Polyhedron
A
polyhedron is a geometric shape which in
mathematics is defined by three related meanings. In the traditional meaning it is a 3-dimensional
polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. Further generalizing the latter, there are topological polyhedra.
 |
A dodecahedron |
In classical mathematics, a
polyhedron (from
Greek πολυεδρον, from
poly-, stem of πολυς, "many," +
-edron, form of εδρον, "base", "seat", or "face") is a three-dimensional shape that is made up of a finite number of
polygonal
faces which are parts of
planes, the faces meet in
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. A polyhedron is a three-dimensional analog of a
polygon. The general term for polygons, polyhedra and even higher dimensional analogs is
polytope.
Names of polyhedra by number of faces are
tetrahedron,
pentahedron,
hexahedron,
octahedron,
decahedron, etc. Such terms are in particular used with "regular" in front or implied (in the five cases in which this is applicable) because for each there are different types which have not much in common except having the same number of faces. For a tetrahedron this applies to a much lesser extent, it is always a triangular pyramid.
Classical polyhedra include the five regular convex polyhedra:
tetrahedron (4 sides),
cube (6 sides),
octahedron (8 sides),
dodecahedron (12 side) and
icosahedron (20 sides), four regular non convex polyhedra (the
Kepler-Poinsot solids), thirteen convex
Archimedean solids and the 53 remaining
uniform polyhedra.
Dual polyhedron can also be considered classical.
Characteristics
A polyhedron is:
*
Convex if the line segment joining any two points of the polyhedron is contained in the polyhedron or its interior
*
Vertex-uniform if all vertices are the same, in the sense that for any two vertices there exists a
symmetry of the polyhedron mapping the first
isometrically onto the second
*
Edge-uniform if all edges are the same, in the sense that for any two edges there exists a symmetry of the polyhedron mapping the first isometrically onto the second
*
Face-uniform if all faces are the same, in the sense that for any two faces there exists a symmetry of the polyhedron mapping the first isometrically onto the second
*
Regular if it is vertex-uniform, edge-uniform and face-uniform; this implies that every face is a regular polygon
*
Quasi-regular if it is vertex-uniform and edge-uniform but not face-uniform, and every face is a regular polygon
*
Semi-regular if it is vertex-uniform but neither edge-uniform nor face-uniform, and every face is a regular polygon
*
Uniform if it is vertex-uniform and every face is a regular polygon, i.e. it is regular, quasi-regular, or semi-regular.
The
Euler characteristic relates the number of edges
E, vertices
V, and faces
F of a simply connected polyhedron:
V -
E +
F = 2.
Symmetry
Many polyhedra are highly
symmetric, their
symmetry groups are all
point groups and include:
*
T -
chiral tetrahedral symmetry; the rotation group for a regular
tetrahedron; order 12.
*
Td -
full tetrahedral symmetry; the symmetry group for a regular
tetrahedron; order 24.
*
Th -
pyritohedral symmetry; order 24. The symmetry of a pyritohedron [
1].
*
O -
chiral octahedral symmetry;the rotation group of the
cube and
octahedron; order 24.
*
Oh -
full octahedral symmetry; the symmetry group of the
cube and
octahedron; order 48.
*
I -
chiral icosahedral symmetry; the rotation group of the
icosahedron and the
dodecahedron; order 60.
*
Ih -
full icosahedral symmetry; the symmetry group of the
icosahedron and the
dodecahedron; order 120.
*
Cnv -
n-fold pyramidal symmetry*
Dnh -
n-fold prismatic symmetry*
Dnv -
n-fold antiprismatic symmetryThose with chiral symmetry do not have
reflection symmetry and hence have two
enantiomorphous forms which are reflections of each other. The
snub polyhedra have this property.
Main article Uniform polyhedron.
Uniform polyhedra are
vertex uniform and every face is a regular polygon.They are either regular, quasi-regular, or
semi-regular but not necessarily convex. The Uniform polyhedra include all the polyhedra mentioned above.
As conjectured by
H. S. M. Coxeter et al. in 1954 and later confirmed by J. Skilling, there are exactly 75 uniform polyhedra, plus an infinite number of
prisms and
antiprisms. Some of the antiprisms are non-convex.
The full
list of uniform polyhedra contains details of all uniform polyhedra and
List of uniform polyhedra by vertex figure exhibits some relations between the polyhedra.
Of the 39
non-convex semiregular polyhedra 17 are stellations of Archimedean solids.
Two examples of non-convex semiregular polyhedra are the
*
Tetrahemihexahedron*
Great dirhombicosidodecahedronRegular polyhedra are
vertex-uniform, edge-uniform and face-uniform -- this implies that every face is a regular polygon and all faces are congruent.
Platonic solids
There are exactly five
regular convex polyhedra. These have been known since ancient times, and are called the
Platonic solids:
*
Tetrahedron*
Hexahedron or
cube*
Octahedron*
Dodecahedron*
IcosahedronKepler-Poinsot solids
There are exactly four
regular non-convex polyhedra: the
Kepler-Poinsot solids:
*
Small stellated dodecahedron *
Great stellated dodecahedron *
Great icosahedron*
Great dodecahedronSemi-regular means
vertex-uniform but not edge-uniform. The convex ones consist of the prisms and antiprisms and the Archimedean solids.Non-convex semi-regular are listed below.
Prisms and antiprisms
4.4.3 |
4.4.4 |
4.4.5 |
4.4.6 |
4.4.8 |
4.4.10 |
4.4.12 |
3.3.3.3 |
3.3.3.4 |
3.3.3.5 |
3.3.3.6 |
3.3.3.8 |
3.3.3.10 |
3.3.3.12 |
3.3.3.17 |
There are infinitely many
semi-regular convex polyhedra in two infinite series:
*
Prisms (with 2
n-gons and
n squares) and
*
Antiprisms (with 2
n-gons and 2
n triangles)
Archimedean solid
There are 13
Archimedean solids:
Two are
quasi-regular convex polyhedra which have the additional property of being
edge-uniform.
*
Cuboctahedron (with triangles and squares)
*
Icosidodecahedron (with triangles and pentagons)
and 11 other convex polyhedra:
*
Truncated tetrahedron*
Truncated cube*
Truncated octahedron*
Truncated dodecahedron*
Truncated icosahedron*
Truncated cuboctahedron*
Truncated icosidodecahedron*
Rhombicuboctahedron*
Rhombicosidodecahedron*
Snub cube or snub cuboctahedron
*
Snub dodecahedron or snub icosidodecahedron
No other convex edge-uniform polyhedra composed of regular polygons exist than the five regular and two quasi-regular convex polyhedra, so edge uniformity and face regularity with convexity implies vertex-uniformity. (There are two other edge-uniform convex polyhedra, the
rhombic dodecahedron and the
rhombic triacontahedron, but they are not face-regular and not vertex-uniform. These are the duals of the quasi-regular convex polyhedra, and are both members of the
Catalan solids.)
For every polyhedron there is a
dual polyhedron which can be obtained, for regular polyhedra, by connecting the midpoints of the faces. For an arbitrary polyhedron, the more complicated process of spherical reciprocation is required (see
dual polyhedron). Face-uniformity of a polyhedron corresponds to vertex-uniformity of the dual and conversely, and edge-uniformity of a polyhedron corresponds to edge-uniformity of the dual.
Thus the regular polyhedra come in natural pairs: the dodecahedron with the icosahedron, the cube with the octahedron, and the tetrahedron with itself.
In most duals of uniform polyhedra, faces are irregular polygons. The exceptions are:
* The
tetrahedron which is self dual.
* The
cube and
octahedron, which are dual to each other.
* The
icosahedron and
dodecahedron, which are dual to each other.
* The
Kepler-Poinsot solids whose duals are other Kepler-Poinsot solids.
Quasi-regular duals
The duals of the quasi-regular polyhedra are
edge- and face-uniform. These are, correspondingly:
*
Rhombic dodecahedron dual of the
cuboctahedron.
*
Rhombic triacontahedron dual of the
icosidodecahedron.
and 13 other, nonconvex ones.
Pyramids and prisms
*
Pyramids are self dual.
*
Trapezohedron (such as the cube), are dual to an
antiprisms.
*
Bipyramids (such as the octahedron) are dual to
prisms.
Semi-regular duals
Main article Semiregular polyhedron.
The duals of the semi-regular polyhedra are
face-uniform. These are, correspondingly:
*
Bipyramids
*
Trapezohedra* 11 of the
Catalan solids
Stellations
Main article Stellation.
Stellation of a polyhedron is the process of extending the faces (within their planes) so that they meet to form a new polyhedron.
It the exact reciprocal to the process of
facetting which is the removal of parts of a polyhedron without creating any new vertices.
Compounds
Polyhedral compounds are formed as compounds of two or more polyhedra. These include
*
Stella octangula: compound of two
tetrahedron,
* Compound of a cube and octahedron,
* Compound of a dodecahedron and an icosahedron,
*
Compound of five tetrahedra* Compound of five octahedra,
* Compound of five tetrahedra,
* Compound of five cubes.These compounds often share the same vertices as other polyhedra and are often formed by stellation. Some are listed in the
list of Wenninger polyhedron models.
Johnson solids
Norman Johnson sought which non-uniform polyhedra had regular faces. In
1966, he published a list of 92 convex solids, now known as the
Johnson solids, and gave them their names and numbers. He did not prove there were only 92, but he did conjecture that there were no others.
Victor Zalgaller in 1969 proved that Johnson's list was complete.
Deltahedron
A
deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. There are infinitely many deltahedra, but only eight of these are convex:
* 3 regular convex polyhedra (3 of the Platonic solids)
** Tetrahedron
** Octahedron
** Icosahedron
* 5 non-uniform convex polyhedra (5 of the Johnson solids)
**
Triangular dipyramid**
Pentagonal dipyramid**
Snub disphenoid**
Triaugmented triangular prism**
Gyroelongated square dipyramidOther polyhedron with regular faces
With regard to polyhedra whose faces are all squares: if
coplanar faces are not allowed, even if they are disconnected, there is only the cube. Otherwise there is also the result of pasting six cubes to the sides of one, all seven of the same size; it has 30 square faces (counting disconnected faces in the same plane as separate). This can be extended in one, two, or three directions: we can consider the union of arbitrarily many copies of these structures, obtained by translations of (expressed in cube sizes) (2,0,0), (0,2,0), and/or (0,0,2), hence with each adjacent pair having one common cube. The result can be any connected set of cubes with positions (
a,
b,
c), with integers
a,
b,
c of which at most one is even.
There is no special name for polyhedra whose faces are all equilateral pentagons or pentagrams. There are infinitely many of these, but only one is convex: the dodecahedron. The rest are assembled by (pasting) combinations of the regular polyhedra described earlier: the dodecahedron, the small stellated dodecahedron, the great stellated dodecahedron and the great icosahedron.
There exists no polyhedron whose faces are all regular polygons with six or more sides.
Catalan solids
a
Catalan solid, or
Archimedean dual, is a
dual polyhedron to an
Archimedean solid. They are face-uniform but not vertex-uniform.
Zonohedron
A
zonohedron is a convex polyhedron where every face is a
polygon with inversion
symmetry or, equivalently, symmetry under
rotations through 180°.
More recently
mathematics has defined a
polyhedron as a set in
real affine (or
Euclidean) space of any dimensional
n that has flat sides. It could be defined as the union of a finite number of convex polyhedra, where a
convex polyhedron is any set that is the intersection of a finite number of
half-spaces. It may be bounded or unbounded. In this meaning, a
polytope is a bounded polyhedron.
All classical polyhedra are general polyhedra, and in addition there are examples like:
* A quadrant in the plane. For instance, the region of the cartesian plane consisting of all points above the horizontal axis and to the right of the vertical axis: { (
x,
y ) : x ≥ 0, y ≥ 0 }. Its sides are the two positive axes.
* An octant in Euclidean 3-space, { (
x,
y,
z ) : x ≥ 0, y ≥ 0, z ≥ 0 }
* A prism of infinite extent. For instance a doubly-infinite square prism in 3-space, consisting of a square in the
xy-plane swept along the
z-axis: { (
x,
y,
z ) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 }
* 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 classical 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.
A
topological polyhedron 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 that needs better description.
Any polyhedron gives rise to a
graph, called
skeleton, with corresponding vertices and edges. Thus
graph terminology and properties can be applied to polyhedra:
* The Archimedean solids give rise to
regular graphs: 7 Archimedean solids are
degree 3, 4 solids are degree 4, and the remaining 2 are chiral pairs of degree 5.
* The octahedron gives rise to a
strongly regular graph, because adjacent vertices have always two common neighbors, and non-adjacent vertices always four.
* Only the tetrahedron gives rise to a
complete graph (K
4).
* Due to
Steinitz theorem convex polyhedra are in one-to-one correspondence with 3-connected planar graphs.
Much of the history of polyhedra is covered in the
history section of the polytopes article.Natural occurrences of polyhedra is covered in
polytopes in nature.
*
Antiprism*
Archimedean solid*
Bipyramid*
Defect*
Deltahedron*
Deltohedron*
M.C. Escher*
Johnson solid*
Kepler-Poinsot solid*
Overview of many polyhedra, with images*
Platonic solid*
Polychoron 4 dimensional analogues to polyhedra.
*
Polyhedral compound*
Polyhedron models
*
Prism*
Semiregular polyhedron*
Spidron*
Tessellation*
Trapezohedron*
Uniform polyhedron*
Zonohedron*
Conway Polyhedron Notation - A notation for describing how to construct polyhedra.
*
Polyhedra Index Page*
Stella: Polyhedron Navigator - Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.
*
The Uniform Polyhedra*
Virtual Reality Polyhedra - The Encyclopedia of Polyhedra
*
Paper Models of Polyhedra Many links
*
Paper Models of Uniform (and other) Polyhedra*
Interactive 3D polyhedra in Java*
World of Polyhedra - comprehensive polyhedra in flash applet, showing vertices and edges (but not shaded faces)
*
Polyhedra software, die-cast models, & posters*
Electronic Geometry Models contains a peer reviewed selection of polyhedra with unusual properties.
*
Symmetry, Crystals and Polyhedra
