Set of dual-uniform n-gonal bipyramids | |
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Example: dual-uniform hexagonal bipyramid (n = 6) | |
Type | dual-uniform in the sense of dual-semiregular polyhedron |
Coxeter diagram | |
Schläfli symbol | { } + {n} ^{[1]} |
Faces | 2n congruent isosceles triangles |
Edges | 3n |
Vertices | 2 + n |
Face configuration | V4.4.n |
Symmetry group | D_{nh}, [n,2], (*n22), order 4n |
Rotation group | D_{n}, [n,2]^{+}, (n22), order 2n |
Dual polyhedron | (convex) uniform n-gonal prism |
Properties | convex, face-transitive, regular vertices^{[2]} |
Net | Example: net of pentagonal bipyramid (n = 5) |
A (symmetric) n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.^{[3]}^{[4]} An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices.
The "n-gonal" in the name of a bipyramid does not refer to a face but to the internal polygon base, lying in the mirror plane that connects the two pyramid halves. (If it were a face, then each of its edges would connect three faces instead of two.)
A "regular" bipyramid has a regular polygon base. It is usually implied to be also a right bipyramid.
A right bipyramid has its two apices right above and right below the center or the centroid of its polygon base.
A "regular" right (symmetric) n-gonal bipyramid has Schläfli symbol { } + {n}.
A right (symmetric) bipyramid has Schläfli symbol { } + P, for polygon base P.
The "regular" right (thus face-transitive) n-gonal bipyramid with regular vertices^{[2]} is the dual of the n-gonal uniform (thus right) prism, and has congruent isosceles triangle faces.
A "regular" right (symmetric) n-gonal bipyramid can be projected on a sphere or globe as a "regular" right (symmetric) n-gonal spherical bipyramid: n equally spaced lines of longitude going from pole to pole, and an equator line bisecting them.
Bipyramid name | Digonal bipyramid | Triangular bipyramid (See: J_{12}) |
Square bipyramid (See: O) |
Pentagonal bipyramid (See: J_{13}) |
Hexagonal bipyramid | Heptagonal bipyramid | Octagonal bipyramid | Enneagonal bipyramid | Decagonal bipyramid | ... | Apeirogonal bipyramid |
---|---|---|---|---|---|---|---|---|---|---|---|
Polyhedron image | ... | ||||||||||
Spherical tiling image | Plane tiling image | ||||||||||
Face config. | V2.4.4 | V3.4.4 | V4.4.4 | V5.4.4 | V6.4.4 | V7.4.4 | V8.4.4 | V9.4.4 | V10.4.4 | ... | V∞.4.4 |
Coxeter diagram | ... |
Only three kinds of bipyramids can have all edges of the same length (which implies that all faces are equilateral triangles, and thus the bipyramid is a deltahedron): the "regular" right (symmetric) triangular, tetragonal, and pentagonal bipyramids. The tetragonal or square bipyramid with same length edges, or regular octahedron, counts among the Platonic solids; the triangular and pentagonal bipyramids with same length edges count among the Johnson solids (J_{12} and J_{13}).
"Regular" right (symmetric) bipyramid name |
Trigonal or Triangular bipyramid J_{12} |
Tetragonal or square bipyramid (Regular octahedron) O |
Pentagonal bipyramid J_{13} |
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Bipyramid image |
A "regular" right (symmetric) n-gonal bipyramid has dihedral symmetry group D_{nh}, of order 4n, except in the case of a regular octahedron, which has the larger octahedral symmetry group O_{h}, of order 48, which has three versions of D_{4h} as subgroups. The rotation group is D_{n}, of order 2n, except in the case of a regular octahedron, which has the larger rotation group O, of order 24, which has three versions of D_{4} as subgroups.
Note: Every "regular" right (symmetric) n-gonal bipyramid has the same (dihedral) symmetry group as the dual-uniform n-gonal bipyramid, for n ≠ 4.
The 4n triangle faces of a "regular" right (symmetric) 2n-gonal bipyramid, projected as the 4n spherical triangle faces of a "regular" right (symmetric) 2n-gonal spherical bipyramid, represent the fundamental domains of dihedral symmetry in three dimensions: D_{nh}, [n,2], (*n22), of order 4n. These domains can be shown as alternately colored spherical triangles:
An n-gonal (symmetric) bipyramid can be seen as the Kleetope of the "corresponding" n-gonal dihedron.
Dihedral Symmetry | D_{1h} | D_{2h} | D_{3h} | D_{4h} | D_{5h} | D_{6h} | ... | D_{nh} |
---|---|---|---|---|---|---|---|---|
Fundamental domains image | ... | |||||||
Coxeter diagram | ... |
Volume of a (symmetric) bipyramid:
This works for any shape of the base, and for any location of the apices, provided that h is measured as the perpendicular distance from the base plane to any apex. Hence:
Volume of a (symmetric) bipyramid whose base is a regular n-sided polygon with side length s and whose height is h:
Non-right bipyramids are called oblique bipyramids.
A concave bipyramid has a concave polygon base.
(*) Its base has no obvious center; but if its apices are right above and right below the centroid of its base, then it is a right bipyramid. Anyway, it is a concave octahedron.
An asymmetric right bipyramid joins two right pyramids with congruent bases but unequal heights, base-to-base.
An inverted right bipyramid joins two right pyramids with congruent bases but unequal heights, base-to-base, but on the same side of their common base.
The dual of an asymmetric/inverted right n-gonal bipyramid is an n-gonal frustum.
A "regular" asymmetric/inverted right n-gonal bipyramid has symmetry group C_{nv}, of order 2n.
Asymmetric | Inverted |
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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,^{[4]} 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:
Note: For at most two particular values of z_{A} = |z_{A'}|, the faces of such a scalene triangle bipyramid may be isosceles.^{[citation needed]}
Double example:
In crystallography, "isotoxal" right (symmetric) "didigonal" (*) (8-faced), ditrigonal (12-faced), ditetragonal (16-faced), and dihexagonal (24-faced) bipyramids exist.^{[4]}^{[3]}
(*) The smallest geometric di-n-gonal bipyramids have eight faces, and are topologically identical to the regular octahedron. In this case (2n = 2×2):
an "isotoxal" right (symmetric) "didigonal" bipyramid is called a rhombic bipyramid,^{[4]}^{[3]} although all its faces are scalene triangles, because its flat polygon base is a rhombus.
A "regular" right "symmetric" di-n-gonal scalenohedron is defined by a regular zigzag skew 2n-gon base, two symmetric apices right above and right below the base center, and triangle faces connecting each basal edge to each apex.
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.^{[3]}
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),^{[4]} 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:
Note: For at most two particular values of z_{A} = |z_{A'}|, 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.^{[4]}^{[3]}
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.^{[4]}^{[3]}
Let us temporarily focus on the "regular" right "symmetric" 8-faced scalenohedra with h = r, i.e. z_{A} = |z_{A'}| = x_{U} = |x_{U'}| = y_{V} = |y_{V'}|. Their two apices can be represented as A = (0,0,1), A' = (0,0,−1), and their four basal vertices as U = (1,0,z), U' = (−1,0,z), V = (0,1,−z), V' = (0,−1,−z), 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 |
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Note: 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:
Note: 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 self-intersecting or star bipyramid has a star polygon base.
A "regular" right symmetric star bipyramid is defined by a regular star polygon base, two symmetric apices right above and right below the base center, and thus one-to-one symmetric triangle faces connecting each basal edge to each apex.
A "regular" right symmetric star bipyramid has congruent isosceles triangle faces, and is isohedral.
Note: For at most one particular value of z_{A} = |z_{A'}|, the faces of such a "regular" star bipyramid may be equilateral.
A p/q-bipyramid has Coxeter diagram .
Star polygon base | 5/2-gon | 7/2-gon | 7/3-gon | 8/3-gon | 9/2-gon | 9/4-gon |
---|---|---|---|---|---|---|
Star bipyramid image | ||||||
Coxeter diagram |
Star polygon base | 10/3-gon | 11/2-gon | 11/3-gon | 11/4-gon | 11/5-gon | 12/5-gon |
---|---|---|---|---|---|---|
Star bipyramid image | ||||||
Coxeter diagram |
An "isotoxal" right symmetric 2p/q-gonal star bipyramid is defined by an isotoxal in-out star 2p/q-gon base, two symmetric apices right above and right below the base center, and thus one-to-one symmetric triangle faces connecting each basal edge to each apex.
An "isotoxal" right symmetric 2p/q-gonal star bipyramid has congruent scalene triangle faces, and is isohedral. It can be seen as another type of a 2p/q-gonal right "symmetric" star scalenohedron, with an isotoxal in-out star polygon base.
Note: For at most two particular values of z_{A} = |z_{A'}|, the faces of such a scalene triangle star bipyramid may be isosceles.
Star polygon base | Isotoxal in-out 8/3-gon |
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Scalene triangle star bipyramid image |
A "regular" right "symmetric" 2p/q-gonal star scalenohedron is defined by a regular zigzag skew star 2p/q-gon base, two symmetric apices right above and right below the base center, and triangle faces connecting each basal edge to each apex.
A "regular" right "symmetric" 2p/q-gonal star scalenohedron has congruent scalene triangle faces, and is isohedral. It can be seen as another type of a right "symmetric" 2p/q-gonal star bipyramid, with a regular zigzag skew star polygon base.
Note: For at most two particular values of z_{A} = |z_{A'}|, the faces of such a star scalenohedron may be isosceles.
Star polygon base | Regular zigzag skew 8/3-gon |
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Star scalenohedron image |
Note: If the star 2p/q-gon base is both isotoxal in-out and zigzag skew, then not all faces of the "isotoxal" right "symmetric" star scalenohedron are congruent.
Star polygon base | Isotoxal in-out zigzag skew 8/3-gon |
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Star scalenohedron image |
Note: For some particular values of z_{A} = |z_{A'}|, half the faces of such a star scalenohedron may be isosceles or equilateral.
Example with four different edge lengths:
Example with three different edge lengths:
The dual of the rectification of each convex regular 4-polytopes is a cell-transitive 4-polytope with bipyramidal cells. In the following, the apex vertex of the bipyramid is A and an equator vertex is E. The distance between adjacent vertices on the equator EE = 1, the apex to equator edge is AE and the distance between the apices is AA. 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. N_{AE} bipyramids meet along each type AE edge. N_{EE} bipyramids meet along each type EE edge. C_{AE} is the cosine of the dihedral angle along an AE edge. C_{EE} is the cosine of the dihedral angle along an EE edge. As cells must fit around an edge, N_{EE} cos^{−1}(C_{EE}) ≤ 2π, N_{AE} cos^{−1}(C_{AE}) ≤ 2π.
4-polytope properties | Bipyramid properties | |||||||||||||
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Dual of | Coxeter diagram |
Cells | V_{A} | V_{E} | N_{A} | N_{E} | N_{AE} | N_{EE} | Cell | Coxeter diagram |
AA | AE** | C_{AE} | C_{EE} |
Rectified 5-cell | 10 | 5 | 5 | 4 | 6 | 3 | 3 | Triangular bipyramid | 0.667 | |||||
Rectified tesseract | 32 | 16 | 8 | 4 | 12 | 3 | 4 | Triangular bipyramid | 0.624 | |||||
Rectified 24-cell | 96 | 24 | 24 | 8 | 12 | 4 | 3 | Triangular bipyramid | 0.745 | |||||
Rectified 120-cell | 1200 | 600 | 120 | 4 | 30 | 3 | 5 | Triangular bipyramid | 0.613 | |||||
Rectified 16-cell | 24* | 8 | 16 | 6 | 6 | 3 | 3 | Square bipyramid | 1 | |||||
Rectified cubic honeycomb | ∞ | ∞ | ∞ | 6 | 12 | 3 | 4 | Square bipyramid | 0.866 | |||||
Rectified 600-cell | 720 | 120 | 600 | 12 | 6 | 3 | 3 | Pentagonal bipyramid | 1.447 |
In general, a bipyramid can be seen as an n-polytope constructed with a (n − 1)-polytope in a hyperplane with two points in opposite directions and equal perpendicular distances from the hyperplane. If the (n − 1)-polytope is a regular polytope, it will have identical pyramidal facets.
A 2-dimensional ("regular") right symmetric (digonal) bipyramid is formed by joining two congruent isosceles triangles base-to-base; its outline is a rhombus, {}+{}.
A polyhedral bipyramid is a 4-polytope with a polyhedron base, and an apex point.
An example is the 16-cell, which is an octahedral bipyramid, {}+{3,4}, and more generally an n-orthoplex is an (n − 1)-orthoplex bipyramid, {}+{3^{n-2},4}.
Other bipyramids include the tetrahedral bipyramid, {}+{3,3}, icosahedral bipyramid, {}+{3,5}, and dodecahedral bipyramid, {}+{5,3}, the first two having all regular cells, they are also Blind polytopes.