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 (pl. apices, 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; [lower-alpha 1] otherwise it is oblique. When the base is a regular polygon, the bipyramid is also called regular.
A bipyramid is a polyhedron constructed by fusing two pyramids which share the same polygonal base; [1] a pyramid is in turn constructed by connecting each vertex of its base to a single new vertex (the apex) not lying in the plane of the base, for an -gonal base forming triangular faces in addition to the base face. An -gonal bipyramid thus has faces, edges, and vertices. More generally, a right pyramid is a pyramid where the apices are on the perpendicular line through the centroid of an arbitrary polygon or the incenter of a tangential polygon, depending on the source. [lower-alpha 1] Likewise, a right bipyramid is a polyhedron constructed by attaching two symmetrical right bipyramid bases; bipyramids whose apices are not on this line are called oblique bipyramids. [2]
When the two pyramids are mirror images, the bipyramid is called symmetric. It is called regular if its base is a regular polygon. [1] When the base is a regular polygon and the apices are on the perpendicular line through its center (a regular right bipyramid) then all of its faces are isosceles triangles; sometimes the name bipyramid refers specifically to symmetric regular right bipyramids, [3] Examples of such bipyramids are the triangular bipyramid, octahedron (square bipyramid) and pentagonal bipyramid. In the case all of their edges are equal in length, these shapes consist of equilateral triangle faces, making them deltahedra; [4] [5] the triangular bipyramid and the pentagonal bipyramid are Johnson solids, and the regular octahedron is a Platonic solid. [6]
The symmetric regular right bipyramids have prismatic symmetry, the dihedral group of of order : their appearance is symmetrical by rotating around the axis of symmetry and reflecting across the mirror plane. [7] Because the appearance looks the same under such symmetries, and all of the faces are congruent, the bipyramids have the property of isohedral. [8] [9] They are the dual polyhedron of prisms and the prisms are the dual of bipyramids as well: the bipyramids vertices correspond to the faces of the prism, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other, and vice versa; [10] the prisms shares the same symmetry as the bipyramids. [11] The regular octahedron is more symmetric still, as its base vertices and apices are indistinguishable and can be exchanged by reflections or rotations: the octahedron and its dual the cube have octahedral symmetry. [12]
The volume of a symmetric bipyramid is
where B is the area of the base and h the height from the base plane to any apex. In the case of a regular -sided polygon with side length and whose altitude is , the volume of such bipyramid is:
A concave bipyramid has a concave polygon base, and one example is a concave tetragonal bipyramid or an irregular concave octahedron. A bipyramid with an arbitrary polygonal base could be considered a right bipyramid if the apices are on a line perpendicular to the base passing through the base's centroid.
An asymmetric bipyramid has apices which are not mirrored across the base plane; for a right bipyramid this only happens if each apex is a different distance from the base.
The dual of an asymmetric right n-gonal bipyramid is an n-gonal frustum.
A regular asymmetric right n-gonal bipyramid has symmetry group Cnv, of order 2n.
An isotoxal right (symmetric) di-n-gonal bipyramid is a right (symmetric) 2n-gonal bipyramid with an isotoxal flat polygon base: its 2n basal vertices are coplanar, but alternate in two radii.
All its faces are congruent scalene triangles, and it is isohedral. It can be seen as another type of a right symmetric di-n-gonal scalenohedron, with an isotoxal flat polygon base.
An isotoxal right (symmetric) di-n-gonal bipyramid has n two-fold rotation axes through opposite basal vertices, n reflection planes through opposite apical edges, an n-fold rotation axis through apices, a reflection plane through base, and an n-fold rotation-reflection axis through apices, [13] representing symmetry group Dnh, [n,2], (*22n), of order 4n. (The reflection about the base plane corresponds to the 0° rotation-reflection. If n is even, then there is an inversion symmetry about the center, corresponding to the 180° rotation-reflection.)
Example with 2n = 2×3:
Example with 2n = 2×4:
For at most two particular values of the faces of such a scalene triangle bipyramid may be isosceles.[ citation needed ]
Double example:
In crystallography, isotoxal right (symmetric) didigonal [lower-alpha 2] (8-faced), ditrigonal (12-faced), ditetragonal (16-faced), and dihexagonal (24-faced) bipyramids exist. [13] [16]
A scalenohedron is similar to a bipyramid; the difference is that the scalenohedra has a zig-zag pattern in the middle edges. [17]
It has two apices and 2n basal vertices, 4n faces, and 6n edges; it is topologically identical to a 2n-gonal bipyramid, but its 2n basal vertices alternate in two rings above and below the center. [16]
All its faces are congruent scalene triangles, and it is isohedral. It can be seen as another type of a right symmetric di-n-gonal bipyramid, with a regular zigzag skew polygon base.
A regular right symmetric di-n-gonal scalenohedron has n two-fold rotation axes through opposite basal mid-edges, n reflection planes through opposite apical edges, an n-fold rotation axis through apices, and a 2n-fold rotation-reflection axis through apices (about which 1n rotations-reflections globally preserve the solid), [13] representing symmetry group Dnv = Dnd, [2+,2n], (2*n), of order 4n. (If n is odd, then there is an inversion symmetry about the center, corresponding to the 180° rotation-reflection.)
Example with 2n = 2×3:
Example with 2n = 2×2:
For at most two particular values of the faces of such a scalenohedron may be isosceles.
Double example:
In crystallography, regular right symmetric didigonal (8-faced) and ditrigonal (12-faced) scalenohedra exist. [13] [16]
The smallest geometric scalenohedra have eight faces, and are topologically identical to the regular octahedron. In this case (2n = 2×2), in crystallography, a regular right symmetric didigonal (8-faced) scalenohedron is called a tetragonal scalenohedron. [13] [16]
Let us temporarily focus on the regular right symmetric 8-faced scalenohedra with h = r, i.e.
Their two apices can be represented as A, A' and their four basal vertices as U, U', V, V':
where z is a parameter between 0 and 1.
At z = 0, it is a regular octahedron; at z = 1, it has four pairs of coplanar faces, and merging these into four congruent isosceles triangles makes it a disphenoid; for z > 1, it is concave.
z = 0.1 | z = 0.25 | z = 0.5 | z = 0.95 | z = 1.5 |
---|---|---|---|---|
If the 2n-gon base is both isotoxal in-out and zigzag skew, then not all faces of the isotoxal right symmetric scalenohedron are congruent.
Example with five different edge lengths:
For some particular values of zA = |zA'|, half the faces of such a scalenohedron may be isosceles or equilateral.
Example with three different edge lengths:
A star bipyramid has a star polygon base, and is self-intersecting. [20]
A regular right symmetric star bipyramid has congruent isosceles triangle faces, and is isohedral.
A p/q-bipyramid has Coxeter diagram .
Base | 5/2-gon | 7/2-gon | 7/3-gon | 8/3-gon |
---|---|---|---|---|
Image |
The dual of the rectification of each convex regular 4-polytopes is a cell-transitive 4-polytope with bipyramidal cells. In the following:
The bipyramid 4-polytope will have VA vertices where the apices of NA bipyramids meet. It will have VE vertices where the type E vertices of NE bipyramids meet.
As cells must fit around an edge,
4-polytope properties | Bipyramid properties | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Dual of rectified polytope | Coxeter diagram | Cells | VA | VE | NA | NE | Bipyramid cell | Coxeter diagram | AA | AE [lower-alpha 3] | ||||
R. 5-cell | 10 | 5 | 5 | 4 | 6 | 3 | 3 | Triangular | 0.667 | |||||
R. tesseract | 32 | 16 | 8 | 4 | 12 | 3 | 4 | Triangular | 0.624 | |||||
R. 24-cell | 96 | 24 | 24 | 8 | 12 | 4 | 3 | Triangular | 0.745 | |||||
R. 120-cell | 1200 | 600 | 120 | 4 | 30 | 3 | 5 | Triangular | 0.613 | |||||
R. 16-cell | 24 [lower-alpha 4] | 8 | 16 | 6 | 6 | 3 | 3 | Square | 1 | |||||
R. cubic honeycomb | ∞ | ∞ | ∞ | 6 | 12 | 3 | 4 | Square | 0.866 | |||||
R. 600-cell | 720 | 120 | 600 | 12 | 6 | 3 | 3 | Pentagonal | 1.447 |
A generalized n-dimensional "bipyramid" is any n-polytope constructed from an (n − 1)-polytope base lying in a hyperplane, with every base vertex connected by an edge to two apex vertices. If the (n − 1)-polytope is a regular polytope and the apices are equidistant from its center along the line perpendicular to the base hyperplane, it will have identical pyramidal facets.
A 2-dimensional analog of a right symmetric bipyramid is formed by joining two congruent isosceles triangles base-to-base to form a rhombus. More generally, a kite is a 2-dimensional analog of a (possibly asymmetric) right bipyramid, and any quadrilateral is a 2-dimensional analog of a general bipyramid.
In geometry, an octahedron 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 each vertex.
In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra.
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, geometry, and chemistry.
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.
In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.
In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.
In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.
In geometry, the triangular bipyramid is the hexahedron with six triangular faces, constructed by attaching two tetrahedrons face-to-face. The same shape is also called the triangular dipyramid or trigonal bipyramid. If these tetrahedrons are regular, all faces of triangular bipyramid are equilateral. It is an example of a deltahedron and of a Johnson solid.
In geometry, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares, while the base and its opposite face are regular polygons, the triangular, square, and pentagonal cupolae all count among the Johnson solids, and can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, and rhombicosidodecahedron, respectively.
In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron, heccaidecadeltahedron, or tetrakis square antiprism; these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid.
In geometry, an n-gonaltrapezohedron, n-trapezohedron, n-antidipyramid, n-antibipyramid, or n-deltohedron is the dual polyhedron of an n-gonal antiprism. The 2n faces of an n-trapezohedron are congruent and symmetrically staggered; they are called twisted kites. With a higher symmetry, its 2n faces are kites.
In geometry, a disdyakis dodecahedron,, is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid creates a polyhedron that looks almost like the disdyakis dodecahedron, and is topologically equivalent to it.
In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.
In geometry, the square cupola the cupola with octagonal base. In the case of edges are equal in length, it is the Johnson solid, a convex polyhedron with faces are regular. It can be considered as half of the rhombicuboctahedron. It can be used to construct many polyhedrons, particularly in other Johnson solids.
In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vertices have four faces and others have five. It is a dodecahedron, one of the eight convex deltahedra, and is the 84th Johnson solid. It can be thought of as a square antiprism where both squares are replaced with two equilateral triangles.
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.
In geometry, a polytope or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.
In geometry of 4 dimensions or higher, a double pyramid, duopyramid, or fusil is a polytope constructed by 2 orthogonal polytopes with edges connecting all pairs of vertices between the two. The term fusil is used by Norman Johnson as a rhombic-shape. The term duopyramid was used by George Olshevsky, as the dual of a duoprism.