![]() Polyhedrons are classified and named according to the number and type of faces. The cross sections of a convex polyhedron are all convex polygons. The intersection of a plane and a polyhedron is called the cross section of the polyhedron. Convex polyhedrons, also known as Euler polyhedrons, can be defined by the equation E = v + f – e = 2, where v is the number of vertices, f is the number of faces, and e is the number of edges. Vertices are connected through the body of the polyhedron by an imaginary line called a diagonal.Ī polyhedron is classified as convex if a diagonal contains only points inside of the polyhedron. The points at which the ends of edges intersect (think of the corner of a cereal box) are the vertices. The line segments along which the faces meet are called the edges. The bounding polygons of a polyhedron are called the faces. ![]() In an interesting exception, however, crystals grow in mathematically perfect -and frequently complex - polyhedrons. Most shapes formed in nature are irregular. The pyramids are a type of polyhedron, as are geodesic domes. The cube is seen in everything from dice to clock radios CD cases and sticks of butter are in the shape of polyhedrons called parallelpipeds. Many common objects are in the shape of polyhedrons. ” A polyhedron is a solid whose boundaries consist of planes. The word polyhedron comes from the Greek prefix poly-, which means “many, ” and the root word hedron which refers to “surface. Louis: Webster Division McGraw-Hill Book Company, 1962.Ī polyhedron is a three-dimensional closed surface or solid, bounded by plane figures called polygons. Modern Geometry: Its Structure and Function. The construction of polyhedron models can help make concepts in geometry easier to learn. 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. A net contains all faces of a polyhedron, some of them separated by angular gaps. The resulting map, similar to a dressmaker's pattern, is called a net. NetsĪ polyhedron can be "opened up" along some of its edges until its surface is spread out like a rug. For example, a truncated dodecahedron is made of the pentagon-pentagon-triangle sequence. Each face is a regular polygon, and around every vertex the same polygons appear in the same sequence. Another common group of polyhedrons is the Archimedean solids, in which two or more different types of polygons appear. The tetrahedron consists of four triangular faces, and is represented as. The illustration below depicts the five Platonic solids (from left to right): tetrahedron, cube, octahedron, dodecahedron, and icosahedron. 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. One common group is known as the Platonic solids, so-called because its five members appeared in the writings of Greek philosopher Plato. There are many groupings of polyhedrons classified by certain characteristics -too many to discuss here. Using the Euler characteristic and knowing two of the three variables, one can calculate the third variable. 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 (1707 –1783). For example, the cube has 8 vertices, 6 faces, and 12 edges, which gives 8 + 6 − 12 = 2. ![]() The relationship between the number of vertices ( v ), faces ( f ), and edges ( e ) is given by the equation v + f − e = 2. ![]() The illustration below indicates these features for a cube, which is a well-known polyhedron comprised of six square faces. A point where three or more edges meet is called a vertex. A straight side that intersects two faces is called an edge. A polygon is considered irregular if its interior angles are not equal or if the lengths of its sides are not equal.Įach of the polygons of a polyhedron is called a face. 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).Ī regular polygon, like the square, is one that contains equal interior angles and equal side lengths. The number of sides of each polygon is the major feature distinguishing polyhedrons from one another. A square and a triangle are two examples of polygons. Polygons are flat, two-dimensional figures (planes) bounded by straight sides. ![]() A polyhedron is a closed, three-dimensional solid bounded entirely by at least four polygons, no two of which are in the same plane. ![]()
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