Polyhedron Totally Explained

Everything about Polyhedron totally explained

Dodecahedron (Regular polyhedron)	Small stellated dodecahedron (Regular star)
Icosidodecahedron (Uniform)	Great cubicuboctahedron (Uniform star)
Rhombic triacontahedron (Uniform dual)	Elongated pentagonal cupola (Convex regular-faced)
Octagonal prism (Uniform prism)	Square antiprism (Uniform antiprism)

A polyhedron (plural polyhedra or polyhedrons) is often defined as a geometric object with flat faces and straight edges (the word polyhedron comes from the Classical Greek πολυεδρον, from poly-, stem of πολυς, "many," + -edron, form of εδρον, "base", "seat", or "face").
This definition of a polyhedron isn't very precise, and to a modern mathematician is quite unsatisfactory. Grünbaum (1994, p.43) observed that:
The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... [inthat] at each stage ... the writers failed to define what are the 'polyhedra' ... Modern mathematicians don't even agree as to exactly what makes something a polyhedron.

What is a polyhedron?

We can at least say that a polyhedron is built up from different kinds of element or entity, each associated with a different number of dimensions:

3 dimensions: The body is bounded by the faces, and is usually the volume inside them.
2 dimensions: A face is bounded by a circuit of edges, and is usually a flat (plane) region called a polygon. The faces together make up the polyhedral surface.
1 dimension: An edge joins one vertex to another and one face to another, and is usually a line of some kind. The edges together make up the polyhedral skeleton.
0 dimensions: A vertex (plural vertices) is a corner point.
-1 dimension: The nullity is a kind of non-entity required by abstract theories.

More generally in mathematics and other disciplines, "polyhedron" is used to refer to a variety of related constructs, some geometric and others purely algebraic or abstract.
A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and doesn't end abruptly or split off in different directions.
A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions.

Characteristics

Naming polyhedra Polyhedra are often named according to the number of faces. The naming system is again based on Classical Greek, for example tetrahedron (4), pentahedron (5), hexahedron (6), heptahedron (7), triacontahedron (30), and so on.
   Often this is qualified by a description of the kinds of faces present, for example the Rhombic dodecahedron vs. the Pentagonal dodecahedron.
   Other common names indicate that some operation has been performed on a simpler polyhedron, for example the truncated cube looks like a cube with its corners cut off, and has 14 faces (so it's also an example of a tetrakaidecahedron).
   Some special polyhedra have grown their own names over the years, such as Miller's monster or the Szilassi polyhedron. Edges Edges have two important characteristics (unless the polyhedron is complex):

An edge joins just two vertices.

An edge joins just two faces. These two characteristics are dual to each other. Euler characteristic The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:
ť χ = V - E + F.

For a simply connected polyhedron, χ = 2. For a detailed discussion, see Proofs and Refutations by Imre Lakatos. Duality For every polyhedron there's a dual polyhedron having faces in place of the original's vertices and vice versa. In most cases the dual can be obtained by the process of spherical reciprocation. Vertex figure For every vertex one can define a vertex figure consisting of the vertices joined to it. The vertex is said to be regular if this is a regular polygon and symmetrical with respect to the whole polyhedron.

Traditional polyhedra

In geometry, a polyhedron is traditionally a three-dimensional shape that's made up of a finite number of polygonal faces which are parts of planes; the faces meet in pairs along edges which are straight-line segments, and the edges meet in points called vertices. Cubes, prisms and pyramids are examples of polyhedra. The polyhedron surrounds a bounded volume in three-dimensional space; sometimes this interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges.
A polyhedron is said to be Convex if its surface (comprising its faces, edges and vertices) doesn't intersect itself and the line segment joining any two points of the polyhedron is contained in the interior and surface.

Symmetrical polyhedra

Many of the most studied polyhedra are highly symmetrical.
Of course it's easy to distort such polyhedra so they're no longer symmetrical. But where a polyhedral name is given, such as icosidodecahedron, the most symmetrical geometry is almost always implied, unless otherwise stated.
Some of the most common names in particular are often used with "regular" in front or implied because for each there are different types which have little in common except for having the same number of faces. These are the tetrahedron, cube, octahedron, dodecahedron and icosahedron:
ť .

Each cell in a Voronoi tessellation is a convex polyhedron. In the Voronoi tessellation of a set S, the cell A corresponding to a point c∈S is bounded (hence a traditional polyhedron) when c lies in the interior of the convex hull of S, and otherwise (when c lies on the boundary of the convex hull of S) A is unbounded.

Hollow faced or skeletal polyhedra

It isn't necessary to fill in the face of a figure before we can call it a polyhedron. For example Leonardo da Vinci devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione. In modern times, Branko Grünbaum (1994) made a special study of this class of polyhedra, in which he developed an early idea of abstract polyhedra. He defined a face as a cyclically ordered set of vertices, and allowed faces to be skew as well as planar.

Tessellations or tilings

Tessellations or tilings of the plane are sometimes treated as polyhedra, because they've quite a lot in common. For example the regular ones can be given Schläfli symbols.

Non-geometric polyhedra

Various mathematical constructs have been found to have properties also present in traditional polyhedra.

Topological polyhedra

A topological polytope is a topological space given along with a specific decomposition into shapes that are topologically equivalent to convex polytopes and that are attached to each other in a regular way.
Such a figure is called simplicial if each of its regions is a simplex, for example in an n-dimensional space each region has n+1 vertices. The dual of a simplicial polytope is called simple. Similarly, a widely studied class of polytopes (polyhedra) is that of cubical polyhedra, when the basic building block is an n-dimensional cube.

Abstract polyhedra

An abstract polyhedron is a partially ordered set (poset) of elements. Theories differ in detail, but essentially the elements of the set correspond to the body, faces, edges and vertices of the polyhedron. The empty set corresponds to the null polytope, or nullitope, which has a dimensionality of -1. These posets belong to the larger family of abstract polytopes in any number of dimensions.

Polyhedra as graphs

Any polyhedron gives rise to a graph, or skeleton, with corresponding vertices and edges. Thus graph terminology and properties can be applied to polyhedra. For example:

Due to Steinitz theorem convex polyhedra are in one-to-one correspondence with 3-connected planar graphs.

The tetrahedron gives rise to a complete graph (K₄). It is the only polyhedron to do so.

The octahedron gives rise to a strongly regular graph, because adjacent vertices always have two common neighbors, and non-adjacent vertices have four.

The Archimedean solids give rise to regular graphs: 7 of the Archimedean solids are of degree 3, 4 of degree 4, and the remaining 2 are chiral pairs of degree 5.

History

Prehistory

Stones carved in shapes showing the symmetries of various polyhedra have been found in Scotland and may be as much a 4,000 years old. These stones show not only the form of various symmetrical polyehdra, but also the relations of duality amongst some of them (that is, that the centres of the faces of the cube gives the vertices of an octahedron, and so on). Examples of these stones are on display in the John Evans room

of the Ashmolean Museum at Oxford University. It is impossible to know why these objects were made, or how the sculptor gained the inspiration for them.
Other polyhedra have of course made their mark in architecture - cubes and cuboids being obvious examples, with the earliest four-sided pyramids of ancient Egypt also dating from the Stone Age.
The Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua (in Northern Italy) in the late 1800s of a dodecahedron made of soapstone, and dating back more than 2,500 years (Lindemann, 1987). Pyritohedric crystals are found in northern Italy.

Greeks

The earliest known written records of these shapes come from Classical Greek authors, who also gave the first known mathematical description of them. The earlier Greeks were interested primarily in the convex regular polyhedra, while Archimedes later expanded his study to the convex uniform polyhedra.

Muslims and Chinese

After the end of the Classical era, Islamic scholars continued to make advances, for example in the tenth century Abu'l Wafa described the convex regular and quasiregular spherical polyhedra. Meanwhile in China, dissection of the cube into its characteristic tetrahedron (orthoscheme) and related solids was used as the basis for calculating volumes of earth to be moved during engineering excavations.

Renaissance

Much to be said here: Piero della Francesca, Pacioli, Leonardo Da Vinci, Wenzel Jamnitzer, Durer, etc. leading up to Kepler.

Star polyhedra

For almost 2,000 years, the concept of a polyhedron had remained as developed by the ancient Greek mathematicians. Johannes Kepler realised that star polygons could be used to build star polyhedra, which have non-convex regular polygons, typically pentagrams as faces. Some of these star polyhedra may have been discovered before Kepler's time, but he was the first to recognise that they could be considered "regular" if one removed the restriction that regular polytopes be convex. Later, Louis Poinsot realised that star vertex figures (circuits around each corner) can also be used, and discovered the remaining two regular star polyhedra. Cauchy proved Poinsot's list complete, and Cayley gave them their accepted English names: (Kepler's) the small stellated dodecahedron and great stellated dodecahedron, and (Poinsot's) the great icosahedron and great dodecahedron. Collectively they're called the Kepler-Poinsot polyhedra.
The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push by H. S. M. Coxeter and others in 1938, with the now famous paper The 59 icosahedra. This work has recently been re-published (Coxeter, 1999).
The reciprocal process to stellation is called facetting (or faceting). Every stellation of one polytope is dual, or reciprocal, to some facetting of the dual polytope. The regular star polyhedra can also be obtained by facetting the Platonic solids. listed the simpler facettings of the dodecahedron, and reciprocated them to discover a stellation of the icosahedron that was missing from the famous "59". More have been discovered since, and the story isn't yet ended. See also:

Regular polyhedron: History

Regular polytope: History of discovery.

Polyhedra in nature

For natural occurrences of regular polyhedra, see Regular polyhedron: History. Irregular polyhedra appear in nature as crystals.

Further Information

Get more info on 'Polyhedron'.

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