·drons or -he·dra / -ˈhēdrə/ ) Geom. a solid figure with many plane faces, typically more than six. DERIVATIVES: pol·y·he·dral / -ˈhēdrəl/ adj. pol·y·he·dric / -ˈhēdrik/ adj.

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## polyhedron

polyhedron In geometry, three-dimensional solid figure whose surface is made up of polygons. The polygons are called the faces of the polyhedron, and the points at which they meet are the vertices.

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