Bipyramid

"Dipyramid" redirects here. For the mountain, see Dipyramid (Alaska).

In geometry, a bipyramid or dipyramid is a polyhedron formed by fusing two pyramids together base-to-base. The base of each pyramid must therefore be the same, and unless otherwise specified, 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 centroid, it is a right bipyramid; otherwise it is oblique. When the base is a regular polygon, the bipyramid is also called regular.

Description

The triangular bipyramid, octahedron, and pentagonal bipyramid.

A bipyramid is a polyhedron constructed by attaching bases of two identical pyramids, and the mirror image in their bases. All of its faces are isosceles triangles. For $n$-sided bipyramid, it has $2n$ faces, $3n$ edges, and $n+2$ vertices. Examples of such bipyramids are the triangular bipyramid and pentagonal bipyramid; in the case of all edges equal in length, they are the Johnson solids and deltahedron, as their faces are regular. Octahedron is another example of a bipyramid with all edges equal in length, and it is a Platonic solid. These three are deltahedron, as they have equilateral triangles as their faces.

The octahedron is dual to the cube

The bipyramids have prismatic symmetry, the dihedral group of $D_{nh}$ of order $2n$: their appearance is symmetrical by rotating around the axis of symmetry and reflecting across the mirror plane. 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; the prisms shares the same symmetry as the bipyramids. One example is the octahedron and the cube.

Volume

Volume of a (symmetric) bipyramid: $$V={\frac {2}{3}}Bh,$$ 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: $$V={\frac {n}{6}}hs^{2}\cot {\frac {\pi }{n}}.$$

Concave bipyramids

A concave bipyramid has a concave polygon base. A concave tetragonal bipyramid is 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

Example concave bipyramid:

Name	concave tetragonal (symmetric) bipyramid
Image

Asymmetric/inverted right bipyramids

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.

Example regular right hexagonal bipyramids:

Asymmetric	Inverted

Scalene triangle bipyramids

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, 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, 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.

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, U', V, V' and right symmetric apices A, A'$${\begin{alignedat}{5}U&=(1,0,0),&\quad V&=(0,2,0),&\quad A&=(0,0,1),\\U'&=(-1,0,0),&\quad V'&=(0,-2,0),&\quad A'&=(0,0,-1),\end{alignedat}}$$ has its faces isosceles. Indeed:
  • Upper apical edge lengths:$${\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {2}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}={\sqrt {5}}\,;\end{aligned}}$$
  • Base edge lengths: $${\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {5}}\,;$$
  • Lower apical edge lengths (equal to upper edge lengths):$${\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {2}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {5}}\,.\end{aligned}}$$

• The bipyramid with same base vertices, but with right symmetric apices $${\begin{aligned}A&=(0,0,2),\\A'&=(0,0,-2),\end{aligned}}$$ also has its faces isosceles. Indeed:
  • Upper apical edge lengths:$${\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {5}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}=2{\sqrt {2}}\,;\end{aligned}}$$
  • Base edge length (equal to previous example): $${\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {5}}\,;$$
  • Lower apical edge lengths (equal to upper edge lengths):$${\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {5}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=2{\sqrt {2}}\,.\end{aligned}}$$

Examples of rhombic bipyramids

In crystallography, isotoxal right (symmetric) didigonal (8-faced), ditrigonal (12-faced), ditetragonal (16-faced), and dihexagonal (24-faced) bipyramids exist.

Scalenohedra

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.

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), 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, 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; it has no center of inversion symmetry.

Examples of disphenoids and of an 8-faced scalenohedron

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, U', V, V' and right symmetric apices A, A'$${\begin{alignedat}{5}U&=(3,0,2),&\quad V&=(0,3,-2),&\quad A&=(0,0,3),\\U'&=(-3,0,2),&\quad V'&=(0,-3,-2),&\quad A'&=(0,0,-3),\end{alignedat}}$$ has its faces isosceles. Indeed:
  • Upper apical edge lengths:$${\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {10}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}={\sqrt {34}}\,;\end{aligned}}$$
  • Base edge length:$${\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {34}}\,;$$
  • Lower apical edge lengths (equal to upper edge lengths swapped):$${\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {34}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {10}}\,.\end{aligned}}$$

• The scalenohedron with same base vertices, but with right symmetric apices$${\begin{aligned}A&=(0,0,7),\\A'&=(0,0,-7),\end{aligned}}$$ also has its faces isosceles. Indeed:
  • Upper apical edge lengths:$${\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {34}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}=3{\sqrt {10}}\,;\end{aligned}}$$
  • Base edge length (equal to previous example): $${\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}={\sqrt {34}}\,;$$
  • Lower apical edge lengths (equal to upper edge lengths swapped):$${\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}=3{\sqrt {10}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {34}}\,.\end{aligned}}$$

In crystallography, regular right symmetric didigonal (8-faced) and ditrigonal (12-faced) scalenohedra exist.

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.

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, A' and their four basal vertices as U, U', V, V': $${\begin{alignedat}{5}U&=(1,0,z),&\quad V&=(0,1,-z),&\quad A&=(0,0,1),\\U'&=(-1,0,z),&\quad V'&=(0,-1,-z),&\quad A'&=(0,0,-1),\end{alignedat}}$$ 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

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, U', V, V' and right symmetric apices A, A' $${\begin{alignedat}{5}U&=(1,0,1),&\quad V&=(0,2,-1),&\quad A&=(0,0,3),\\U'&=(-1,0,1),&\quad V'&=(0,-2,-1),&\quad A'&=(0,0,-3),\end{alignedat}}$$ has congruent scalene upper faces, and congruent scalene lower faces, but not all its faces are congruent. Indeed:
  • Upper apical edge lengths:$${\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {5}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}=2{\sqrt {5}}\,;\end{aligned}}$$
  • Base edge length:$${\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}=3;$$
  • Lower apical edge lengths:$${\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {17}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=2{\sqrt {2}}\,.\end{aligned}}$$

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, U', V, V' and right symmetric apices A, A' $${\begin{alignedat}{5}U&=(3,0,2),&\quad V&=\left(0,{\sqrt {65}},-2\right),&\quad A&=(0,0,7),\\U'&=(-3,0,2),&\quad V'&=\left(0,-{\sqrt {65}},-2\right),&\quad A'&=(0,0,-7),\end{alignedat}}$$ has congruent scalene upper faces, and congruent equilateral lower faces; thus not all its faces are congruent. Indeed:
  • Upper apical edge lengths:$${\begin{aligned}{\overline {AU}}&={\overline {AU'}}={\sqrt {34}}\,,\\[2pt]{\overline {AV}}&={\overline {AV'}}={\sqrt {146}}\,;\end{aligned}}$$
  • Base edge length:$${\overline {UV}}={\overline {VU'}}={\overline {U'V'}}={\overline {V'U}}=3{\sqrt {10}}\,;$$
  • Lower apical edge length(s): $${\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}=3{\sqrt {10}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=3{\sqrt {10}}\,.\end{aligned}}$$

Regular star bipyramids

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.

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 .

Example regular right symmetric star bipyramids:

Base	5/2-gon	7/2-gon	7/3-gon	8/3-gon
Image
Base	9/2-gon	9/4-gon	10/3-gon	11/2-gon
Image
Base	11/3-gon	11/4-gon	11/5-gon	12/5-gon
Image

Scalene triangle star bipyramids

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.

For at most two particular values of $z_{A}=|z_{A'}|,$ the faces of such a scalene triangle star bipyramid may be isosceles.

Example isotoxal right symmetric star bipyramid:

Base	Isotoxal in-out 8/3-gon
Image

Star scalenohedra

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.

For at most two particular values of $z_{A}=|z_{A'}|,$ the faces of such a star scalenohedron may be isosceles.

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.

Examples of right symmetric star scalenohedra:

Base	Regular zigzag skew 8/3-gon	Isotoxal in-out zigzag skew 8/3-gon
Image

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, U₁, U₂, U₃, V₀, V₁, V₂, V₃ and with right symmetric apices A, A' $${\begin{alignedat}{5}U_{0}&=(1,0,1),&\quad V_{0}&=(2,2,-1),&\quad A&=(0,0,3),\\[3mu]U_{1}&=(0,1,1),&\quad V_{1}&=(-2,2,-1),&\quad A'&=(0,0,-3),\\[3mu]U_{2}&=(-1,0,1),&\quad V_{2}&=(-2,-2,-1),\\[3mu]U_{3}&=(0,-1,1),&\quad V_{3}&=(2,-2,-1),{\vphantom {x_{|}}}\end{alignedat}}$$ has congruent scalene upper faces, and congruent isosceles lower faces; thus not all its faces are congruent. Indeed:
  • Upper apical edge lengths: $${\begin{aligned}{\overline {AU_{0}}}&={\overline {AU_{1}}}={\overline {AU_{2}}}={\overline {AU_{3}}}={\sqrt {5}}\,,\\[2pt]{\overline {AV_{0}}}&={\overline {AV_{1}}}={\overline {AV_{2}}}={\overline {AV_{3}}}=2{\sqrt {6}}\,;\end{aligned}}$$
  • Base edge length: $${\overline {U_{0}V_{1}}}={\overline {V_{1}U_{3}}}={\overline {U_{3}V_{0}}}={\overline {V_{0}U_{2}}}={\overline {U_{2}V_{3}}}={\overline {V_{3}U_{1}}}={\overline {U_{1}V_{2}}}={\overline {V_{2}U_{0}}}={\sqrt {17}}\,;$$
  • Lower apical edge lengths:$${\begin{aligned}{\overline {A'U_{0}}}&={\overline {A'U_{1}}}={\overline {A'U_{2}}}={\overline {A'U_{3}}}={\sqrt {17}}\,,\\[2pt]{\overline {A'V_{0}}}&={\overline {A'V_{1}}}={\overline {A'V_{2}}}={\overline {A'V_{3}}}=2{\sqrt {3}}\,.\end{aligned}}$$

Example with three different edge lengths:

• The star scalenohedron with isotoxal in-out zigzag skew 8/3-gon base vertices U₀, U₁, U₂, U₃, V₀, V₁, V₂, V₃ and with right symmetric apices A, A' $${\begin{alignedat}{5}U_{0}&=\left(4,0,{\sqrt {2}}\right),&\quad V_{0}&=\left(6,6,-{\sqrt {2}}\right),&\quad A&=\left(0,0,7{\sqrt {2}}\right),\\U_{1}&=\left(0,4,{\sqrt {2}}\right),&\quad V_{1}&=\left(-6,6,-{\sqrt {2}}\right),&\quad A'&=\left(0,0,-7{\sqrt {2}}\right),\\U_{2}&=\left(-4,0,{\sqrt {2}}\right),&\quad V_{2}&=\left(-6,-6,-{\sqrt {2}}\right),\\U_{3}&=\left(0,-4,{\sqrt {2}}\right),&\quad V_{3}&=\left(6,-6,-{\sqrt {2}}\right),\end{alignedat}}$$ has congruent scalene upper faces, and congruent equilateral lower faces; thus not all its faces are congruent. Indeed:
  • Upper apical edge lengths: $${\begin{aligned}{\overline {AU_{0}}}&={\overline {AU_{1}}}={\overline {AU_{2}}}={\overline {AU_{3}}}=2{\sqrt {22}}\,,\\[2pt]{\overline {AV_{0}}}&={\overline {AV_{1}}}={\overline {AV_{2}}}={\overline {AV_{3}}}=10{\sqrt {2}}\,;\end{aligned}}$$
  • Base edge length: $${\overline {U_{0}V_{1}}}={\overline {V_{1}U_{3}}}={\overline {U_{3}V_{0}}}={\overline {V_{0}U_{2}}}={\overline {U_{2}V_{3}}}={\overline {V_{3}U_{1}}}={\overline {U_{1}V_{2}}}={\overline {V_{2}U_{0}}}=12;$$
  • Lower apical edge length(s): $${\begin{aligned}{\overline {A'U_{0}}}&={\overline {A'U_{1}}}={\overline {A'U_{2}}}={\overline {A'U_{3}}}=12,\\[2pt]{\overline {A'V_{0}}}&={\overline {A'V_{1}}}={\overline {A'V_{2}}}={\overline {A'V_{3}}}=12.\end{aligned}}$$

4-polytopes with bipyramidal cells

The dual of the rectification of each convex regular 4-polytopes is a cell-transitive 4-polytope with bipyramidal cells. In the following:

• A is the apex vertex of the bipyramid;
• E is an equator vertex;
• EE is the distance between adjacent vertices on the equator (equal to 1);
• AE is the apex-to-equator edge length;
• AA is the distance between the apices.

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.

• ⁠$N_{\overline {AE}}$⁠ bipyramids meet along each type AE edge.
• ⁠$N_{\overline {EE}}$⁠ bipyramids meet along each type EE edge.
• ⁠$C_{\overline {AE}}$⁠ is the cosine of the dihedral angle along an AE edge.
• ⁠$C_{\overline {EE}}$⁠ is the cosine of the dihedral angle along an EE edge.

As cells must fit around an edge, $${\begin{aligned}N_{\overline {EE}}\arccos C_{\overline {EE}}&\leq 2\pi ,\\[4pt]N_{\overline {AE}}\arccos C_{\overline {AE}}&\leq 2\pi .\end{aligned}}$$

4-polytopes with bipyramidal cells

4-polytope properties									Bipyramid properties
Dual of rectified polytope	Coxeter diagram	Cells	VA	VE	NA	NE	⁠$N_{\overline {\!AE}}$⁠	⁠$N_{\overline {\!EE}}$⁠	Bipyramid cell	Coxeter diagram	AA	AE	⁠$C_{\overline {AE}}$⁠	⁠$C_{\overline {EE}}$⁠
R. 5-cell		10	5	5	4	6	3	3	Triangular		${\textstyle {\frac {2}{3}}}$	0.667	${\textstyle -{\frac {1}{7}}}$	${\textstyle -{\frac {1}{7}}}$
R. tesseract		32	16	8	4	12	3	4	Triangular		${\textstyle {\frac {\sqrt {2}}{3}}}$	0.624	${\textstyle -{\frac {2}{5}}}$	${\textstyle -{\frac {1}{5}}}$
R. 24-cell		96	24	24	8	12	4	3	Triangular		${\textstyle {\frac {2{\sqrt {2}}}{3}}}$	0.745	${\textstyle {\frac {1}{11}}}$	${\textstyle -{\frac {5}{11}}}$
R. 120-cell		1200	600	120	4	30	3	5	Triangular		${\textstyle {\frac {{\sqrt {5}}-1}{3}}}$	0.613	${\textstyle -{\frac {10+9{\sqrt {5}}}{61}}}$	${\textstyle -{\frac {7-12{\sqrt {5}}}{61}}}$
R. 16-cell		24	8	16	6	6	3	3	Square		${\textstyle {\sqrt {2}}}$	1	${\textstyle -{\frac {1}{3}}}$	${\textstyle -{\frac {1}{3}}}$
R. cubic honeycomb		∞	∞	∞	6	12	3	4	Square		${\textstyle 1}$	0.866	${\textstyle -{\frac {1}{2}}}$	${\textstyle 0}$
R. 600-cell		720	120	600	12	6	3	3	Pentagonal		${\textstyle {\frac {5+3{\sqrt {5}}}{5}}}$	1.447	${\textstyle -{\frac {11+4{\sqrt {5}}}{41}}}$	${\textstyle -{\frac {11+4{\sqrt {5}}}{41}}}$

Other dimensions

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, { } + { }.

Polyhedral bipyramids

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, { } + {3ⁿ⁻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.

See also

• Trapezohedron

References

General references

• Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 4: Duals of the Archimedean polyhedra, prisms and antiprisms
• Spencer, Leonard James (1911). "Crystallography" . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 07 (11th ed.). Cambridge University Press. pp. 569–591.

External links

• 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.