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Plane tiling corresponding to a polyhedron
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations
Projective_polyhedron
Partition of a toroidal surface into polygons
self-intersecting and topologically self-dual. Projective polyhedron Skew apeirohedron (infinite skew polyhedron) Spherical polyhedron Toroidal graph Whiteley (1979);
Toroidal_polyhedron
Abstract regular polyhedron with 3 square faces
four vertices) on a projective plane. The hemicube should not be confused with the demicube – the hemicube is a projective polyhedron, while the demicube
Hemicube_(geometry)
Abstract regular polyhedron with 6 pentagonal faces
polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective
Hemi-dodecahedron
Polyhedron associated with another by swapping vertices for faces
here is closely related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra
Dual_polyhedron
regular octahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 4 triangles and 3 square), which can
Hemi-cuboctahedron
Abstract regular polyhedron with 4 triangular faces
as a projective polyhedron (a tessellation of the real projective plane by 4 triangles), which can be visualized by constructing the projective plane
Hemi-octahedron
Topics referred to by the same term
Projective connection Projective Hilbert space Projective morphism Projective polyhedron Projective resolution Projective test Projective techniques Projection
Projective
Type of geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that
Projective_geometry
Convex polyhedron with regular faces
Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons and that is not a uniform polyhedron. There are 92 such solids: 48 composed
Johnson_solid
Partition of a sphere's surface into polygons
universal covering space of the projective plane, where any tiling on a sphere has a double covering map on the real projective plane R P 2 {\displaystyle
Spherical_polyhedron
Polyhedron with regular congruent polygons as faces
regular polyhedron is a polyhedron with regular and congruent polygons as faces. Its symmetry group acts transitively on its flags. A regular polyhedron is
Regular_polyhedron
Abstract regular polyhedron with 10 triangular faces
polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane
Hemi-icosahedron
Flat-sided three-dimensional shape
In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many' and ἕδρον (-hedron) 'base, seat') is a three-dimensional figure
Polyhedron
Polyhedron with 7 faces
as the abstract polyhedron, the hemi-cuboctahedron. The tetrahemihexahedron is a non-orientable surface. It is projective polyhedron, yielding a representation
Tetrahemihexahedron
Type of polyhedron with many holes
Leonardo polyhedron is a polyhedron with a Platonic solid's rotational symmetry and genus g ≥ 2 {\displaystyle g\geq 2} . Here, a polyhedron is the unbounded
Leonardo_polyhedron
Topological invariant in mathematics
non-convex Kepler–Poinsot polyhedra. Projective polyhedra all have Euler characteristic 1, like the real projective plane, while the surfaces of toroidal
Euler_characteristic
Any of the five regular polyhedra
Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical
Platonic_solid
Catalan solid with 12 faces
convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron
Rhombic_dodecahedron
Hungarian mathematician
a professor of mathematics at the University of Szeged who worked in projective and non-Euclidean geometry, applying his research to computer generated
Lajos_Szilassi
dodecahedron Great stellated dodecahedron Abstract regular polyhedra (Projective polyhedron) Hemicube Hemi-octahedron Hemi-dodecahedron Hemi-icosahedron Archimedean
List_of_mathematical_shapes
Subspace of n-space whose dimension is (n-1)
the solution of a single linear equation. Projective hyperplanes are used in projective geometry. A projective subspace is a set of points with the property
Hyperplane
Polyhedron made from triangles that approximates a sphere
A geodesic polyhedron is a convex polyhedron made from triangles which approximates a sphere. They usually have icosahedral symmetry, such that they have
Geodesic_polyhedron
Uniform star polyhedron whose faces pass through its center
number of faces of that other polyhedron – hence the "hemi" prefix. This prefix is also used to refer to certain projective polyhedra, such as the hemi-cube
Hemipolyhedron
Method of describing higher-order polyhedra
Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified
Conway_polyhedron_notation
Great icosahedron, Great dodecahedron Abstract regular polyhedra (Projective polyhedron) Hemicube (geometry), hemi-octahedron, hemi-dodecahedron, hemi-icosahedron
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Polyhedron with seven sides
A heptahedron (pl.: heptahedra) is a polyhedron having seven sides, or faces. A heptahedron can take a large number of different basic forms, or topologies
Heptahedron
Solid with four equal triangular faces
tetrahedron is a polyhedron with four equilateral triangular faces. A regular tetrahedron is a tetrahedron (that is, a four-sided polyhedron) in which all
Regular_tetrahedron
Right-angled non-convex polyhedron
certain algebraic variety associated with the polyhedron would be a projective variety if the polyhedron could be made convex in this way. However, Adrien
Jessen's_icosahedron
which can be seen in projective symmetry of the polytopes. The Witting polytope, 3{3}3{3}3{3}3, contains the Hessian polyhedron as cells and vertex figures
Hessian_polyhedron
Quadrilateral with sides of equal length
dodecahedron is a convex polyhedron with 12 congruent rhombi as its faces. The rhombic triacontahedron is a convex polyhedron with 30 golden rhombi (rhombi
Rhombus
Polyhedron with some pattern of nonconvexity
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general
Star_polyhedron
Sphere tangent to every edge of a polyhedron
or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform
Midsphere
Polyhedra in which all vertices are the same
elongated square gyrobicupola or pseudorhombicuboctahedron is an extra polyhedron with regular faces and congruent vertices. Still, it is not generally
Archimedean_solid
Archimedean solid with 32 faces
In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi-) triangular faces and twelve (dodeca-) pentagonal faces
Icosidodecahedron
Overview of and topical guide to geometry
infinity Projective line Projective plane Oval (projective plane) Roman surface Projective space Complex projective line Complex projective plane Fundamental
Outline_of_geometry
57-cell, {5,3,5}, which have regular projective polyhedra as cells and vertex figures. The elements of an abstract polyhedron are its body (the maximal element)
List_of_regular_polytopes
Convex polyhedron projected from hypercube
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon)
Zonohedron
Archimedean solid with 26 faces
[6] and [8] projective symmetry, and numerous [2] symmetries can be constructed from various projected planes relative to the polyhedron elements. The
Truncated_cuboctahedron
Archimedean solid with 26 faces
The rhombicuboctahedron or small rhombicuboctahedron is a polyhedron with 26 faces, consisting of 8 equilateral triangles and 18 squares. It was named
Rhombicuboctahedron
General concept and operation in mathematics
electric fields. In some projective planes, it is possible to find geometric transformations that map each point of the projective plane to a line, and each
Duality_(mathematics)
Poset representing certain properties of a polytope
the projective counterparts of the Platonic solids, and can be realized as (globally) projective polyhedra – they tessellate the real projective plane
Abstract_polytope
In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces (8 triangles, 6 squares, and 6 octagons), 48 edges
Small_cubicuboctahedron
Polygonal chain whose vertices are not all coplanar
Petrie polygon Skew quadrilateral Regular skew polyhedron Skew apeirohedron (infinite skew polyhedron) Skew lines Coxeter 1973, §1.1 Regular polygons;
Skew_polygon
Archimedean solid with 62 faces
and 120 edges. Johannes Kepler in Harmonices Mundi (1618) named this polyhedron a rhombicosidodecahedron, being short for truncated icosidodecahedral
Rhombicosidodecahedron
In geometry, a monostatic polytope or unistable polyhedron is a d {\displaystyle d} -polytope which "can stand on only one face". They were described in
Monostatic_polytope
Representation of 3D and 4D polytopes
the polyhedron. The Schlegel diagram completely represents the morphology of the polyhedron. It is sometimes convenient to project the polyhedron from
Schlegel_diagram
Solid with six equal square faces
these edges form six square faces of the same size. It is an example of a polyhedron. It is a special case of a cuboid, a parallelepiped, and a rhombohedron
Cube
Graph-theoretic description of polyhedra
from the polyhedron. When the triple intersection point is not on the far side of this face, a projective transformation of the polyhedron suffices to
Steinitz's_theorem
Notation for a polyhedron's vertex figure
geometry, a vertex configuration is a shorthand notation for representing a polyhedron or tiling as the sequence of faces around a vertex. It has variously been
Vertex_configuration
Space with one dimension
{\displaystyle K} is a one-dimensional vector space over itself. The projective line over K , {\displaystyle K,} denoted P 1 ( K ) , {\displaystyle \mathbf
One-dimensional_space
Archimedean solid with 62 faces
icosidodecahedron has rectangles instead of squares. This nonuniform polyhedron is topologically equivalent to the Archimedean solid. Alternate interchangeable
Truncated_icosidodecahedron
(1623–1662) – projective geometry Christiaan Huygens (1629–1695) – evolute Giordano Vitale (1633–1711) Philippe de La Hire (1640–1718) – projective geometry
List_of_geometers
Polygon constructed from another
isosceles triangle. In the Dorman Luke construction, each face of a dual polyhedron is the dual polygon of the corresponding vertex figure. As an example
Dual_polygon
Polyhedral compromise map projection
and two Waterman projections from the W5 convex hull. To project the sphere to the polyhedron, the Earth is divided into eight octants. Each meridian is
Waterman_butterfly_projection
Uniform star polyhedron with 12 faces
geometry, the octahemioctahedron or octatetrahedron is a nonconvex uniform polyhedron, indexed as U3. It contains twelve faces (eight triangles and four hexagons)
Octahemioctahedron
Part of a straight line that is bounded by two distinct end points
points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or a
Line_segment
Catalan solid with 30 faces
the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of
Rhombic_triacontahedron
Prism with a 3-sided base
other polyhedra, examples are some of the Johnson solids and Schönhardt polyhedron. It has a relationship with the honeycombs and polytopes. It can be found
Triangular_prism
Euclidean geometry without distance and angles
synthetic finite geometry. In projective geometry, affine space means the complement of a hyperplane at infinity in a projective space. Affine space can also
Affine_geometry
Geometric system with a finite number of points
Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field (that is, the
Finite_geometry
Branch of mathematics
form only in projective space. For these reasons, projective space plays a fundamental role in algebraic geometry. Nowadays, the projective space Pn of
Algebraic_geometry
Property of a mathematical space
(3-dimensional) represented using a variety of strategies, such as a polyhedron consisting of connected polygon faces. The software is expected to use
Dimension
Tiling of euclidean or hyperbolic space of three or more dimensions
honeycomb is said to be a space-filling polyhedron. A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be
Honeycomb_(geometry)
Catalan solid with 120 faces
most faces of any other strictly convex polyhedron where every face of the polyhedron has the same shape. Projected into a sphere, the edges of a disdyakis
Disdyakis_triacontahedron
Solid with twenty equal triangular faces
The regular icosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from a pentagonal antiprism by attaching two pentagonal
Regular_icosahedron
Computer programmer and YouTuber
Rupert property in geometry. A polyhedron with this property can be made to drill through another copy of the same polyhedron without breaking the exterior
Tom_Murphy_VII
it can be described in terms of geometrically finite groups. A convex polyhedron C in hyperbolic space is called geometrically finite if its closure C
Geometric_finiteness
Type of map projection
projection based on a spherical polyhedron. Typically, the polyhedron is overlaid on the globe, and each face of the polyhedron is transformed to a polygon
Polyhedral_map_projection
Polyhedron with 24 faces
In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its
Dodecadodecahedron
Field of mathematics dealing with three-dimensional Euclidean spaces
volumes of various solids, including pyramids, prisms, cubes (and other polyhedrons), cylinders, cones (including truncated) and other solids of revolution
Solid_geometry
Polyhedral compound
icosicosahedron, is one of the five regular polyhedral compounds. This polyhedron can be seen as either a stellation of the icosahedron or a compound. This
Compound_of_ten_tetrahedra
Straight figure with zero width and depth
introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry. In the Greek deductive geometry of Euclid's Elements
Line_(geometry)
Antiprism with a five-sided base
faces of the pentagonal antiprism are all regular, it is a semiregular polyhedron. It can also be considered as a parabidiminished icosahedron, a shape
Pentagonal_antiprism
Spherical shell structure based on a geodesic polyhedron
hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron. The rigid triangular elements of the dome distribute stress throughout
Geodesic_dome
Polyhedron with 8 rhombic and 4 hexagonal faces
parallelohedron identified by Evgraf Fedorov. In other words, it is a space-filling polyhedron, meaning the elongated dodecahedron and its copy can tile space face-to-face
Elongated_dodecahedron
Two tetrahedra crossing each other
octahedron is a faceting of the cube, meaning that it is a polyhedron or compound polyhedron within a cube that uses only the vertices of the cube. Faceting
Stellated_octahedron
Branch of mathematics
of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which
Geometry
Spatial grid based on a geodesic polyhedron
A geodesic grid is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron. The earliest use of the (icosahedral) geodesic grid in geophysical
Geodesic_grid
Undirected graph with 14 vertices
maximal. The map can be faithfully realized as the Szilassi polyhedron, the only known polyhedron apart from the tetrahedron such that every pair of faces
Heawood_graph
Solid with eight equal triangular faces
In geometry, a regular octahedron is an eight-sided polyhedron with equilateral triangles as its faces. Known for its highly symmetrical form, the regular
Regular_octahedron
Geometric object with flat sides
example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of
Polytope
Catalan solid with 48 faces
dodecahedron can be drawn in a number of symmetric orthogonal projective orientations. Between a polyhedron and its dual, vertices and faces are swapped in positions
Disdyakis_dodecahedron
3D symmetry group
projective special linear group, see here for a proof; S 5 ≅ PGL ( 2 , 5 ) , {\displaystyle S_{5}\cong \operatorname {PGL} (2,5),} the projective general
Icosahedral_symmetry
Polyhedron with 22 faces
dodecahemicosahedron (or small dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U65. It has 22 faces (12 pentagons and 10 hexagons), 60 edges
Great_dodecahemicosahedron
Compact Riemann surface of genus 3
the "Klein quartic" referred specifically to the subset of the complex projective plane P2(C) defined by an algebraic equation. This has a specific Riemannian
Klein_quartic
Coordinate system that is defined by points instead of vectors
coordinate-free definition of the projective completion of an affine space, and a definition of a projective frame. The projective completion of an affine space
Barycentric_coordinate_system
Unique point and line of a conic section
plane into its pole. In projective geometry, this affords a one-to-one correspondence between points and lines in the projective plane. This correspondence
Pole_and_polar
Mathematical space with two coordinates
meaningfully compared, as they can in a more general symplectic surface. The projective plane does away with both distance and parallelism. A two-dimensional
Two-dimensional_space
Polygon that is the boundary of a convex set
curve Concave polygon – Simple polygon which is not convex Convex polyhedron – Polyhedron that is the boundary of a convex set Convex polytope – Convex hull
Convex_polygon
Straight line segment that passes through the centre of a circle
Geometry Projecting a sphere to a plane Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic
Diameter
Relation between sides of a right triangle
in n-dimensional space (Rn) onto which the m-dimensional objects are projected (m ≤ n): x = ( n m ) = n ! m ! ( n − m ) ! {\displaystyle x={\binom {n}{m}}={\frac
Pythagorean_theorem
Method to solve optimization problems
any problem, the convex hull of the solutions is an integral polyhedron; if this polyhedron has a nice/compact description, then we can efficiently find
Linear_programming
Study of complex manifolds and several complex variables
not in general affine or projective. By Serre's GAGA theorem, every projective complex analytic variety is actually a projective complex algebraic variety
Complex_geometry
Non-Euclidean geometry
passing through the origin. In the projective model of elliptic geometry, the points of n-dimensional real projective space are used as points of the model
Elliptic_geometry
Catalan solid with 60 faces
In geometry, a pentakis dodecahedron or kisdodecahedron is a polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron;
Pentakis_dodecahedron
Study of graphs defined by geometric means
1-skeleton of a polyhedron or polytope is the set of vertices and edges of said polyhedron or polytope. The skeleton of any convex polyhedron is a planar
Geometric_graph_theory
Polyhedron with parallel bases connected by triangles
In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an n-sided polygon, connected
Antiprism
Geometric configuration of 9 points and 12 lines
transitive) with 432 automorphisms. It can be realized in the complex projective plane as the set of inflection points of an elliptic curve, but it has
Hesse_configuration
Polyhedron with 12 congruent golden rhombus faces
In geometry, the Bilinski dodecahedron is a convex polyhedron with twelve congruent golden rhombus faces. It has the same topology as the face-transitive
Bilinski_dodecahedron
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PROJECTIVE POLYHEDRON
PROJECTIVE POLYHEDRON
Girl/Female
Irish
Protective.
Girl/Female
German, Swedish
Protective Victory
Boy/Male
German
Protective
Boy/Male
Polish
Protective shield.
Girl/Female
Indian
Protective Angel
Boy/Male
Greek
Productive.
Girl/Female
Indian
Protective Angel
Boy/Male
Arabic, Indian, Muslim, Sindhi
Protective; Safety
Girl/Female
Muslim/Islamic
Protective angel
Boy/Male
Christian & English(British/American/Australian)
Protective Friend
Girl/Female
Muslim
Protective Angel
Girl/Female
Muslim
Protective Angel
Girl/Female
Celtic, French, German, Irish
Strong; Protective
Girl/Female
Muslim/Islamic
Protective angel
Boy/Male
British, English, Netherlands
Protective
Girl/Female
Irish
Protective.
Girl/Female
German, Italian, Swedish
Protective; Victorious Shield
Girl/Female
German American
Protective.
Boy/Male
Christian & English(British/American/Australian)
Protective Grace
Boy/Male
German
Protective
PROJECTIVE POLYHEDRON
PROJECTIVE POLYHEDRON
PROJECTIVE POLYHEDRON
PROJECTIVE POLYHEDRON
PROJECTIVE POLYHEDRON
PROJECTIVE POLYHEDRON
PROJECTIVE POLYHEDRON
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