Bipyramid

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.

"Regular" right (symmetric) n-gonal bipyramids:

Bipyramid name	Digonal bipyramid	Triangular bipyramid (See: J12)	Square bipyramid (See: O)	Pentagonal bipyramid (See: J13)	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 (J₁₂ and J₁₃).

Equilateral triangle bipyramids:

"Regular" right (symmetric) bipyramid name	Trigonal or Triangular bipyramid J12	Tetragonal or square bipyramid (Regular octahedron) O	Pentagonal bipyramid J13
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 D₄ 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:

• across a reflection plane through cocyclic edges, mirror image domains are in different colors (indirect isometry);
• about an n-fold or a 2-fold rotation axis through opposite vertices, a domain and its image are in the same color (direct isometry).

An n-gonal (symmetric) bipyramid can be seen as the Kleetope of the "corresponding" n-gonal dihedron.

Fundamental domains of dihedral symmetry in three dimensions:

Dihedral Symmetry	D1h	D2h	D3h	D4h	D5h	D6h	...	Dnh
Fundamental domains image							...
Coxeter diagram							...

Volume of a (symmetric) bipyramid:

$${\displaystyle V={\frac {2}{3}}Bh,}$$

where B is the area of the base and h the height from the base plane to any apex.

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:

$${\displaystyle V={\frac {n}{6}}hs^{2}\cot {\frac {\pi }{n}}.}$$

Non-right bipyramids are called oblique bipyramids.

A concave bipyramid has a concave polygon base.

Example: concave (symmetric) tetragonal bipyramid (n = 4) (*)

(*) 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.

Examples: "regular" asymmetric/inverted right hexagonal bipyramids (n = 6):

Asymmetric	Inverted

Example: ditetragonal bipyramid (2n = 2×4)

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:

An "isotoxal" right (symmetric) ditrigonal bipyramid has three similar vertical planes of symmetry, intersecting in a (vertical) 3-fold rotation axis; perpendicular to them is a fourth plane of symmetry (horizontal); at the intersection of the three vertical planes with the horizontal plane are three similar (horizontal) 2-fold rotation axes; there is no center of inversion symmetry,[5] but there is a center of symmetry: the intersection point of the four axes.

Example with 2n = 2×4:

An "isotoxal" right (symmetric) ditetragonal bipyramid has four vertical planes of symmetry of two kinds, intersecting in a (vertical) 4-fold rotation axis; perpendicular to them is a fifth plane of symmetry (horizontal); at the intersection of the four vertical planes with the horizontal plane are four (horizontal) 2-fold rotation axes of two kinds, each perpendicular to a plane of symmetry; two vertical planes bisect the angles between two horizontal axes; and there is a centre of inversion symmetry.[6]

Note: For at most two particular values of z_A = |z_A'|, the faces of such a scalene triangle bipyramid may be isosceles.

Double example:

• The bipyramid with isotoxal 2×2-gon base vertices:

U = (1,0,0), U′ = (−1,0,0), V = (0,2,0), V′ = (0,−2,0),

and with "right" symmetric apices:

A = (0,0,1), A′ = (0,0,−1),

has its faces isosceles. Indeed:

upper apical edge lengths:

AU = AU′ = ${\displaystyle {\sqrt {2}},}$

AV = AV′ = ${\displaystyle {\sqrt {5}};}$

base edge length:

UV = VU′ = U′V' = V′U = ${\displaystyle {\sqrt {5}};}$

lower apical edge lengths = upper ones.

• The bipyramid with same base vertices, but with "right" symmetric apices:

A = (0,0,2), A′ = (0,0,−2),

also has its faces isosceles. Indeed:

upper apical edge lengths:

AU = AU′ = ${\displaystyle {\sqrt {5}},}$

AV = AV′ = 2${\displaystyle {\sqrt {2}};}$

base edge length = previous one = ${\displaystyle {\sqrt {5}};}$

lower apical edge lengths = upper ones.

In crystallography, "isotoxal" right (symmetric) "didigonal" (*) (8-faced), ditrigonal (12-faced), ditetragonal (16-faced), and dihexagonal (24-faced) bipyramids exist.[4][3]

Examples of rhombic bipyramids

(*) 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.

Example: ditrigonal scalenohedron (2n = 2×3)

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:

A "regular" right "symmetric" ditrigonal scalenohedron has three similar vertical planes of symmetry inclined to one another at 60° and intersecting in a (vertical) 3-fold rotation axis, three similar horizontal 2-fold rotation axes, each perpendicular to a plane of symmetry, a center of inversion symmetry,[7] and a vertical 6-fold rotation-reflection axis.

Example with 2n = 2×2:

A "regular" right "symmetric" "didigonal" scalenohedron has only one vertical and two horizontal 2-fold rotation axes, two vertical planes of symmetry, which bisect the angles between the horizontal pair of axes, and a vertical 4-fold rotation-reflection axis;[8] it has no center of inversion symmetry.

Examples of disphenoids and of an 8-faced scalenohedron

Note: For at most two particular values of z_A = |z_A'|, the faces of such a scalenohedron may be isosceles.

Double example:

• The scalenohedron with regular zigzag skew 2×2-gon base vertices:

U = (3,0,2), U' = (−3,0,2), V = (0,3,−2), V' = (0,−3,−2),

and with "right" symmetric apices:

A = (0,0,3), A' = (0,0,−3),

has its faces isosceles. Indeed:

upper apical edge lengths:

AU = AU' = ${\displaystyle {\sqrt {10}},}$

AV = AV' = ${\displaystyle {\sqrt {34}};}$

base edge length:

UV = VU' = U'V' = V'U = ${\displaystyle {\sqrt {34}};}$

lower apical edge lengths = (swapped) upper ones.

• The scalenohedron with same base vertices, but with "right" symmetric apices:

A = (0,0,7), A' = (0,0,−7),

also has its faces isosceles. Indeed:

upper apical edge lengths:

AU = AU' = ${\displaystyle {\sqrt {34}},}$

AV = AV' = 3${\displaystyle {\sqrt {10}};}$

base edge length = previous one = ${\displaystyle {\sqrt {34}};}$

lower apical edge lengths = (swapped) upper ones.

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.

Example: geometric variations with "regular" right "symmetric" 8-faced scalenohedra:

z = 0.1	z = 0.25	z = 0.5	z = 0.95	z = 1.5

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:

The scalenohedron with isotoxal in-out zigzag skew 2×2-gon base vertices:

U = (1,0,1), U′ = (−1,0,1), V = (0,2,−1), V′ = (0,−2,−1),

and with "right" symmetric apices:

A = (0,0,3), A′ = (0,0,−3),

has congruent scalene upper faces, and congruent scalene lower faces, but not all its faces are congruent. Indeed:

upper apical edge lengths:

AU = AU′ = ${\displaystyle {\sqrt {5}},}$

AV = AV′ = 2${\displaystyle {\sqrt {5}};}$

base edge length:

UV = VU′ = U′V' = V′U = 3;

lower apical edge lengths:

A′U = A′U′ = ${\displaystyle {\sqrt {17}},}$

A′V = A′V′ = 2${\displaystyle {\sqrt {2}}.}$

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:

The scalenohedron with isotoxal in-out zigzag skew 2×2-gon base vertices:

U = (3,0,2), U' = (−3,0,2), V = (0,${\displaystyle {\sqrt {65}}}$,−2), V' = (0,−${\displaystyle {\sqrt {65}}}$,−2),

and with "right" symmetric apices:

A = (0,0,7), A' = (0,0,−7),

has congruent scalene upper faces, and congruent equilateral lower faces; thus not all its faces are congruent. Indeed:

upper apical edge lengths:

AU = AU' = ${\displaystyle {\sqrt {34}},}$

AV = AV' = ${\displaystyle {\sqrt {146}};}$

base edge length:

UV = VU' = U'V' = V'U = 3${\displaystyle {\sqrt {10}};}$

lower apical edge length(s):

A'U = A'U' = 3${\displaystyle {\sqrt {10}},}$

A'V = A'V' = 3${\displaystyle {\sqrt {10}}.}$

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 .

Examples of "regular" right symmetric star bipyramids:

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

Examples of "regular" right symmetric star bipyramids:

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.

Example of an "isotoxal" right symmetric 2p/q-gonal star bipyramid:

Star polygon base	Isotoxal in-out 8/3-gon
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.

Example of a "regular" right "symmetric" 2p/q-gonal star scalenohedron:

Star polygon base	Regular zigzag skew 8/3-gon
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.

Example of an "isotoxal" right "symmetric" 2p/q-gonal star scalenohedron:

Star polygon base	Isotoxal in-out zigzag skew 8/3-gon
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:

The star scalenohedron with isotoxal in-out zigzag skew 8/3-gon base vertices:

U₀ = (1,0,1), U₁ = (0,1,1), U₂ = (−1,0,1), U₃ = (0,−1,1),

V₀ = (2,2,−1), V₁ = (−2,2,−1), V₂ = (−2,−2,−1), V₃ = (2,−2,−1),

and with "right" symmetric apices:

A = (0,0,3), A′ = (0,0,−3),

has congruent scalene upper faces, and congruent isosceles lower faces; thus not all its faces are congruent. Indeed:

upper apical edge lengths:

AU₀ = AU₁ = AU₂ = AU₃ = ${\displaystyle {\sqrt {5}},}$

AV₀ = AV₁ = AV₂ = AV₃ = 2${\displaystyle {\sqrt {6}};}$

base edge length:

U₀V₁ = V₁U₃ = U₃V₀ = V₀U₂ = U₂V₃ = V₃U₁ = U₁V₂ = V₂U₀ = ${\displaystyle {\sqrt {17}};}$

lower apical edge lengths:

A′U₀ = A′U₁ = A′U₂ = A′U₃ = ${\displaystyle {\sqrt {17}},}$

A′V₀ = A′V₁ = A′V₂ = A′V₃ = 2${\displaystyle {\sqrt {3}}.}$

Example with three different edge lengths:

The star scalenohedron with isotoxal in-out zigzag skew 8/3-gon base vertices:

U₀ = (4,0,${\displaystyle {\sqrt {2}}}$), U₁ = (0,4,${\displaystyle {\sqrt {2}}}$), U₂ = (−4,0,${\displaystyle {\sqrt {2}}}$), U₃ = (0,−4,${\displaystyle {\sqrt {2}}}$),

V₀ = (6,6,−${\displaystyle {\sqrt {2}}}$), V₁ = (−6,6,−${\displaystyle {\sqrt {2}}}$), V₂ = (−6,−6,−${\displaystyle {\sqrt {2}}}$), V₃ = (6,−6,−${\displaystyle {\sqrt {2}}}$),

and with "right" symmetric apices:

A = (0,0,7${\displaystyle {\sqrt {2}}}$), A' = (0,0,−7${\displaystyle {\sqrt {2}}}$),

has congruent scalene upper faces, and congruent equilateral lower faces; thus not all its faces are congruent. Indeed:

upper apical edge lengths:

AU₀ = AU₁ = AU₂ = AU₃ = 2${\displaystyle {\sqrt {22}},}$

AV₀ = AV₁ = AV₂ = AV₃ = 10${\displaystyle {\sqrt {2}};}$

base edge length:

U₀V₁ = V₁U₃ = U₃V₀ = V₀U₂ = U₂V₃ = V₃U₁ = U₁V₂ = V₂U₀ = 12;

lower apical edge length(s):

A'U₀ = A'U₁ = A'U₂ = A'U₃ = 12,

A'V₀ = A'V₁ = A'V₂ = A'V₃ = 12.

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-polytopes with bipyramidal cells

4-polytope properties									Bipyramid properties
Dual of	Coxeter diagram	Cells	VA	VE	NA	NE	NAE	NEE	Cell	Coxeter diagram	AA	AE**	CAE	CEE
Rectified 5-cell		10	5	5	4	6	3	3	Triangular bipyramid		${\textstyle {\frac {2}{3}}}$	0.667	${\textstyle -{\frac {1}{7}}}$	${\textstyle -{\frac {1}{7}}}$
Rectified tesseract		32	16	8	4	12	3	4	Triangular bipyramid		${\textstyle {\frac {\sqrt {2}}{3}}}$	0.624	${\textstyle -{\frac {2}{5}}}$	${\textstyle -{\frac {1}{5}}}$
Rectified 24-cell		96	24	24	8	12	4	3	Triangular bipyramid		${\textstyle {\frac {2{\sqrt {2}}}{3}}}$	0.745	${\textstyle {\frac {1}{11}}}$	${\textstyle -{\frac {5}{11}}}$
Rectified 120-cell		1200	600	120	4	30	3	5	Triangular bipyramid		${\textstyle {\frac {{\sqrt {5}}-1}{3}}}$	0.613	${\textstyle -{\frac {10+9{\sqrt {5}}}{61}}}$	${\textstyle -{\frac {7-12{\sqrt {5}}}{61}}}$
Rectified 16-cell		24*	8	16	6	6	3	3	Square bipyramid		${\textstyle {\sqrt {2}}}$	1	${\textstyle -{\frac {1}{3}}}$	${\textstyle -{\frac {1}{3}}}$
Rectified cubic honeycomb		∞	∞	∞	6	12	3	4	Square bipyramid		${\textstyle 1}$	0.866	${\textstyle -{\frac {1}{2}}}$	${\textstyle 0}$
Rectified 600-cell		720	120	600	12	6	3	3	Pentagonal bipyramid		${\textstyle {\frac {5+3{\sqrt {5}}}{5}}}$	1.447	${\textstyle -{\frac {11+4{\sqrt {5}}}{41}}}$	${\textstyle -{\frac {11+4{\sqrt {5}}}{41}}}$

* The rectified 16-cell is the regular 24-cell and vertices are all equivalent – octahedra are regular bipyramids.

** Given numerically due to more complex form.

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, {}+{}.

• Trapezohedron

Wikimedia Commons has media related to Bipyramids.

• Weisstein, Eric W. "Dipyramid". MathWorld.
• Weisstein, Eric W. "Isohedron". MathWorld.
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
  • VRML models (George Hart) <3> <4> <5> <6> <7> <8> <9> <10>
    • Conway Notation for Polyhedra Try: "dPn", where n = 3, 4, 5, 6, ... Example: "dP4" is an octahedron.