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 on th ...
formed by
joining Join may refer to:
* Join (law), to include additional counts or additional defendants on an indictment
*In mathematics:
** Join (mathematics), a least upper bound of sets orders in lattice theory
** Join (topology), an operation combining two topo ...
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, quadrilat ...
and its
mirror image
A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...
base-to-base.
An -gonal bipyramid has
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...
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
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
'' 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
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
) -gonal bipyramid with regular vertices
is the
dual of the -gonal uniform (thus right)
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
, and has
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
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 t ...
on a sphere or
globe
A globe is a spherical model of Earth, of some other celestial body, or of the celestial sphere. Globes serve purposes similar to maps, but unlike maps, they do not distort the surface that they portray except to scale it down. A model 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 letter l ...
going from
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
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 als ...
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 triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
s, and thus the bipyramid is a
deltahedron
In geometry, a deltahedron (plural ''deltahedra'') is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle. There are infinitely many d ...
): the "regular" right (symmetric)
triangular
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- collinea ...
,
tetragonal
In crystallography, the tetragonal crystal system is one of the 7 crystal systems. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square ...
, and
pentagonal
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
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 at ea ...
, 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 edges c ...
s; the triangular and pentagonal bipyramids with same length edges count among the
Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...
s and .
Kaleidoscopic symmetry
A ''"regular" right'' (symmetric) -gonal bipyramid has
dihedral symmetry
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
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 at ea ...
, which has the larger
octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
group , of order , which has three versions of as subgroups. The
rotation group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
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
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
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 grea ...
faces of a "regular" right (symmetric) -gonal
spherical
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
bipyramid, represent the fundamental domains of
dihedral symmetry in three dimensions
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dih''n'' (for ''n'' ≥ 2).
Types
Ther ...
: , [], (), 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
In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope is another polyhedron or polytope formed by replacing each facet of with a shallow pyramid. Kleetopes are named after Victor Klee.
Exam ...
of the "corresponding" -gonal
dihedron
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...
.
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). The de ...
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 can ...
distance from the base
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
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 toge ...
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.
:
![Concave quadrilateral bipyramid](https://upload.wikimedia.org/wikipedia/commons/e/ef/Concave_quadrilateral_bipyramid.png)
(*) Its base has no obvious center; but if its apices are ''right'' above and ''right'' below the
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
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
In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are ...
.
A "regular" asymmetric/inverted right -gonal bipyramid has symmetry group , of order .
Scalene triangle bipyramids
![EB1911 Crystallography Fig](https://upload.wikimedia.org/wikipedia/commons/f/fb/EB1911_Crystallography_Fig._46_Ditetragonal_Bipyramid.jpg)
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
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
.
All its faces are
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
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-collinear ...
s, and it is
isohedral
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
. 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
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
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 A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space.
For an object, any unique centre and, more ...
: 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
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
, "isotoxal" right (symmetric) "didigonal" (*) (-faced), ditrigonal (-faced), ditetragonal (-faced), and dihexagonal (-faced) bipyramids exist.
![EB1911 Crystallography Figs](https://upload.wikimedia.org/wikipedia/commons/4/42/EB1911_Crystallography_Figs._54_%26_55_Orthorhombic_Bipyramids.jpg)
(*) The smallest geometric di--gonal bipyramids have eight faces, and are topologically identical to the 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 at ea ...
. 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
![EB1911 Crystallography Fig](https://upload.wikimedia.org/wikipedia/commons/2/20/EB1911_Crystallography_Fig._68.%E2%80%94Scalenohedron.jpg)
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
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
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-collinear ...
s, and it is
isohedral
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
. 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
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
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.
![EB1911 Crystallography Figs](https://upload.wikimedia.org/wikipedia/commons/3/38/EB1911_Crystallography_Figs._50_%26_51.jpg)
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
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
, "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 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 at ea ...
. 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 sh ...
; for , it is concave.
Note: If the -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 -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
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
isosceles
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
triangle faces, and is
isohedral
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
.
Note: For at most one particular value of , the faces of such a "regular" star bipyramid may be equilateral.
A -bipyramid has
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
.
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
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
scalene triangle faces, and is
isohedral
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
. 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
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
scalene triangle faces, and is
isohedral
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
. 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-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 of each
convex regular 4-polytope
In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
There are six convex and ten star regu ...
s is a
cell-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congrue ...
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
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
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
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
constructed with a (''n'' − 1)-polytope in a
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
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
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, quadrilater ...
facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
.
A 2-dimensional ("regular") right symmetric (digonal) bipyramid is formed by joining two
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
s base-to-base; its outline is a
rhombus
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
, +.
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 on th ...
base, and an apex point.
An example is the
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mi ...
, which is an octahedral bipyramid, +, and more generally an ''n''-
orthoplex
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
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 vert ...
, +,
icosahedral bipyramid, +, and
dodecahedral bipyramid
In 4-dimensional geometry, the dodecahedral bipyramid is the direct sum of a dodecahedron and a segment, + . Each face of a central dodecahedron is attached with two pentagonal pyramids, creating 24 pentagonal pyramidal cells, 72 isosceles trian ...
, +, the first two having all regular cells, they are also
Blind polytopes.
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
VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graphi ...
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.
{{Polyhedron navigator
Polyhedra