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A hexagonal bipyramid is a
The ''hexagonal bipyramid'', dt, can be in sequence rectified, rdt, truncated, trdt and alternated ( snubbed), srdt:
:
It is the first polyhedra in a sequence defined by the
Virtual Reality Polyhedra
The Encyclopedia of Polyhedra **
hexagonal dipyramid
Try: dP6 Polyhedra Pyramids and bipyramids
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 ...
formed from two hexagonal pyramids
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 ...
joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical 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.
Although it is 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 congrue ...
, it is not a 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 ...
because some vertices have four faces meeting and others have six faces, and it is not a 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 each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johns ...
because its faces cannot be 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 oth ...
s; 6 equilateral triangles would make a flat vertex.
It is one of an infinite set of bipyramid
A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid 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 ...
s. Having twelve faces, it is a type of dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentag ...
, although that name is usually associated with the regular polyhedral form with pentagonal faces.
The hexagonal bipyramid has a plane of symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D t ...
(which is horizontal
Horizontal may refer to:
*Horizontal plane, in astronomy, geography, geometry and other sciences and contexts
*Horizontal coordinate system, in astronomy
*Horizontalism, in monetary circuit theory
*Horizontalism, in sociology
*Horizontal market, ...
in the figure to the right) where the bases of the two pyramids are joined. This plane is a regular hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A ''regular hexagon'' h ...
. There are also six planes of symmetry crossing through the two apices. These planes are rhombic
Rhombic may refer to:
* Rhombus, a quadrilateral whose four sides all have the same length (often called a diamond)
*Rhombic antenna, a broadband directional antenna most commonly used on shortwave frequencies
* polyhedra formed from rhombuses, suc ...
and lie at 30° angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
s to each other, 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 ...
to the horizontal plane.
Images
It can be drawn as a tiling on a sphere which also represents the fundamental domains of ,2 *322dihedral 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, ...
:
:
Related polyhedra
The hexagonal bipyramid, dt, can be in sequence truncated, tdt and alternated ( snubbed), sdt: :
The ''hexagonal bipyramid'', dt, can be in sequence rectified, rdt, truncated, trdt and alternated ( snubbed), srdt:
:
It is the first polyhedra in a sequence defined by the face configuration
In geometry, a vertex configurationCrystallography ...
''V4.6.2n''. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any
With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.
Each face on these domains also corresponds to the fundamental domain of a symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
with order 2,3,n mirrors at each triangle face vertex.
See also
* hexagonal trapezohedron A similar 12-sided polyhedron with a twist andkite
A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...
faces.
* Snub disphenoid
In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some ve ...
Another 12-sided polyhedron with 2-fold symmetry and only triangular faces.
External links
* {{mathworld , urlname = Dipyramid , title = DipyramidVirtual Reality Polyhedra
The 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 graph ...
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