Polyhedra

geometrical shape

In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term polyhedron is often used to refer implicitly to the whole structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices.

In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of irregular octahedra also exist, including both convex and non-convex shapes.

A hexahedron (: hexahedra or hexahedrons) or sexahedron (: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex.

in geometry, a planar surface that forms part of the boundary of a solid object

In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two pyramids together base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise specified the base vertices are usually coplanar and a bipyramid is usually symmetric, meaning the two pyramids are mirror images across their common base plane. When each apex (, the off-base vertices) of the bipyramid is on a line perpendicular to the base and passing through its center, it is a right bipyramid; otherwise it is oblique. When the base is a regular polygon, the bipyramid i

In geometry, a pentahedron (: pentahedra) is a polyhedron with five faces or sides. There are no face-transitive polyhedra with five sides, and there are two distinct topological types. Notable polyhedra with regular polygon faces are: File:Square pyramid.png|Square pyramid with four triangles and one square. Pyramids with any quadrilateral base have the same number of faces. File:Triangular prism.png|Triangular prism with three rectangles and two triangular bases. In the case of a right triangular prism, it is a special case of wedge with connecting parallel edges between triangles; the wedg

polyhedron whose vertices correspond to the faces of another one

thumb|upright=1.6|The eight convex deltahedra. First row: regular tetrahedron, triangular bipyramid, [[regular octahedron, pentagonal bipyramid. Second row: gyroelongated square bipyramid, regular icosahedron, triaugmented triangular prism, snub disphenoid.]] A deltahedron is a polyhedron whose faces are all equilateral triangles. The deltahedron was named by Martyn Cundy, after the Greek capital letter delta resembling a triangular shape Δ.

arrangement of joined polytopes which can be folded to become the facets of a higher-dimensional polytope

In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, -deltohedron, or -antitegum, is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called twisted kites. With a higher symmetry, its faces are kites (sometimes also called trapezoids, or deltoids). They are topologically related to the scalenohedra, which are half-symmetry variants with irregular trigons.

variously-defined concept in geometry

thumb|A 10-sided Dice|die

A dihedron (pl. dihedra) 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 faces can be thought of as a lens, an example of which is the fundamental domain of a lens space L(p,q). Dihedra have also been called bihedra, flat polyhedra, or doubly covered polygons.

sphere contained within a polyhedron, tangent to each of its faces

figure exposed when a corner of a polyhedron or polytope is sliced off

geometric property of an object which cannot be mapped to its mirror image by rotations and translations alone

polyhedron defined by two triangles and three trapezoid faces

thumb|alt=Alt text|A scutoid compared with a Prism (geometry)|prism, [[frustum, and prismatoid]] thumb|Two 5-6 scutoids, flipped and attached A scutoid is a particular type of geometric solid between two parallel surfaces. The boundary of each of the surfaces (and of all the other parallel surfaces between them) either is a polygon or resembles a polygon, but is not necessarily planar, and the vertices of the two end polygons are joined by either a curve or a Y-shaped connection on at least one of the edges, but not necessarily all of the edges. Scutoids present at least one vertex between the

thumb|upright|A Chestahedron, realized with 4 equilateral-triangle and 3 kite faces, all having the same area,

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). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.

thumb|A polyhedron and its midsphere. A viewer situated at a polyhedron vertex would see the red circle surrounding that vertex as the horizon on the sphere.|alt=An opaque white polyhedron with four triangular faces and four quadrilateral faces is crossed by a transparent blue sphere of approximately the same size, tangent to each edge of the polyhedron. The visible portions of the sphere, outside the polyhedron, form circular caps on each face of the polyhedron, of two sizes: smaller in the triangular faces, and larger in the quadrilateral faces. Red circles on the surface of the sphere, pass

vertex which is in some sense the highest of the figure to which it belongs

thumb|Construction of a stellated dodecagon: a regular polygon with [[Schläfli symbol {12/5}]] In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred",

notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex

vertex-transitive polytope

inequality which involves a linear function

thumb|This beach ball would be a hosohedron with 6 [[spherical lune faces, if the 2 white caps on the ends were removed and the lunes extended to meet at the poles.]]

polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality

thumb|right|240px|A tetradecahedron with D2d-symmetry, existing in the Weaire–Phelan structure

method for constructing a uniform polyhedron or plane tiling

polyhedron or other geometric figure all of whose edges are symmetric to each other

tiling of the plane by squares

polytope or tiling with identical faces

In geometry, an enneahedron (or nonahedron) is a polyhedron with nine faces. There are 2606 types of convex enneahedra, each having a different pattern of vertex, edge, and face connections. None of them are regular.

polyhedron with 10 faces

failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would

tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions

a three-dimensional mathematical foam structure with equal-volume cells and low surface area

spherical triangle that can be used to tile a sphere

the problem of partitioning a given shape into pieces that can be rearranged to form a second given shape

polyhedron formed by six congruent rhombi

notation used to describe polyhedra based on a seed polyhedron modified by various operations

thumb|Example of a hexahedronIn geometry, a cuboid is a hexahedron with quadrilateral faces, meaning it is a polyhedron with six faces; it has eight vertices and twelve edges. A rectangular cuboid (sometimes also called a "cuboid") has all right angles and equal opposite rectangular faces. Etymologically, "cuboid" means "like a cube", in the sense of a convex solid which can be transformed into a cube (by adjusting the lengths of its edges and the angles between its adjacent faces). A cuboid is a convex polyhedron whose polyhedral graph is the same as that of a cube.

symbol representing a construction by way of Wythoff's construction applied to Schwarz triangles

In geometry, a bicupola is a solid formed by connecting two cupolae on their bases. Here, two classes of bicupola are included because each cupola (bicupola half) is bordered by alternating triangles and squares. If similar faces are attached together the result is an orthobicupola; if squares are attached to triangles it is a gyrobicupola.

nonconvex uniform polyhedron

polyhedron with 18 faces

\right)^2+h^2} \\[2pt] & \ \ +\ n \frac{b^2}{2 \tan{\frac{\pi}{n} \end{align} | volume = n \frac{a^2+b^2+ab}{6 \tan{\frac{\pi}{n}h | angle = | dual = Elongated bipyramids | properties = convex | vertex_figure = | net = | net_caption =

outermost stellation of the icosahedron

feature of a polytope in the next-lower dimension

operation on a polyhedron or tiling that removes alternate vertices

polyhedron with 26 faces

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Image:CubeAndStel.svg Stella octangula as a faceting of the cube

geometric polyhedral group

structure that resembles a geodesic dome

operation that bevels a regular polytope at its edges and vertices

Group realized geometrically by reflections across the sides of a triangle

trapezohedron with 8 faces