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Bipyramid

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From Wikipedia, the free encyclopedia

Polyhedron formed by joining mirroring pyramids base-to-base

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

In geometry, abipyramid,dipyramid, ordouble pyramid is apolyhedron formed by fusing twopyramids togetherbase-to-base. Thepolygonal base of each pyramid must therefore be the same, and unless otherwise specified the basevertices are usuallycoplanar and a bipyramid is usuallysymmetric, meaning the two pyramids aremirror images across their common base plane. When eachapex (pl. apices, the off-base vertices) of the bipyramid is on a line perpendicular to the base and passing through its center, it is aright bipyramid;[a] otherwise it isoblique. When the base is aregular polygon, the bipyramid is also calledregular.

Definition and properties

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Thetriangular bipyramid,square bipyramid, andpentagonal bipyramid.

A bipyramid is a polyhedron constructed by fusing twopyramids which share the samepolygonal base;^[1] a pyramid is in turn constructed by connecting each vertex of its base to a single newvertex (theapex) not lying in the plane of the base, for ann-gonal base formingn triangular faces in addition to the base face. Ann-gonal bipyramid thus has2n faces,3n edges, andn + 2 vertices.More generally, a right pyramid is a pyramid where the apices are on the perpendicular line through thecentroid of an arbitrary polygon or theincenter of atangential polygon, depending on the source.^[a] Likewise, aright bipyramid is a polyhedron constructed by attaching two symmetrical right bipyramid bases; bipyramids whose apices are not on this line are calledoblique bipyramids.^[2]

When the two pyramids are mirror images, the bipyramid is calledsymmetric. It is calledregular if its base is aregular polygon.^[1] When the base is a regular polygon and the apices are on the perpendicular line through its center (aregular right bipyramid) then all of its faces areisosceles triangles; sometimes the namebipyramid refers specifically to symmetric regular right bipyramids,^[3] Examples of such bipyramids are thetriangular bipyramid,octahedron (square bipyramid) andpentagonal bipyramid. If all their edges are equal in length, these shapes consist ofequilateral triangle faces, making themdeltahedra;^[4]^[5] the triangular bipyramid and the pentagonal bipyramid areJohnson solids, and the regular octahedron is aPlatonic solid.^[6]

The symmetric regular right bipyramids haveprismatic symmetry, withdihedral symmetry groupD_nh of order4n: they are unchanged when rotated⁠1/n⁠ of a turn around theaxis of symmetry, reflected across any plane passing through both apices and a base vertex or both apices and the center of a base edge, or reflected across the mirror plane.^[7] Because their faces are transitive under these symmetry transformations, they areisohedral.^[8]^[9] They are thedual polyhedra ofprisms 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 share the same symmetry as the bipyramids.^[11] Theregular octahedron is more symmetric still, as its base vertices and apices are indistinguishable and can be exchanged by reflections orrotations; the regular octahedron and its dual, thecube, haveoctahedral symmetry.^[12]

Thevolume of a symmetric bipyramid is ${\frac {2}{3}}Bh,$ whereB is the area of the base andh the perpendicular distance from the base plane to either apex. In the case of a regularn-sided polygon with side lengths and whose altitude ish, the volume of such a bipyramid is: ${\frac {n}{6}}hs^{2}\cot {\frac {\pi }{n}}.$

Related and other types of bipyramid

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A concave tetragonal bipyramid

An asymmetric hexagonal bipyramid

Concave bipyramids

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Aconcave bipyramid has aconcave 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 aright bipyramid if the apices are on a line perpendicular to the base passing through the base'scentroid.

Asymmetric bipyramids

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Anasymmetric 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.

Thedual of an asymmetric rightn-gonal bipyramid is ann-gonalfrustum.

A regular asymmetric rightn-gonal bipyramid has symmetry groupC_nv, of order2n.

Scalene triangle bipyramids

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An isotoxal right (symmetric)di-n-gonal bipyramid is a right (symmetric)2n-gonal bipyramid with anisotoxal flat polygon base: its2n basal vertices are coplanar, but alternate in tworadii.

All its faces arecongruent scalene triangles, and it isisohedral. It can be seen as another type of a right symmetric di-n-gonalscalenohedron, with an isotoxal flat polygon base.

An isotoxal right (symmetric) di-n-gonal bipyramid hasn two-fold rotation axes through opposite basal vertices,n reflection planes through opposite apical edges, ann-fold rotation axis through apices, a reflection plane through base, and ann-foldrotation-reflection axis through apices,^[13] representing symmetry groupD_nh, [n,2], (*22n), of order4n. (The reflection about the base plane corresponds to the0° rotation-reflection. Ifn is even, then there is aninversion symmetry about the center, corresponding to the180° rotation-reflection.)

Example with2n = 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,^[14] but there is acenter of symmetry: the intersection point of the four axes.

Example with2n = 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.^[15]

Double example:

The bipyramid with isotoxal2×2-gon base verticesU, U', V, V' and right symmetric apicesA, 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:

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): ${\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}={\sqrt {5}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}=2{\sqrt {2}}\,.\end{aligned}}$

Incrystallography, isotoxal right (symmetric) didigonal^[b] (8-faced), ditrigonal (12-faced), ditetragonal (16-faced), and dihexagonal (24-faced) bipyramids exist.^[13]^[16]

Scalenohedra

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Ascalenohedron 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 and2n basal vertices,4n faces, and6n edges; it is topologically identical to a2n-gonal bipyramid, but its2n basal vertices alternate in two rings above and below the center.^[16]

All its faces arecongruent scalene triangles, and it isisohedral. 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 hasn two-fold rotation axes through opposite basal mid-edges,n reflection planes through opposite apical edges, ann-fold rotation axis through apices, and a2n-foldrotation-reflection axis through apices (about which1n rotations-reflections globally preserve the solid),^[13] representing symmetry groupD_nv = D_nd, [2⁺,2n], (2*n), of order4n. (Ifn is odd, then there is aninversion symmetry about the center, corresponding to the180° rotation-reflection.)

Example with2n = 2×3:

A regular right symmetric ditrigonal scalenohedron has three similar vertical planes of symmetry inclined to one another at60° and intersecting in a (vertical)3-fold rotation axis, three similar horizontal2-fold rotation axes, each perpendicular to a plane of symmetry, a center of inversion symmetry,^[18] and a vertical6-fold rotation-reflection axis.

Example with2n = 2×2:

A regular right symmetric didigonal scalenohedron has only one vertical and two horizontal2-fold rotation axes, two vertical planes of symmetry, which bisect the angles between the horizontal pair of axes, and a vertical4-fold rotation-reflection axis;^[19] it has no center of inversion symmetry.

Examples of disphenoids and of an8-faced scalenohedron

For at most two particular values of $z_{A}=|z_{A'}|,$ the faces of such ascalenohedron may beisosceles.

Double example:

The scalenohedron with regular zigzag skew2×2-gon base verticesU, U', V, V' and right symmetric apicesA, 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:

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): ${\begin{aligned}{\overline {A'U}}&={\overline {A'U'}}=3{\sqrt {10}}\,,\\[2pt]{\overline {A'V}}&={\overline {A'V'}}={\sqrt {34}}\,.\end{aligned}}$

Incrystallography, 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 regularoctahedron. In this case (2n = 2×2), in crystallography, a regular right symmetric didigonal (8-faced) scalenohedron is called atetragonal scalenohedron.^[13]^[16]

Let us temporarily focus on the regular right symmetric8-faced scalenohedra withh =r, i.e. $z_{A}=|z_{A'}|=x_{U}=|x_{U'}|=y_{V}=|y_{V'}|.$ Their two apices can be represented asA, A' and their four basal vertices asU, U', V, V': ${\begin{alignedat}{5}U&=(1,0,z),&\quad V&=(0,1,-z),&\quad A&=(0,0,1),\\U'&=(-1,0,z),&\quad V'&=(0,-1,-z),&\quad A'&=(0,0,-1),\end{alignedat}}$ wherez is a parameter between0 and1.

Atz = 0, it is a regular octahedron; atz = 1, it has four pairs of coplanar faces, and merging these into four congruent isosceles triangles makes it adisphenoid; forz > 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 the2n-gon base is bothisotoxal in-out andzigzag skew, thennot all faces of the isotoxal right symmetric scalenohedron are congruent.

Example with five different edge lengths:

The scalenohedron with isotoxal in-out zigzag skew2×2-gon base verticesU, U', V, V' and right symmetric apicesA, 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:

For some particular values ofz_A = |z_A'|, half the faces of such ascalenohedron may beisosceles orequilateral.

Example with three different edge lengths:

The scalenohedron with isotoxal in-out zigzag skew2×2-gon base verticesU, U', V, V' and right symmetric apicesA, 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:

Star bipyramids

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Astar bipyramid has astar polygon base, and is self-intersecting.^[20]

A regular right symmetric star bipyramid hascongruent isosceles triangle faces, and isisohedral.

Ap/q-bipyramid hasCoxeter diagram.

Example star bipyramids:
Base	5/2-gon	7/2-gon	7/3-gon	8/3-gon
Image

4-polytopes with bipyramidal cells

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Thedual of therectification of eachconvex regular 4-polytopes is acell-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 haveV_A vertices where the apices ofN_A bipyramids meet. It will haveV_E vertices where the typeE vertices ofN_E bipyramids meet.

⁠ $N_{\overline {AE}}$ ⁠ bipyramids meet along each typeAE edge.
⁠ $N_{\overline {EE}}$ ⁠ bipyramids meet along each typeEE edge.
⁠ $C_{\overline {AE}}$ ⁠ is the cosine of thedihedral angle along anAE edge.
⁠ $C_{\overline {EE}}$ ⁠ is the cosine of the dihedral angle along anEE 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	V_A	V_E	N_A	N_E	⁠ $N_{\overline {\!AE}}$ ⁠	⁠ $N_{\overline {\!EE}}$ ⁠	Bipyramid cell	Coxeter diagram	AA	AE^[c]	⁠ $C_{\overline {AE}}$ ⁠	⁠ $C_{\overline {EE}}$ ⁠
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^[d]	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

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A rhombus is a 2-dimensional analog of a right symmetric bipyramid

A generalizedn-dimensional "bipyramid" is anyn-polytope constructed from an(n − 1)-polytopebase lying in ahyperplane, with every base vertex connected by an edge to twoapex 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 identicalpyramidal facets.

A 2-dimensional analog of a right symmetric bipyramid is formed by joining twocongruent isosceles triangles base-to-base to form arhombus. More generally, akite is a 2-dimensional analog of a (possibly asymmetric) right bipyramid, and any quadrilateral is a 2-dimensional analog of a general bipyramid.

Notes

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^^a ^bThe center of a regular polygon is unambiguous, but for irregular polygons sources disagree. Some sources only allow a right pyramid to have a regular polygon as a base. Others define a right pyramid as having its apices on a line perpendicular to the base and passing through itscentroid.Polya (1954) restricts right pyramids to those with atangential polygon for a base, with the apices on a line perpendicular to the base and passing through theincenter.
^The smallest geometric di-n-gonal bipyramids have eight faces, and are topologically identical to the regularoctahedron. In this case (2n = 2×2):
an isotoxal right (symmetric) didigonal bipyramid is called arhombic bipyramid,^[13]^[16] although all its faces are scalene triangles, because its flat polygon base is a rhombus.
^Given numerically due to more complex form.
^The rectified 16-cell is the regular 24-cell and vertices are all equivalent – octahedra are regular bipyramids.

Citations

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^^a ^bAarts, J. M. (2008).Plane and Solid Geometry. Springer. p. 303.doi:10.1007/978-0-387-78241-6.ISBN 978-0-387-78241-6.
^Polya, G. (1954).Mathematics and Plausible Reasoning: Induction and analogy in mathematics. Princeton University Press. p. 138.ISBN 0-691-02509-6.{{cite book}}:ISBN / Date incompatibility (help)
^Montroll, John (2009).Origami Polyhedra Design. A K Peters.p. 6.ISBN 9781439871065.
^Trigg, Charles W. (1978). "An infinite class of deltahedra".Mathematics Magazine.51 (1):55–57.doi:10.1080/0025570X.1978.11976675.JSTOR 2689647.MR 1572246.
^Uehara, Ryuhei (2020).Introduction to Computational Origami: The World of New Computational Geometry. Springer. p. 62.doi:10.1007/978-981-15-4470-5.ISBN 978-981-15-4470-5.S2CID 220150682.
^Cromwell, Peter R. (1997).Polyhedra.Cambridge University Press.ISBN 978-0-521-55432-9.
^Flusser, Jan; Suk, Tomas; Zitofa, Barbara (2017).2D and 3D Image Analysis by Moments. John & Sons Wiley. p. 126.ISBN 978-1-119-03935-8.
^Chang, Ch.; Patzer, A. B. C.; Sülzle, D.; Hauer, H."Onion-Like Inorganic Fullerenes from a Polyhedral Perspective". In Sattler, Klaus D. (ed.).21st Century Nanoscience: A Handbook. Taylor & Francis. p. 15-4.
^McLean, K. Robin (1990). "Dungeons, dragons, and dice".The Mathematical Gazette.74 (469):243–256.doi:10.2307/3619822.JSTOR 3619822.S2CID 195047512.
^Sibley, Thomas Q. (2015).Thinking Geometrically: A Survey of Geometries. Mathematical Association of American. p. 53.ISBN 978-1-939512-08-6.
^King, Robert B. (1994)."Polyhedral Dynamics". In Bonchev, Danail D.; Mekenyan, O.G. (eds.).Graph Theoretical Approaches to Chemical Reactivity. Springer.doi:10.1007/978-94-011-1202-4.ISBN 978-94-011-1202-4.
^Armstrong, M. A. (1988).Group and Symmetry. Undergraduate Texts in Mathematics. Springer. p. 39.doi:10.1007/978-1-4757-4034-9.ISBN 978-1-4757-4034-9.
^^a ^b ^c ^d ^e ^f"Crystal Form, Zones, Crystal Habit".Tulane.edu. Retrieved16 September 2017.
^Spencer 1911, 6. Hexagonal system,rhombohedral division, ditrigonal bipyramidal class, p. 581 (p. 603 on Wikisource).
^Spencer 1911, 2. Tegragonal system, holosymmetric class, fig. 46, p. 577 (p. 599 on Wikisource).
^^a ^b ^c ^d ^e"The 48 Special Crystal Forms". 18 September 2013. Archived fromthe original on 18 September 2013. Retrieved18 November 2020.
^Klein, Cornelis; Philpotts, Anthony R. (2013).Earth Materials: Introduction to Mineralogy and Petrology. Cambridge University Press. p. 108.ISBN 978-0-521-14521-3.
^Spencer 1911, 6. Hexagonal system,rhombohedral division, holosymmetric class, fig. 68, p. 580 (p. 602 on Wikisource).
^Spencer 1911, p. 2. Tetragonal system, scalenohedral class, fig. 51, p. 577 (p. 599 on Wikisource).
^Rankin, John R. (1988). "Classes of polyhedra defined by jet graphics".Computers & Graphics.12 (2):239–254.doi:10.1016/0097-8493(88)90036-2.

Works cited

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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" . InChisholm, Hugh (ed.).Encyclopædia Britannica. Vol. 07 (11th ed.). Cambridge University Press. pp. 569–591.