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A hexagonal bipyramid is a polyhedron formed from two hexagonal
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 ''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
Virtual Reality Polyhedra
The Encyclopedia of Polyhedra ** VRML mode
hexagonal dipyramid
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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, 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 triangles; 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, although that name is usually associated with the regular polyhedral form with pentagonal faces.
The hexagonal bipyramid has a plane of symmetry (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 *322 dihedral symmetry: :
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 ''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 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 mode
hexagonal dipyramid
Try: dP6 Polyhedra Pyramids and bipyramids