
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a uniform polyhedron has
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s as
faces and is
vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of fa ...
(i.e., there is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
mapping any vertex onto any other). It follows that all vertices are
congruent.
Uniform
polyhedra may be
regular
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrum ...
(if also
face- and
edge-transitive),
quasi-regular (if also edge-transitive but not face-transitive), or
semi-regular (if neither edge- nor face-transitive). The faces and vertices need not be
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
, so many of the uniform polyhedra are also
star polyhedra.
There are two infinite classes of uniform polyhedra, together with 75 other polyhedra:
*Infinite classes:
**
prisms,
**
antiprisms.
* Convex exceptional:
** 5
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s: regular convex polyhedra,
** 13
Archimedean solids: 2
quasiregular and 11
semiregular convex polyhedra.
* Star (nonconvex) exceptional:
** 4
Kepler–Poinsot polyhedra: regular nonconvex polyhedra,
** 53
uniform star polyhedra: 14
quasiregular and 39 semiregular.
Hence 5 + 13 + 4 + 53 = 75.
There are also many degenerate uniform polyhedra with pairs of edges that coincide, including one found by John Skilling called the
great disnub dirhombidodecahedron (Skilling's figure).
Dual polyhedra to uniform polyhedra are
face-transitive (isohedral) and have regular
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
s, and are generally classified in parallel with their dual (uniform) polyhedron. The dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a
Catalan solid.
The concept of uniform polyhedron is a special case of the concept of
uniform polytope, which also applies to shapes in higher-dimensional (or lower-dimensional) space.
Definition
define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property. By a polygon they implicitly mean a polygon in 3-dimensional Euclidean space; these are allowed to be non-convex and to intersect each other.
There are some generalizations of the concept of a uniform polyhedron. If the connectedness assumption is dropped, then we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate, then we get the so-called degenerate uniform polyhedra. These require a more general definition of polyhedra. gave a rather complicated definition of a polyhedron, while
gave a simpler and more general definition of a polyhedron: in their terminology, a polyhedron is a 2-dimensional
abstract polytope with a non-degenerate 3-dimensional realization. Here an abstract polytope is a poset of its "faces" satisfying various condition, a realization is a function from its vertices to some space, and the realization is called non-degenerate if any two distinct faces of the abstract polytope have distinct realizations.
Some of the ways they can be degenerate are as follows:
*Hidden faces. Some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are usually not counted as uniform polyhedra.
*Degenerate compounds. Some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges.
*Double covers. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron. There double covers have doubled faces, edges and vertices. They are usually not counted as uniform polyhedra.
*Double faces. There are several polyhedra with doubled faces produced by Wythoff's construction. Most authors do not allow doubled faces and remove them as part of the construction.
*Double edges. Skilling's figure has the property that it has double edges (as in the degenerate uniform polyhedra) but its faces cannot be written as a union of two uniform polyhedra.
History
Regular convex polyhedra
* The
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s date back to the classical Greeks and were studied by the
Pythagoreans,
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institutio ...
(c. 424 – 348 BC),
Theaetetus Theaetetus (Θεαίτητος) is a Greek name which could refer to:
* Theaetetus (mathematician) (c. 417 BC – 369 BC), Greek geometer
* ''Theaetetus'' (dialogue), a dialogue by Plato, named after the geometer
* Theaetetus (crater), a lunar imp ...
(c. 417 BC – 369 BC),
Timaeus of Locri
Timaeus of Locri (; grc, Τίμαιος ὁ Λοκρός, Tímaios ho Lokrós; la, Timaeus Locrus) is a character in two of Plato's dialogues, ''Timaeus'' and '' Critias''. In both, he appears as a philosopher of the Pythagorean school. If there ...
(ca. 420–380 BC) and
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ...
(fl. 300 BC). The
Etruscans
The Etruscan civilization () was developed by a people of Etruria in ancient Italy with a common language and culture who formed a federation of city-states. After conquering adjacent lands, its territory covered, at its greatest extent, roug ...
discovered the regular dodecahedron before 500 BC.
Nonregular uniform convex polyhedra
* The
cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
was known by
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institutio ...
.
*
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scienti ...
(287 BC – 212 BC) discovered all of the 13
Archimedean solids. His original book on the subject was lost, but
Pappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
(c. 290 – c. 350 AD) mentioned Archimedes listed 13 polyhedra.
*
Piero della Francesca (1415 – 1492) rediscovered the five truncations of the Platonic solids: truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron, and included illustrations and calculations of their metric properties in his book ''
De quinque corporibus regularibus''. He also discussed the cuboctahedron in a different book.
*
Luca Pacioli
Fra Luca Bartolomeo de Pacioli (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as accountin ...
plagiarized Francesca's work in ''
De divina proportione'' in 1509, adding the
rhombicuboctahedron
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at ...
, calling it a ''icosihexahedron'' for its 26 faces, which was drawn by
Leonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially re ...
.
*
Johannes Kepler (1571–1630) was the first to publish the complete list of
Archimedean solids, in 1619. He also identified the infinite families of uniform
prisms and antiprisms.
Regular star polyhedra
*
Kepler (1619) discovered two of the regular
Kepler–Poinsot polyhedra, the
small stellated dodecahedron and
great stellated dodecahedron.
*
Louis Poinsot (1809) discovered the other two, the
great dodecahedron
In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel penta ...
and
great icosahedron.
*The set of four were proven complete by
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...
in 1813, and named by
Arthur Cayley in 1859.
Other 53 nonregular star polyhedra
* Of the remaining 53,
Edmund Hess (1878) discovered two, Albert Badoureau (1881) discovered 36 more, and Pitsch (1881) independently discovered 18, of which 3 had not previously been discovered. Together these gave 41 polyhedra.
* The geometer
H.S.M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
discovered the remaining twelve in collaboration with
J. C. P. Miller
Jeffrey Charles Percy Miller (31 August 1906 – 24 April 1981) was an English mathematician and computing pioneer. He worked in number theory and on geometry, particularly polyhedra, where Miller's monster refers to the great dirhombicosidodec ...
(1930–1932) but did not publish.
M.S. Longuet-Higgins
A Master of Science ( la, Magisterii Scientiae; abbreviated MS, M.S., MSc, M.Sc., SM, S.M., ScM or Sc.M.) is a master's degree in the field of science awarded by universities in many countries or a person holding such a degree. In contrast to ...
and
H.C. Longuet-Higgins independently discovered eleven of these. Lesavre and Mercier rediscovered five of them in 1947.
* published the list of uniform polyhedra.
* proved their conjecture that the list was complete.
* In 1974,
Magnus Wenninger published his book
''Polyhedron models'', which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by
Norman Johnson.
* independently proved the completeness, and showed that if the definition of uniform polyhedron is relaxed to allow edges to coincide then there is just one extra possibility (the
great disnub dirhombidodecahedron).
* In 1987,
Edmond Bonan drew all the uniform polyhedra and their duals in 3D, with a Turbo Pascal program called Polyca: almost of them were shown during the International Stereoscopic Union Congress held at the Congress Theatre, Eastbourne, United Kingdom..
* In 1993, Zvi Har'El (1949–2008) produced a complete kaleidoscopic construction of the uniform polyhedra and duals with a computer program called Kaleido, and summarized in a paper ''Uniform Solution for Uniform Polyhedra'', counting figures 1-80.
* Also in 1993, R. Mäder ported this Kaleido solution to
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimi ...
with a slightly different indexing system.
* In 2002 Peter W. Messer discovered a minimal set of closed-form expressions for determining the main combinatorial and metrical quantities of any uniform polyhedron (and its dual) given only its
Wythoff symbol.
Uniform star polyhedra

The 57 nonprismatic nonconvex forms, with exception of the
great dirhombicosidodecahedron, are compiled by Wythoff constructions within
Schwarz triangle
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere ( spherical tiling), possibly overlapping, through reflections in its edges. They were classified in .
These can be defin ...
s.
Convex forms by Wythoff construction

The convex uniform polyhedra can be named by
Wythoff construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.
Construction process
...
operations on the regular form.
In more detail the convex uniform polyhedron are given below by their Wythoff construction within each symmetry group.
Within the Wythoff construction, there are repetitions created by lower symmetry forms. The cube is a regular polyhedron, and a square prism. The
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at e ...
is a regular polyhedron, and a triangular antiprism. The
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at e ...
is also a ''rectified tetrahedron''. Many polyhedra are repeated from different construction sources, and are colored differently.
The Wythoff construction applies equally to uniform polyhedra and
uniform tilings on the surface of a sphere, so images of both are given. The spherical tilings including the set of
hosohedrons and
dihedrons which are degenerate polyhedra.
These symmetry groups are formed from the reflectional
point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries t ...
, each represented by a fundamental triangle (''p'' ''q'' ''r''), where ''p'' > 1, ''q'' > 1, ''r'' > 1 and .
*
Tetrahedral symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection ...
(3 3 2) – order 24
*
Octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhed ...
(4 3 2) – order 48
*
Icosahedral symmetry
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of t ...
(5 3 2) – order 120
*
Dihedral symmetry (''n'' 2 2), for ''n'' = 3,4,5,... – order 4''n''
The remaining nonreflective forms are constructed by
alternation operations applied to the polyhedra with an even number of sides.
Along with the prisms and their
dihedral symmetry, the spherical Wythoff construction process adds two ''regular'' classes which become degenerate as polyhedra : the ''
dihedra'' and the ''
hosohedra'', the first having only two faces, and the second only two vertices. The truncation of the regular ''hosohedra'' creates the prisms.
Below the convex uniform polyhedra are indexed 1–18 for the nonprismatic forms as they are presented in the tables by symmetry form.
For the infinite set of prismatic forms, they are indexed in four families:
#
Hosohedra ''H''
2... (only as spherical tilings)
#
Dihedra ''D''
2... (only as spherical tilings)
#
Prisms ''P''
3... (truncated hosohedra)
#
Antiprisms ''A''
3... (snub prisms)
Summary tables
And a sampling of dihedral symmetries:
(The sphere is not cut, only the tiling is cut.) (On a sphere, an edge is the arc of the great circle, the shortest way, between its two vertices. Hence, a digon whose vertices are not polar-opposite is flat: it looks like an edge.)
(3 3 2) Td tetrahedral symmetry
The
tetrahedral symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection ...
of the sphere generates 5 uniform polyhedra, and a 6th form by a snub operation.
The tetrahedral symmetry is represented by a fundamental triangle with one vertex with two mirrors, and two vertices with three mirrors, represented by the symbol (3 3 2). It can also be represented by the
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
A
2 or
,3 as well as a
Coxeter diagram: .
There are 24 triangles, visible in the faces of the
tetrakis hexahedron, and in the alternately colored triangles on a sphere:
:

(4 3 2) Oh octahedral symmetry
The
octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhed ...
of the sphere generates 7 uniform polyhedra, and a 7 more by alternation. Six of these forms are repeated from the tetrahedral symmetry table above.
The octahedral symmetry is represented by a fundamental triangle (4 3 2) counting the mirrors at each vertex. It can also be represented by the
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
B
2 or
,3 as well as a
Coxeter diagram: .
There are 48 triangles, visible in the faces of the
disdyakis dodecahedron, and in the alternately colored triangles on a sphere:
:

(5 3 2) Ih icosahedral symmetry
The
icosahedral symmetry
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of t ...
of the sphere generates 7 uniform polyhedra, and a 1 more by alternation. Only one is repeated from the tetrahedral and octahedral symmetry table above.
The icosahedral symmetry is represented by a fundamental triangle (5 3 2) counting the mirrors at each vertex. It can also be represented by the
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
G
2 or
,3 as well as a
Coxeter diagram: .
There are 120 triangles, visible in the faces of the
disdyakis triacontahedron, and in the alternately colored triangles on a sphere:

(p 2 2) Prismatic ,2 I2(p) family (D''p''h dihedral symmetry)
The
dihedral symmetry of the sphere generates two infinite sets of uniform polyhedra, prisms and antiprisms, and two more infinite set of degenerate polyhedra, the hosohedra and dihedra which exist as tilings on the sphere.
The dihedral symmetry is represented by a fundamental triangle (p 2 2) counting the mirrors at each vertex. It can also be represented by the
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
I
2(p) or
,2 as well as a prismatic
Coxeter diagram: .
Below are the first five dihedral symmetries: D
2 ... D
6. The dihedral symmetry D
p has order ''4n'', represented the faces of a
bipyramid
A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices.
The "-gonal" in the name of a bipyramid does ...
, and on the sphere as an equator line on the longitude, and n equally-spaced lines of longitude.
(2 2 2) Dihedral symmetry
There are 8 fundamental triangles, visible in the faces of the
square bipyramid (Octahedron) and alternately colored triangles on a sphere:
:
(3 2 2) D3h dihedral symmetry
There are 12 fundamental triangles, visible in the faces of the
hexagonal bipyramid and alternately colored triangles on a sphere:
:
(4 2 2) D4h dihedral symmetry
There are 16 fundamental triangles, visible in the faces of the
octagonal bipyramid
The octagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If an octagonal bipyramid is to be face-transitive, all faces must be isosceles triangles. 16-sided dice are often octagonal bipyramids.
Images
It ca ...
and alternately colored triangles on a sphere:
:
(5 2 2) D5h dihedral symmetry
There are 20 fundamental triangles, visible in the faces of the
decagonal bipyramid
In geometry, a decagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If a decagonal bipyramid is to be face-transitive, all faces must be isosceles triangles. It is an icosahedron
In geometry, an icosa ...
and alternately colored triangles on a sphere:
:
(6 2 2) D6h dihedral symmetry
There are 24 fundamental triangles, visible in the faces of the
dodecagonal bipyramid and alternately colored triangles on a sphere.
Wythoff construction operators
See also
*
Polyhedron
**
Regular polyhedron
**
Quasiregular polyhedron
**
Semiregular polyhedron
*
List of uniform polyhedra
**
List of uniform polyhedra by vertex figure
**
List of uniform polyhedra by Wythoff symbol
**
List of uniform polyhedra by Schwarz triangle
*
List of Johnson solids
*
List of Wenninger polyhedron models
*
Polyhedron model
A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material.
Since there are 75 uniform polyhedra, including the five regular c ...
*
Uniform tiling
*
Uniform tilings in hyperbolic plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive ( transitive on it ...
*
Pseudo-uniform polyhedron
*
List of shapes
Notes
References
*
Brückner, M. ''Vielecke und vielflache. Theorie und geschichte.''. Leipzig, Germany: Teubner, 1900
*
*
*
*
*
*
External links
*
Uniform Solution for Uniform PolyhedraThe Uniform PolyhedraUniform Polyhedra
Uniform polyhedron gallery''Has a visual chart of all 75''
{{Polytopes
Uniform polyhedra, *