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;^{[a]} 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.^{[a]} 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 are isohedral.^{[8]}^{[9]} They are the dual polyhedra 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
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 C_{nv}, 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 D_{nh}, [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:
Double example:
In crystallography, isotoxal right (symmetric) didigonal^{[b]} (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 D_{nv} = D_{nd}, [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.
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 z_{A} = |z_{A'}|, 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 |
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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 V_{A} vertices where the apices of N_{A} bipyramids meet. It will have V_{E} vertices where the type E vertices of N_{E} bipyramids meet.
As cells must fit around an edge,
4-polytope properties | Bipyramid properties | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Dual of rectified polytope |
Coxeter diagram |
Cells | V_{A} | V_{E} | N_{A} | N_{E} | Bipyramid cell |
Coxeter diagram |
AA | AE^{[c]} | ||||
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 ^{[d]} | 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.