Regular polyhedron

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Regular polyhedron

A regular polyhedron is a polyhedron

Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
whose faces are congruent

Congruence (geometry)

In geometry, two sets of point are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translation s, rotations and reflection s....
regular polygons which are assembled in the same way around each vertex

Vertex

Vertex may refer to:...
. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive

Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if all its vertex are the same. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces....
and face-transitive - i.e. it is transitive on its flag

Flag (geometry)

In geometry, a flag is a sequence of faces of a Abstract polytope, each contained in the next, with just one face from each dimension.More formally, a flag ψ of an n-polytope is a set such that Fi ≤ Fi+1 and there is precisely one Fi in ψ for each i, ....
s. This last alone is a sufficient definition.

A regular polyhedron is identified by its Schläfli symbol

Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
of the form , where n is the number of sides of each face and m the number of faces meeting at each vertex.

and four regular star polyhedra

Star polyhedron

In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvex polygon giving it a star-like visual quality.There are two general kinds of star polyhedron:...
, the Kepler-Poinsot polyhedra:

quivalent properties

The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition:

Concentric spheres

A regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:

Symmetry

The regular polyhedra are the most symmetrical

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
of all the polyhedra.

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Encyclopedia

A regular polyhedron is a polyhedron

Polyhedron

Congruence (geometry)

Vertex

Vertex may refer to:...
. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive

Vertex-transitive

Flag (geometry)

Schläfli symbol

The nine regular polyhedra

There are five convex

Convex polygon

In geometry, a polygon can be either convex or concave....
regular polyhedra, known as the Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s:


Tetrahedron Tetrahedron A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....	Cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....	Octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	Dodecahedron Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....	Icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....

and four regular star polyhedra

Star polyhedron


Small stellated dodecahedron Small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex....	Great stellated dodecahedron Great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra.It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex....	Great dodecahedron Great dodecahedron In geometry, the great dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces , with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path....	Great icosahedron Great icosahedron In geometry, the great icosahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 20 intersecting triangular faces, with five triangles meeting at each vertex in a pentagrammic sequence....

Characteristics

Equivalent properties

The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition:

The vertices of the polyhedron all lie on a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
.
All the dihedral angle
Dihedral angle

In geometry, the angle between two Plane s is called their dihedral or torsion angle.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection....
s of the polyhedron are equal.
All the vertex figure
Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
s of the polyhedron are regular polygon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
s.
All the solid angle
Solid angle

The solid angle, O, is the angle in three-dimensional space that an object subtends at a point. It is a measure of how big that object appears to an observer looking from that point....
s of the polyhedron are congruent. (Cromwell, 1997)

Concentric spheres

A regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre:

An insphere, tangent to all faces.
An intersphere or midsphere
Midsphere

In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point....
, tangent to all edges.
A circumsphere, tangent to all vertices.

Symmetry

The regular polyhedra are the most symmetrical

Symmetry

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
s, which are named after them:

Tetrahedral
Octahedral (or cubic)
Icosahedral (or dodecahedral)

Euler characteristic

The five Platonic solids have an Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
of 2. Some of the regular stars have a different value.

Duality of the regular polyhedra

The regular polyhedra come in natural pairs, with each twin being dual

Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
to the other (i.e. the vertices of one polyhedron correspond to the faces of the other, and vice versa):

The tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
is self dual, i.e. it pairs with itself.
The cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
and octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
are dual to each other.
The icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
and dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
are dual to each other.
The small stellated dodecahedron
Small stellated dodecahedron

In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex....
and great dodecahedron
Great dodecahedron

In geometry, the great dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces , with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path....
are dual to each other.
The great stellated dodecahedron
Great stellated dodecahedron

In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra.It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex....
and great icosahedron
Great icosahedron

In geometry, the great icosahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 20 intersecting triangular faces, with five triangles meeting at each vertex in a pentagrammic sequence....
are dual to each other.

The Schläfli symbol of the dual is just the original written backwards, for example the dual of is .

For further information please see the individual articles or the general polyhedron

Polyhedron

History

Prehistory

Stones carved in shapes showing the symmetry of all five of the Platonic solids have been found in Scotland

Scotland

conventional_long_name = ScotlandAlba|common_name= Scotland|image_flag = Flag of Scotland.svg|flag_width = 130px...
and may be as much as 4,000 years old. These stones show not only the form of each of the five Platonic solids, but also the relations of duality amongst 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 of the Ashmolean Museum

Ashmolean Museum

The Ashmolean Museum on Beaumont Street, Oxford, England, is the world's first university museum. Its first building is sometimes attributed to Christopher Wren, though there is no good evidence for this claim, and was built in 1678?1683 to house the collection or cabinet of curiosities Elias Ashmole gave Oxford University in 1677....
at Oxford University. Why these objects were made, or how their creators gained the inspiration for them, is a mystery.

It is also possible that the Etruscans

Etruscan civilization

Etruscan civilization is the modern English name given to the culture and way of life of a people of ancient Italy and Corsica whom the ancient Romans called Etrusci or Tusci....
preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua

Padua

Padua is a city in the Veneto, northern Italy. It is the capital of the province of Padua and the economic and communications hub of the area. Padua's population is 212,500 ....
(in Northern Italy

Italy

Italy , officially the Italian Republic , is a country located on the Italian Peninsula in Southern Europe and on the two largest islands in the Mediterranean Sea, Sicily and Sardinia....
) in the late 1800s of a dodecahedron

Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
made of soapstone

Soapstone

Soapstone is a metamorphic rock, a talc-schist. It is largely composed of the mineral talc and is rich in magnesium. It is produced by dynamothermal metamorphism and metasomatism, which occurs at the areas where tectonic plates are subduction, changing rocks by heat and pressure, with influx of fluids, but without melting....
, and dating back more than 2,500 years (Lindemann, 1987). Pyritohedric crystals are found in northern Italy.

The earliest known written records of these shapes do come from Greek authors, who also gave the first known mathematical description of them.

Greeks

The Greeks were the first to make written records of the regular Platonic solids. Some authors (Sanford, 1930) credit Pythagoras

Pythagoras

Pythagoras of Samos was an Ionians Ancient Greeks mathematician and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician, mysticism and scientist; however some have questioned the scope of his contributions to mathematics and natural philosophy....
(550 BC) with being familiar with them all, whereas others indicate that he may only have been familiar with the tetrahedron, cube, and dodecahedron, crediting the discovery of the other two to Theaetetus

Theaetetus (mathematician)

Theaetetus of Athens, son of Euphronius, of the Athenian deme Sunium, was a classical Greece mathematician. His principal contributions were on irrational number lengths, which was included in Book X of Euclid's Elements, and proving that there are precisely five Platonic solid....
(an Athenian

Athens

Athens , the Capital and largest city of Greece, dominates the Attica periphery; as one of the List of cities by time of continuous habitation, its recorded history spans around 3,400 years....
), who in any case gave a mathematical description of all five (Van der Waerden, 1954), (Euclid, book XIII). H.S.M. Coxeter (Coxeter, 1948, Section 1.9) credits Plato

Plato

Plato , was a Classical Greece Greeks philosopher, mathematician, writer of philosophical dialogues, and founder of the Platonic Academy in Ancient Athens, the first institution of higher learning in the western world....
(400 BC) with having made models of them, and mentions that one of the earlier Pythagoreans, Timaeus of Locri

Timaeus of Locri

Timaeus of Locri was a Greeks Pythagorean philosopher living in the 5th century BC.He features in Plato's Timaeus , where he is said to come from Locri in Italy, thus of Locrians origin....
, used all five in a correspondence between the polyhedra and the nature of the universe as it was then perceived - this correspondence is recorded in Plato's dialogue Timaeus

Timaeus (dialogue)

Timaeus is a theoretical treatise of Plato in the form of a Socratic dialogue, written circa 360 Before Christ. The work puts forward speculation on the nature of the physical world....
. It is from Plato's name that the term Platonic solids is derived.

Regular star polyhedra

For almost 2000 years, the concept of a regular polyhedron remained as developed by the ancient Greek mathematicians. One might characterise the Greek definition as follows:

A regular polygon is a (convex
Convex polygon

In geometry, a polygon can be either convex or concave....
) planar figure with all edges equal and all corners equal
A regular polyhedron is a solid (convex) figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex.

This definition rules out, for example, the square pyramid

Square pyramid

In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry....
(since although all the faces are regular, the square base is not congruent to the triangular sides), or the shape formed by joining two tetrahedra together (since although all faces would be equilateral triangles, that is, congruent and regular, some vertices have 3 triangles and others have 4).

However, in addition to the Platonic solids, the modern definition of regular polyhedra also includes the regular star polyhedra

Star polyhedron

Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
and Louis Poinsot

Louis Poinsot

Louis Poinsot was a France mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a couple ....
. Star polygons were first described in the 14th century by Thomas Bradwardine

Thomas Bradwardine

Thomas Bradwardine , often called "the Profound Doctor", was an English scholar and courtier and, very briefly, Archbishop of Canterbury....
(Cromwell, 1997). Johannes Kepler realised that star polygons could be used to build star polyhedra, which have non-convex regular polygons

Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
, typically pentagram

Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
s as faces. Some of these star polyhedra may have been discovered by others before Kepler's time, but he was the first to recognise that they could be considered "regular" if one removed the restriction that regular polyhedra be convex. Later, Poinsot realised that star vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
s (circuits around each corner) can also be used, and discovered the remaining two star polyhedra. Cayley gave them English names which have become accepted. They are: (Kepler's) the small stellated dodecahedron

Small stellated dodecahedron

In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex....
and great stellated dodecahedron

Great stellated dodecahedron

In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra.It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex....
, and (Poinsot's) the great icosahedron

Great icosahedron

In geometry, the great icosahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 20 intersecting triangular faces, with five triangles meeting at each vertex in a pentagrammic sequence....
and great dodecahedron

Great dodecahedron

In geometry, the great dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces , with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path....
.

The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation

Stellation

Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again....
. The reciprocal process to stellation is called facetting

Facetting

|}In geometry, facetting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.Facetting is the reciprocal or dual process to stellation....
(or faceting). Every stellation of one polyhedron is dual

Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
, or reciprocal, to some facetting of the dual polyhedron. The regular star polyhedra can also be obtained by facetting the Platonic solids. This was first done by Bertrand.

See also Regular polytope: History of discovery

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension = n....
.

Regular polyhedra in nature

Each of the Platonic solids occurs naturally in one form or another.

The tetrahedron, cube, and octahedron all occur as crystal

Crystal

A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions....
s. These by no means exhaust the numbers of possible forms of crystals (Smith, 1982, p212), of which there are 48. Neither the regular icosahedron nor the regular dodecahedron are amongst them, although one of the forms, called the pyritohedron

Pyritohedron

In geometry, a pyritohedron is an irregular dodecahedron. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 corners....
, has twelve pentagonal faces arranged in the same pattern as the faces of the regular dodecahedron. The faces of the pyritohedron are, however, not regular, so the pyritohedron is also not regular.

Polyhedra appear in biology as well. In the early 20th century, Ernst Haeckel

Ernst Haeckel

'Ernst Heinrich Philipp August Haeckel' ,also written 'von Haeckel', was an eminent Germany biologist, natural history, philosopher, physician, professor and artist who discovered, described and named thousands of new species, mapped a genealogical tree relating all life forms, and coined many terms in biology, including phylum, ph...
described a number of species of Radiolaria, some of whose skeletons are shaped like various regular polyhedra. (Haeckel, 1904) Examples include Circoporus octahedrus, Circogonia icosahedra, Lithocubus geometricus and Circorrhegma dodecahedra; the shapes of these creatures are indicated by their names. The outer protein shells of many virus

Virus

A virus is a Optical microscope#Limitations of light microscopes infectious agent that is unable to grow or reproduce outside a host cell . Viruses infect all cellular life....
es form regular polyhedra. For example, HIV

HIV

Human immunodeficiency virus is a lentivirus that can lead to AIDS , a condition in humans in which the immune system begins to fail, leading to life-threatening opportunistic infections....
is enclosed in a regular icosahedron.

A more recent discovery is of a series of new types of carbon

Carbon

Carbon is a chemical element with chemical symbol C and atomic number 6. As a member of group 14 on the periodic table, it is nonmetallic and tetravalence?making four electrons available to form covalent bond chemical bonds....
molecule, known as the fullerene

Fullerene

Fullerene are a family of carbon Allotropy, molecules composed entirely of carbon, in the form of a hollow sphere, ellipsoid, cylinder , or plane....
s (see (Curl, 1991) for an exposition of this discovery). Although C₆₀, the most easily produced fullerene, looks more or less spherical, some of the larger varieties (such as C₂₄₀, C₄₈₀ and C₉₆₀) are hypothesised to take on the form of slightly rounded icosahedra, a few nanometres across.

In ancient times the Pythagorean

Pythagorean

Pythagorean means of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras. See:...
s believed that there was a harmony between the regular polyhedra and the orbits of the planet

Planet

A planet , as 2006 definition of planet by the International Astronomical Union , is a celestial body orbiting a star or Stellar evolution#Stellar remnants that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared the neighbourhood of planetesimals....
s. In the 17th century, Johannes Kepler

Johannes Kepler

Tycho Brahe

Tycho Brahe, born Tyge Ottesen Brahe , was a Danish nobility known for his accurate and comprehensive astronomy observations. Coming from Sk?neland, then part of Denmark, now part of modern-day Sweden, Brahe was well known in his lifetime as an astronomy and alchemy....
and for a decade tried to establish the Pythagorean ideal by finding a match between the sizes of the polyhedra and the sizes of the planets' orbits. His search failed in its original objective, but out of this research came Kepler's discoveries of the Kepler solids as regular polytopes, the realisation that the orbits of planets are not circles, and the laws of planetary motion

Kepler's laws of planetary motion

In astronomy, Kepler's three laws of planetary motion are*"The orbit of every planet is an ellipse with the sun at a Focus ."*"A line joining a planet and the sun sweeps out equal areas during equal intervals of time."...
for which he is now famous. In Kepler's time only five planets (excluding the earth) were known, nicely matching the number of Platonic solids. Kepler's work, and the discovery since that time of Uranus and Neptune, have invalidated the Pythagorean idea.