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 (globally) projective polyhedron is a
tessellation
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
of the
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has bas ...
. These are projective analogs of
spherical polyhedra
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most ...
– tessellations of the
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
– and
toroidal polyhedra
In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topological genus () of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.
Variations in definition
Toroidal polyhedr ...
– tessellations of the toroids.
Projective polyhedra are also referred to as elliptic tessellations or elliptic tilings, referring to the projective plane as (projective)
elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
, by analogy with
spherical tiling
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most c ...
,
a synonym for "spherical polyhedron". However, the term
elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
applies to both spherical and projective geometries, so the term carries some ambiguity for polyhedra.
As
cellular decomposition In geometric topology, a cellular decomposition ''G'' of a manifold ''M'' is a decomposition of ''M'' as the disjoint union of cells (spaces homeomorphic to ''n''-balls ''Bn'').
The quotient space ''M''/''G'' has points that correspond to the ce ...
s of the projective plane, they have
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
1, while spherical polyhedra have Euler characteristic 2. The qualifier "globally" is to contrast with ''locally'' projective polyhedra, which are
defined
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional defini ...
in the theory of
abstract polyhedra.
Non-overlapping projective polyhedra (
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
1) correspond to
spherical polyhedra
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most ...
(equivalently,
convex polyhedra
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
) with
central symmetry
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
. This is elaborated and extended below in
relation with spherical polyhedra and
relation with traditional polyhedra
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
.
Examples
![Hemicube](https://upload.wikimedia.org/wikipedia/commons/8/8d/Hemicube.svg)
The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the
centrally symmetric
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
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 edges c ...
s, as well as two infinite classes of even
dihedra
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...
and
hosohedra
In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.
A regular -gonal hosohedron has Schläfli symbol with each spherical lune havi ...
:
[Coxeter, ''Introduction to geometry'', 1969, Second edition, sec 21.3 ''Regular maps'', p. 386-388]
*
Hemi-cube, /2
*
Hemi-octahedron
A hemi-octahedron is an abstract regular polyhedron, containing half the faces of a regular octahedron.
It has 4 triangular faces, 6 edges, and 3 vertices. Its dual polyhedron is the hemicube.
It can be realized as a projective polyhedron (a tes ...
, /2
*
Hemi-dodecahedron
A hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 6 pentagons), which can be visualized by const ...
, /2
*
Hemi-icosahedron
A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 10 triangles), which can be visualized by constr ...
, /2
* Hemi-dihedron, /2, p>=1
* Hemi-hosohedron, /2, p>=1
These can be obtained by taking the quotient of the associated spherical polyhedron by the
antipodal map
In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true d ...
(identifying opposite points on the sphere).
On the other hand, the tetrahedron does not have central symmetry, so there is no "hemi-tetrahedron". See
relation with spherical polyhedra below on how the tetrahedron is treated.
Hemipolyhedra
![Tetrahemihexahedron](https://upload.wikimedia.org/wikipedia/commons/8/81/Tetrahemihexahedron.png)
Note that the prefix "hemi-" is also used to refer to
hemipolyhedra
In geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center. These "hemi" faces lie parallel to the faces of some other symmetrical polyhedron, and their count is half the number of faces of that other p ...
, which are
uniform polyhedra
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also fa ...
having some faces that pass through the center of symmetry. As these do not define spherical polyhedra (because they pass through the center, which does not map to a defined point on the sphere), they do not define projective polyhedra by the quotient map from 3-space (minus the origin) to the projective plane.
Of these uniform hemipolyhedra, only the
tetrahemihexahedron
In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin dia ...
is topologically a projective polyhedron, as can be verified by its
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
and visually obvious connection to the
Roman surface
In mathematics, the Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; ...
. It is 2-covered by 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 ...
, and can be realized as the quotient of the spherical cuboctahedron by the antipodal map. It is the only uniform (traditional) polyhedron that is projective – that is, the only uniform projective polyhedron that
immerses in Euclidean three-space as a uniform traditional polyhedron.
Relation with spherical polyhedra
There is a 2-to-1
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete sp ...
of the sphere to the projective plane, and under this map, projective polyhedra correspond to spherical polyhedra with
central symmetry
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
– the 2-fold cover of a projective polyhedron is a centrally symmetric spherical polyhedron. Further, because a
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete sp ...
is a
local homeomorphism
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
If f : X \to Y is a local homeomorphism, X is said to be an ...
(in this case a
local isometry), both the spherical and the corresponding projective polyhedra have the same
abstract vertex figure
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines.
A geometric polytope is said to be ...
.
For example, the 2-fold cover of the (projective)
hemi-cube is the (spherical) cube. The hemi-cube has 4 vertices, 3 faces, and 6 edges, each of which is covered by 2 copies in the sphere, and accordingly the cube has 8 vertices, 6 faces, and 12 edges, while both these polyhedra have a 4.4.4 vertex figure (3 squares meeting at a vertex).
Further, the
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...
(of
isometries
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 ...
) of a projective polyhedron and covering spherical polyhedron are related: the symmetries of the projective polyhedron are naturally identified with the ''rotation'' symmetries of the spherical polyhedron, while the full symmetry group of the spherical polyhedron is the product of its rotation group (the symmetry group of the projective polyhedron) and the cyclic group of order 2, . See
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...
below for elaboration and other dimensions.
Spherical polyhedra without central symmetry do not define a projective polyhedron, as the images of vertices, edges, and faces will overlap. In the language of tilings, the image in the projective plane is a degree 2 tiling, meaning that it covers the projective plane twice – rather than 2 faces in the sphere corresponding to 1 face in the projective plane, covering it twice, each face in the sphere corresponds to a single face in the projective plane, accordingly covering it twice.
The correspondence between projective polyhedra and centrally symmetric spherical polyhedra can be extended to a
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the funda ...
including all spherical polyhedra (not necessarily centrally symmetric) if the classes are extended to include degree 2 tilings of the projective plane, whose covers are not polyhedra but rather the
polyhedral compound
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.
The outer vertices of a compound can be connected ...
of a non-centrally symmetric polyhedron, together with its central inverse (a compound of 2 polyhedra). This geometrizes the Galois connection at the level of finite subgroups of O(3) and PO(3), under which the adjunction is "union with central inverse". For example, the tetrahedron is not centrally symmetric, and has 4 vertices, 6 edges, and 4 faces, and vertex figure 3.3.3 (3 triangles meeting at each vertex). Its image in the projective plane has 4 vertices, 6 edges (which intersect), and 4 faces (which overlap), covering the projective plane twice. The cover of this is the
stellated octahedron
The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicte ...
– equivalently, the compound of two tetrahedra – which has 8 vertices, 12 edges, and 8 faces, and vertex figure 3.3.3.
Generalizations
In the context of
abstract polytope
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines.
A geometric polytope is said to be ...
s, one instead refers to "''locally'' projective polytopes" – see
Abstract polytope: Local topology. For example, the
11-cell
In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedr ...
is a "locally projective polytope", but is not a globally projective polyhedron, nor indeed tessellates ''any'' manifold, as it is not locally Euclidean, but rather locally projective, as the name indicates.
Projective polytopes can be defined in higher dimension as tessellations of projective space in one less dimension. Defining ''k''-dimensional projective polytopes in ''n''-dimensional projective space is somewhat trickier, because the usual definition of polytopes in Euclidean space requires taking
convex combination
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other word ...
s of points, which is not a projective concept, and is infrequently addressed in the literature, but has been defined, such as in .
Symmetry group
The symmetry group of a projective polytope is a finite (hence discrete)
[Since PO is ]compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, finite and discrete sets are identical – infinite sets have an accumulation point. subgroup of the
projective orthogonal group In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space ''V'' = (''V'',''Q'')A quadratic space is a vector space ''V'' together with a quadratic form ''Q''; ...
, PO, and conversely every finite subgroup of PO is the symmetry group of a projective polytope by taking the polytope given by images of a
fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
for the group.
The relevant dimensions are as follows: ''n''-dimensional real projective space is the projectivization of (''n''+1)-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
,
so the projective orthogonal group of an ''n''-dimensional projective space is denoted
:PO(''n''+1) = P(O(''n''+1)) = O(''n''+1)/.
If ''n''=2''k'' is even (so ''n''+1 = 2''k''+1 is odd), then O(2''k''+1) = SO(2''k''+1)× decomposes as a product, and thus
[The ]isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
/equality distinction in this equation is because the context is the 2-to-1 quotient map – PSO(2''k''+1) and PO(2''k''+1) are equal subsets of the target (namely, the whole space), hence the equality, while the induced map is an isomorphism but the two groups are subsets of different spaces, hence the isomorphism rather than an equality.
See for an example of this distinction being made. so the group of projective isometries can be identified with the group of rotational isometries.
Thus in particular the symmetry group of a projective polyhedron is the ''rotational'' symmetry group of the covering spherical polyhedron; the full symmetry group of the spherical polyhedron is then just the direct product with
reflection through the origin
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
, which is the kernel on passage to projective space. The projective plane is non-orientable, and thus there is no distinct notion of "orientation-preserving isometries of a projective polyhedron", which is reflected in the equality PSO(3) = PO(3).
If ''n''=2''k'' + 1 is odd, then O(''n''+1) = O(2''k''+2) does not decompose as a product, and thus the symmetry group of the projective polytope is not simply the rotational symmetries of the spherical polytope, but rather a 2-to-1 quotient of the full symmetry group of the corresponding spherical polytope (the spherical group is a
central extension of the projective group). Further, in odd projective dimension (even vector dimension)
and is instead a proper (index 2) subgroup, so there is a distinct notion of orientation-preserving isometries.
For example, in ''n'' = 1 (polygons), the symmetries of a 2''r''-gon is the
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
Dih
2''r'' (of order 4''r''), with rotational group the cyclic group ''C''
2''r'', these being subgroups of O(2) and SO(2), respectively. The projectivization of a 2''r''-gon (in the circle) is an ''r''-gon (in the projective line), and accordingly the quotient groups, subgroups of PO(2) and PSO(2) are Dih
''r'' and ''C''
''r''. Note that the same
commutative square of subgroups occurs for the square of
Spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a L ...
and
Pin group
The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies.
History and shareholding
The PIN Group originally traded under ...
– Spin(2), Pin
+(2), SO(2), O(2) – here going up to a 2-fold cover, rather than down to a 2-fold quotient.
Lastly, by the
lattice theorem In group theory, the correspondence theorem
(also the lattice theorem,W.R. Scott: ''Group Theory'', Prentice Hall, 1964, p. 27. and variously and ambiguously the third and fourth isomorphism theorem
) states that if N is a normal subgroup of ...
there is a
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the funda ...
between subgroups of O(''n'') and subgroups of PO(''n''), in particular of finite subgroups. Under this connection, symmetry groups of centrally symmetric polytopes correspond to symmetry groups of the corresponding projective polytope, while symmetry groups of spherical polytopes without central symmetry correspond to symmetry groups of degree 2 projective polytopes (tilings that cover projective space twice), whose cover (corresponding to the adjunction of the connection) is a compound of two polytopes – the original polytope and its central inverse.
These symmetry groups should be compared and contrasted with
binary polyhedral group
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 that ...
s – just as Pin
±(''n'') → O(''n'') is a 2-to-1 cover, and hence there is a Galois connection between binary polyhedral groups and polyhedral groups, O(''n'') → PO(''n'') is a 2-to-1-cover, and hence has an analogous Galois connection between subgroups. However, while discrete subgroups of O(''n'') and PO(''n'') correspond to symmetry groups of spherical and projective polytopes, corresponding geometrically to the covering map
there is no covering space of
(for
) as the sphere is
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
, and thus there is no corresponding "binary polytope" for which subgroups of Pin are symmetry groups.
See also
*
Spherical polyhedron
In geometry, a spherical polyhedron or spherical tiling is a tessellation, tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhe ...
*
Toroidal polyhedron
In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topological genus () of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.
Variations in definition
Toroidal polyhedr ...
Notes
References
Footnotes
General references
*
*
*
*
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