A (symmetric) -gonal bipyramid or dipyramid is a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all o ...
formed by
joining an -gonal
pyramid
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
and its
mirror image base-to-base.
An -gonal bipyramid has
triangle faces, edges, and vertices.
The "-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.)
"Regular", right bipyramids
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) -gonal bipyramid has Schläfli symbol .
A right (symmetric) bipyramid has Schläfli symbol , for polygon base .
The "regular" right (thus
face-transitive) -gonal bipyramid with regular vertices
is the
dual of the -gonal uniform (thus right)
prism, and has
congruent isosceles triangle faces.
A "regular" right (symmetric) -gonal bipyramid can be
projected
Projected is an American rock supergroup consisting of Sevendust members John Connolly and Vinnie Hornsby, Alter Bridge and Creed drummer Scott Phillips, and former Submersed and current Tremonti guitarist Eric Friedman. The band released the ...
on a sphere or
globe as a "regular" right (symmetric) -gonal spherical bipyramid: equally spaced lines of
longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
going from
pole to pole, and an
equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can also ...
line
bisecting them.
Equilateral triangle bipyramids
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
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...
, counts among the
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s; the triangular and pentagonal bipyramids with same length edges count among the
Johnson solids and .
Kaleidoscopic symmetry
A ''"regular" right'' (symmetric) -gonal bipyramid has
dihedral symmetry group , of order , except in the case of a
regular octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...
, which has the larger
octahedral symmetry group , of order , which has three versions of as subgroups. The
rotation group is , of order , except in the case of a regular octahedron, which has the larger rotation group , of order , which has three versions of as subgroups.
Note: Every "regular" right (symmetric) -gonal bipyramid has the same (dihedral) symmetry group as the
dual-uniform -gonal bipyramid, for .
The triangle
faces of a "regular" right (symmetric) -gonal bipyramid, projected as the
spherical triangle
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gre ...
faces of a "regular" right (symmetric) -gonal
spherical bipyramid, represent the fundamental domains of
dihedral symmetry in three dimensions: , [], (), of order . 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 -fold or a -fold rotation axis through opposite vertices, a domain and its image are in the same color (direct isometry).
An -gonal (symmetric) bipyramid can be seen as the
Kleetope of the "corresponding" -gonal
dihedron.
Volume
Volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of a (symmetric) bipyramid:
where is the area of the base and 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 is measured as the
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
distance from the base
plane to any apex. Hence:
Volume of a (symmetric) bipyramid whose base is a ''regular'' -sided
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
with side length and whose height is :
Oblique bipyramids
Non-right bipyramids are called oblique bipyramids.
Concave 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.
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 -gonal bipyramid is an -gonal
frustum.
A "regular" asymmetric/inverted right -gonal bipyramid has symmetry group , of order .
Scalene triangle bipyramids
An "''isotoxal''" ''right'' (symmetric) di--gonal bipyramid is a ''right'' (symmetric) -gonal bipyramid with an
''isotoxal'' flat polygon base: its basal vertices are coplanar, but alternate in two
radii.
All its faces are
congruent scalene triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collin ...
s, and it is
isohedral. It can be seen as another type of a right "symmetric" di--gonal
''scalenohedron'', with an isotoxal flat polygon base.
An "isotoxal" right (symmetric) di--gonal bipyramid has two-fold rotation axes through opposite basal vertices, reflection planes through opposite apical edges, an -fold rotation axis through apices, a reflection plane through base, and an -fold
rotation-reflection axis through apices,
representing symmetry group of order . (The reflection about the base plane corresponds to the rotation-reflection. If is even, then there is an
inversion symmetry about the center, corresponding to the rotation-reflection.)
Example with :
:An "isotoxal" right (symmetric) ditrigonal bipyramid has three similar vertical planes of symmetry, intersecting in a (vertical) -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) -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 :
:An "isotoxal" right (symmetric) ditetragonal bipyramid has four vertical planes of symmetry of two kinds, intersecting in a (vertical) -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) -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.
Note: For at most two particular values of , the faces of such a scalene triangle bipyramid may be
isosceles.
Double example:
*The bipyramid with isotoxal -gon base vertices:
::
:and with "right" symmetric apices:
::
:has its faces isosceles. Indeed:
::upper apical edge lengths:
:::
:::
::base edge length:
:::
::lower apical edge lengths upper ones.
*The bipyramid with same base vertices, but with "right" symmetric apices:
::
:also has its faces isosceles. Indeed:
::upper apical edge lengths:
:::
:::
::base edge length previous one
::lower apical edge lengths upper ones.
In
crystallography, "isotoxal" right (symmetric) "didigonal" (*) (-faced), ditrigonal (-faced), ditetragonal (-faced), and dihexagonal (-faced) bipyramids exist.
(*) The smallest geometric di--gonal bipyramids have eight faces, and are topologically identical to the regular
octahedron. In this case ():
an "isotoxal" right (symmetric) "didigonal" bipyramid is called a ''rhombic bipyramid'',
although all its faces are scalene triangles, because its flat polygon base is a rhombus.
Scalenohedra
A ''"regular" right "symmetric"'' di--gonal scalenohedron is defined by a
''regular'' zigzag skew -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 basal vertices, faces, and edges; it is topologically identical to a -gonal bipyramid, but its basal vertices alternate in two rings above and below the center.
All its faces are
congruent scalene triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collin ...
s, and it is
isohedral. It can be seen as another type of a right "symmetric" di--gonal bipyramid, with a regular zigzag skew polygon base.
A "regular" right "symmetric" di--gonal scalenohedron has two-fold rotation axes through opposite basal mid-edges, reflection planes through opposite apical edges, an -fold rotation axis through apices, and a -fold
rotation-reflection axis through apices (about which rotations-reflections globally preserve the solid),
representing symmetry group of order . (If is odd, then there is an
inversion symmetry about the center, corresponding to the rotation-reflection.)
Example with :
:A "regular" right "symmetric" ditrigonal scalenohedron has three similar vertical planes of symmetry inclined to one another at and intersecting in a (vertical) -fold rotation axis, three similar horizontal -fold rotation axes, each perpendicular to a plane of symmetry, a center of inversion symmetry, and a vertical -fold rotation-reflection axis.
Example with :
:A "regular" right "symmetric" "didigonal" scalenohedron has only one vertical and two horizontal -fold rotation axes, two vertical planes of symmetry, which bisect the angles between the horizontal pair of axes, and a vertical -fold rotation-reflection axis; it has no center of inversion symmetry.
Note: For at most two particular values of , the faces of such a scalenohedron may be
isosceles.
Double example:
*The scalenohedron with regular zigzag skew -gon base vertices:
::
:and with "right" symmetric apices:
::
:has its faces isosceles. Indeed:
::upper apical edge lengths:
:::
:::
::base edge length:
:::
::lower apical edge lengths (swapped) upper ones.
*The scalenohedron with same base vertices, but with "right" symmetric apices:
::
:also has its faces isosceles. Indeed:
::upper apical edge lengths:
:::
:::
::base edge length previous one
::lower apical edge lengths (swapped) upper ones.
In
crystallography, "regular" right "symmetric" "didigonal" (-faced) and ditrigonal (-faced) scalenohedra exist.
The smallest geometric scalenohedra have eight faces, and are topologically identical to the regular
octahedron. In this case (), in crystallography, a "regular" right "symmetric" "didigonal" (-faced) scalenohedron is called a ''tetragonal scalenohedron''.
Let us temporarily focus on the "regular" right "symmetric" -faced scalenohedra with i.e. . Their two apices can be represented as and their four basal vertices as where is a parameter between and .
At , it is a regular octahedron; at , it has four pairs of coplanar faces, and merging these into four congruent isosceles triangles makes it a
disphenoid
In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same ...
; for , it is concave.
Note: If the -gon base is both
isotoxal
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given ...
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 -gon base vertices:
::
:and with "right" symmetric apices:
::
:has congruent scalene upper faces, and congruent scalene lower faces, but not all its faces are congruent. Indeed:
::upper apical edge lengths:
:::
:::
::base edge length:
:::
::lower apical edge lengths:
:::
:::
Note: For some particular values of , 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 -gon base vertices:
::
:and with "right" symmetric apices:
::
:has congruent scalene upper faces, and congruent equilateral lower faces; thus not all its faces are congruent. Indeed:
::upper apical edge lengths:
:::
:::
::base edge length:
:::
::lower apical edge length(s):
:::
:::
"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.
Note: For at most one particular value of , the faces of such a "regular" star bipyramid may be equilateral.
A -bipyramid has
Coxeter diagram .
Scalene triangle star bipyramids
An ''"isotoxal" right symmetric'' -gonal ''star'' bipyramid is defined by an
''isotoxal'' in-out ''star'' -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 -gonal star bipyramid has
congruent scalene
Scalene may refer to:
* A scalene triangle, one in which all sides and angles are not the same.
* A scalene ellipsoid, one in which the lengths of all three semi-principal axes are different
* Scalene muscles of the neck
* Scalene tubercle
The sc ...
triangle faces, and is
isohedral. It can be seen as another type of a -gonal right "symmetric"
''star scalenohedron'', with an isotoxal in-out star polygon base.
Note: For at most two particular values of , the faces of such a scalene triangle star bipyramid may be
isosceles.
Star scalenohedra
A ''"regular" right "symmetric"'' -gonal ''star'' scalenohedron is defined by a
''regular'' zigzag skew ''star'' -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" -gonal star scalenohedron has
congruent scalene
Scalene may refer to:
* A scalene triangle, one in which all sides and angles are not the same.
* A scalene ellipsoid, one in which the lengths of all three semi-principal axes are different
* Scalene muscles of the neck
* Scalene tubercle
The sc ...
triangle faces, and is
isohedral. It can be seen as another type of a right "symmetric" -gonal star bipyramid, with a regular zigzag skew star polygon base.
Note: For at most two particular values of , the faces of such a star scalenohedron may be
isosceles.
Note: If the star -gon base is both
isotoxal
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given ...
in-out and zigzag skew, then not all faces of the "isotoxal" right "symmetric" star scalenohedron are congruent.
Note: For some particular values of , 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 -gon base vertices:
::
::
:and with "right" symmetric apices:
::
:has congruent scalene upper faces, and congruent isosceles lower faces; thus not all its faces are congruent. Indeed:
::upper apical edge lengths:
:::
:::
::base edge length:
:::
::lower apical edge lengths:
:::
:::
Example with three different edge lengths:
:The star scalenohedron with isotoxal in-out zigzag skew -gon base vertices:
::
::
:and with "right" symmetric apices:
::
:has congruent scalene upper faces, and congruent equilateral lower faces; thus not all its faces are congruent. Indeed:
::upper apical edge lengths:
:::
:::
::base edge length:
:::
::lower apical edge length(s):
:::
:::
4-polytopes with bipyramidal cells
The
dual of the
rectification
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Recti ...
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,
: * 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.
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
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all o ...
base, and an apex point.
An example is the
16-cell, which is an octahedral bipyramid, +, and more generally an ''n''-
orthoplex is an (''n'' − 1)-orthoplex bipyramid, +.
Other bipyramids include the
tetrahedral bipyramid
In 4-dimensional geometry, the tetrahedral bipyramid is the direct sum of a tetrahedron and a segment, + . Each face of a central tetrahedron is attached with two tetrahedra, creating 8 tetrahedral cells, 16 triangular faces, 14 edges, and 6 verti ...
, +,
icosahedral bipyramid
In 4-dimensional geometry, the icosahedral bipyramid is the direct sum of a icosahedron and a segment, + . Each face of a central icosahedron is attached with two tetrahedra, creating 40 tetrahedral cells, 80 triangular faces, 54 edges, and 14 ve ...
, +, and
dodecahedral bipyramid, +, the first two having all regular cells, they are also
Blind polytope In geometry, a Blind polytope is a convex polytope composed of regular polytope facets.
The category was named after the German couple Gerd and Roswitha Blind, who described them in a series of papers beginning in 1979.
It generalizes the set of s ...
s.
See also
*
Trapezohedron
In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...
References
Citations
General references
* Chapter 4: Duals of the Archimedean polyhedra, prisms and antiprisms
*
External links
*
*
The Uniform PolyhedraThe Encyclopedia of Polyhedra
**
VRML model
(George Hart)<3><4><5><6><7><8><9><10>
*
Try: "dP''n''", where ''n'' = 3, 4, 5, 6, ... Example: "dP4" is an octahedron.
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