Polyhedron

History

The polyhedrons were studied formally by the former Greeks, and nowadays continue to fascinate the students, the mathematicians and the artists.

The definition above can seem sufficiently clear for the majority among us, but not for a Mathématicien. In a remark often quoted but seldom observed, Grü nbaum (1994) foot-note that:

“the Original sin in the theory of the polyhedrons goes back to Euclide, then through Kepler, Poinsot, Cauchy and much of others… that each stage… the authors failed has to define what are the “polyhedrons”…”

And for this day, there has not existed universally approved definition on with the result that something is a polyhedral .

We can at least say that a polyhedron is built starting from various kinds of elements or entities, each one associated with a number different of dimensions:

3 dimensions: the body is limited by the faces, and usually corresponds to volume included/understood inside.
2 dimensions: a face is limited by a circuit of edge, and is usually a plane area called a Polygone. The faces put together form the polyhedric surface .
1 dimension: a edge joint a top with another and vis-a-vis another, and is usually a line of a certain kind. The edges put together form the polyhedric skeleton .
0 dimension: a top is a point of corner.
-1 dimension: the nullity is a kind of not-entity required by the abstract theories .

More generally in Mathematical and in other disciplines, the polyhedral term is used to refer to a variety of constructions connected, unquestionable geometrical and others purely algebraic or abstract.

A polyhedron is an example with 3 dimensions of a more general Polytope in an unspecified number of dimensions.

Characteristics

Nomenclature

The polyhedrons are often named according to the number of faces. The nomenclature is based again on the traditional Greek, for example the Tétraèdre (4), Pentaèdre (5), Hexaèdre (6), Heptaèdre (7), triacontahedral (30) and so on.

The page Polygone contains a list of the Greek prefixes used to name the polygons, the polyhedrons and the polytopes. It is obviously enough to replace - gone by - èdre.

Edges

The edges have two important characteristics (unless the polyhedron is complex):

an edge joint simply two tops.
an edge joint simply two faces.

These two characteristics are dual one of the other.

Characteristic of Euler

That is to say a convex polyhedron , one notes:

$F$ the number of faces of this one,
$has$ the number of edges of this one,
$S$ the number of tops of this one,

One can show that one always has the relation of Euler :

F - has + S = 2 \,

for a convex polyhedron. This number is noted

\ chi \,

Duality

For each polyhedron, there exists a dual Polyèdre having faces in the place of the tops original and vice versa. In the majority of the cases, the dual one can be obtained by the process of reciprocity spherical. The dual one of a polyhedron, is obtained by connecting the centers of the adjacent faces.

Traditional polyhedrons

A polyhedral is traditionally a three-dimensional form which is composed of a number finished of polygonal faces which are parts of plans; the faces meet per pairs along the Arête S which are segments of right, and the edges meet at the named points top S . The Cubic S, the prisms and the Pyramide S are examples of polyhedrons. The polyhedron surrounds a volume limited in space to three dimensions; sometimes this interior volume is considered to be part of the polyhedron, sometimes, only surface is considered.

The traditional polyhedrons include the five regular convex polyhedrons which one names the solid of Plato : the Tetrahedron (4 faces), the Cubic (or hexahedron) (6 faces), the Octahedral (8 faces), the Dodecahedron (12 faces) and the Icosahedral (20 faces). The other traditional polyhedrons are the four regular not-convex polyhedrons (the solid of Kepler-Poinsot ), the thirteen solid of convex Archimedes and the 53 polyhedral uniforms remaining.

Smaller polyhedron

A polyhedron has at least 4 faces, 4 tops and 6 edges. The smallest polyhedron is the tetrahedron.

Convexity, concavity

A polyhedron is known as convex being if its border (including its faces and its edges) does not cut itself and if the segment uniting two unspecified points of the polyhedron belongs to this one or of its interior. In other words, a polyhedron is convex if all its diagonals are entirely contained in its interior. It is possible to give a barycentric definition of such a polyhedron: That is to say $A_1$ , $A_2$ , $\ cdots$ , $A_n$ , $n$ points not Coplanar S; the convex polyhedron $A_1A_2 {\ cdots} A_n$ is the whole of the points M barycentres of: $A_1$ , $A_2$ , $\ cdots$ , $A_n$ affected of coefficients $\ alpha_1$ , $\ alpha_2$ , $\ cdots$ , $\ alpha_n$ where each $\ alpha_i$ is positive.

Symmetrical polyhedrons

The majority of the studied polyhedrons are strongly symmetrical. There exist various classes of these polyhedrons:

uniform Summit : if all the tops are the same ones, with the direction where for two unspecified tops, there exists a symmetry isométriquement polyhedron applying the first to the second.

uniform Edge : if all the edges are the same ones, with the direction where for two unspecified edges, there isométriquement exists a symmetry of the polyhedron applying the first to the second.

uniform Face : if all the faces are the same ones, with the direction where for two unspecified faces, there isométriquement exists a symmetry of the polyhedron applying the first to the second.

Quasi-regular : if the polyhedron is of edge uniform but not that is to say of uniform face or uniform top.

Semi-regular : if the polyhedron is of top uniform but not uniform face and each face is a regular polygon. (it is one of the many definitions of the term, depend on the author, which overlaps the quasi-regular category).

A polyhedron is semi-regular if its faces consist of several kinds of regular polygons, and that all its tops are identical. Thus are for example the solid of Archimedes, the regular prisms and antiprisms. The terminology does not appear not completely stopped. One speaks sometimes about solid semi-regular of the first species to indicate those of these solids which are convex , and of solid uniforms for the general case. The polyhedral of Catalan are not semi-regular, but have identical faces and regular tops. One says sometimes such polyhedrons that they are semi-regular second species .

Regular : if the polyhedron is of top uniform, uniform edge and uniform face. (the uniformity of the tops and the uniformity of the combined edges imply that the faces are regular).

Let us start from a top and take the points located at a distance given on each edge. These points, we connect obtain the polygon of the top. If this one is regular one says that the top is regular. A polyhedron is regular if it consists of identical and regular faces all, and that all its tops are identical. They nine, are classically divided into two families:

five solids of Plato: Tetrahedron, Cubic, Octahedral, Icosahedral Dodecahedron and regular. Plato regarded these solids as the image of the perfection. Modern mathematics attaches these examples to the concept of group.
the four polyhedrons of Kepler-Poinsot, which are not convex.

Regular polyhedrons

A regular Polyèdre has regular faces and regular tops. The dual one of a regular polyhedron is also regular.

the convex regular polyhedral are also called the solid of Plato .

Quasi-regular and duaux polyhedrons

the quasi-regular polyhedral are with regular faces, of uniform Sommet and uniform Arête. There are two convex:

The quasi-regular duaux polyhedral are of uniform Arête and uniform Face . There are two convex, in correspondence with the two precedents:

Semi-regular polyhedrons and their duaux

The semi-regular term ''' ''' is variously defined. A definition consists of " polyhedrons of uniform Summit with two kinds or more faces polygonales". They are indeed the uniform polyhedrons which are neither regular, nor quasi-regular.

The convex polyhedrons and their duaux include the whole of:

There exists also much of polyhedral uniforms not-convex , including examples of various kinds of prisms.

Noble polyhedrons

A noble polyhedron is at the same time Isoèdrique (equal faces) and Isogonal (of equal corners). In addition to the regular polyhedrons, there exists much of other examples.

The dual of a noble polyhedron is also a noble polyhedron.

Other polyhedrons with regular faces

Regular equal faces

Some families of polyhedrons, where each face is an of the same polygon left:

the deltaèdres have equilateral Triangles for faces.

With regard to the polyhedrons whose faces all are of the squares: there exists only the cube, if the faces Coplanaire S are not allowed, even if they are disconnected. Otherwise, there exists also the result of the joining of six cubes on the faces of only one, all the seven of the same size; it has 30 square faces (cash for faces disconnected in the same plan like separate). This can be wide to one, two or three directions: we can consider the union of a great arbitrary number of copies of these structures, obtained by translations of (expressed in the faces of cubes) (2,0,0), (0,2,0), and/or (0,0,2), consequently with each adjacent pair having a joint cube. The result can be an unspecified whole of cubes connected with the positions ( has , B , C ), with the entireties has , B , C or one with more is even.

There does not exist particular name for the polyhedrons which have all the faces in the form of equilateral pentagons or pentagrams. There exists an infinity of between-them, but only one is convex: the dodecahedron. The remainder is assembled by (joining) combinations of regular polyhedrons describes previously: the dodecahedron, the small spangled dodecahedron, the large spangled dodecahedron and the large icosahedron.

There does not exist polyhedron whose faces all are identical and who are regular polygons with six with dimensions or more because the point of meeting of three Hexagone S regular defines a plan. (see Polyhedral oblique infinite for the exceptions).

Deltaèdres

A Deltaèdre is a polyhedron whose faces all are of the equilateral triangles. There is an infinity, but only eights are convex:

3 convex regular polyhedrons (3 of the solids of Plato)

Octahedral Tetrahedron
Icosahedral

5 polyhedron non-uniform convex (5 of the solids of Johnson)

triangular Diamond
pentagonal Diamond
Disphénoïde softened
triangular Prism triaugmenté
square Diamond gyroallongé

Solids of Johnson

Other families of polyhedrons

Pyramids

Stellations and facetings

Truncations

It is the operation which consists in planing a top or an edge. It preserves symmetries of the solid.

Truncation of the tops

This operation makes it possible to obtain seven of the solids of Archimedes starting from the solids of Plato. It is noticed indeed that while planing more and more the edges of a cube one obtains successively the truncated cube, the cuboctaèdre, the octahedral one truncated and finally the octahedral one. One can also follow this series in the other direction.

On the basis of the dodecahedron one obtains the truncated dodecahedron, the icosidodécaèdre, the truncated icosahedron, then the octahedral one.

The tetrahedron gives the truncated tetrahedron.

One can apply this operation to the large dodecahedron or the large icosahedron and obtain concave uniform solids.

Truncation of the edges

Starting from a cube, this operation gives successively a cuboctaèdre, then a rhombic dodecahedron.

Starting from a dodecahedron, one obtains the triacontahedral icosidodécaèdre then rhombic one.

Compounds

Zonoèdres

A Zonoèdre is a convex polyhedron where each face is a Polygone with a opposite Symétrie or, in an equivalent way, Rotation S with 180°.

Generalizations of polyhedrons

The “polyhedral” word was employed for a variety of objects having structural properties similar to the traditional polyhedrons.

Complex polyhedrons

A polyhedral complex is a polyhedron which is built in a complex space with three dimensions. This space has six dimensions: three real dimensions corresponding to ordinary space, with an imaginary dimension accompanying each one. See for example Coxeter (1974).

Curved polyhedrons

Certain fields of study make it possible the polyhedrons to have curved faces and edges.

Spherical polyhedrons

The surface of a sphere can be divided by segments into limited areas, to form spherical polyhedrons. Most of the theory of the symmetrical polyhedrons is derived in a more practical way of this manner.

Curved polyhedrons filling space

The two important types are:

bubbles in foams and scum.
forms filling space used in the Architecture. See for example Pearce (1978).

General polyhedrons

More recently, the Mathématiques defined a polyhedral like a whole in a space refines real (or Euclidean) of unspecified size N which has with dimensions dishes. It can be defined commen the union of a finished number of convex polyhedrons, where a convex polyhedral is an unspecified unit which is the intersection of a finished number of Demi-espace S. It can be limited or not-limited. In this direction, a Polytope is a limited polyhedron.

All the traditional polyhedrons are general polyhedrons, and moreover, there exist examples such as:

a quadrant in the plan. For example, the area of the Cartesian plan made up of all the points above and on the right the y-axis x-axis: {( X , there ): X ≥ 0, there ≥ 0}. Its with dimensions is the two positive axes.
an octant in Euclidean space with three dimensions, {( X , there , Z ): X ≥ 0, there ≥ 0, Z ≥ 0}.
a prism of infinite extension. For example, a square prism infinite doubling in the three-dimensional space, made up of a square in the plan xy swept along the axis Z : {( X , there , Z ): 0 ≤ X ≤ 1,0 ≤ there ≤ 1}.
Each cell in a paving of Voronoï is a convex polyhedron. In the paving of Voronoï of a unit S , the cell has corresponding to a point C ∈ S is limited (consequently a traditional polyhedron) when C is placed in the interior of the convex Enveloppe of S , and differently (when C is placed on the border of the convex envelope of S ) has not-is limited.

See too

External bonds

A. Javary, Treated descriptive geometry , 1881 (on Gallica): the straight line, the plan, the polyhedrons
Applet Java of projection of polytopes 4D in space 3D (in English)
very complete Pages, in French, with applet LiveGraphics3D
the encyclopedia of the polyhedrons, in English
With applet Java LiveGraphics3D
on the solids of Kepler-Poinsot
Polygons, polyhedrons and polytopes (page of the Mathcurve.com site)
the garden of the polyhedrons
All polyhedrons… or almost
Dodecahedron, maths and…

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