Hexagonal trapezohedron

Last updated
Hexagonal trapezohedron
Hexagonal trapezohedron.png
Type trapezohedron
Faces 12 kites
Edges 24
Vertices 14
Vertex configuration V6.3.3.3
Coxeter diagram CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 12.pngCDel node.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 6.pngCDel node fh.png
Symmetry group D6d, [2+,12], (2*6), order 24
Rotation group D6, [2,6]+, (66), order 12
Dual polyhedron hexagonal antiprism
Properties convex, face-transitive

In geometry, a hexagonal trapezohedron or deltohedron is the fourth in an infinite series of trapezohedra which are dual polyhedra to the antiprisms. It has twelve faces which are congruent kites. It can be described by the Conway notation dA6.

Contents

It is an isohedral (face-transitive) figure, meaning that all its faces are the same. More specifically, all faces are not merely congruent but also transitive, i.e. lie within the same symmetry orbit . Convex isohedral polyhedra are the shapes that will make fair dice. [1]

Symmetry

The symmetry a hexagonal trapezohedron is D6d of order 24. The rotation group is D6 of order 12.

Variations

One degree of freedom within D6 symmetry changes the kites into congruent quadrilaterals with 3 edges lengths. In the limit, one edge of each quadrilateral goes to zero length, and these become bipyramids.

Crystal arrangements of atoms can repeat in space with a hexagonal trapezohedral configuration around one atom, which is always enantiomorphous, [2] and comprises space groups 177–182. [3] Beta-quartz is the only common mineral with this crystal system. [4]

If the kites surrounding the two peaks are of different shapes, it can only have C6v symmetry, order 12. These can be called unequal trapezohedra. The dual is an unequal antiprism , with the top and bottom polygons of different radii. If it twisted and unequal its symmetry is reduced to cyclic symmetry, C6 symmetry, order 6.

Example variations
TypeTwisted trapezohedra (isohedral)Unequal trapezohedraUnequal and twisted
Symmetry D6, (662), [6,2]+, order 12C6v, (*66), [6], order 12C6, (66), [6]+, order 6
Image
(n=6)
Twisted hexagonal trapezohedron.png Twisted hexagonal trapezohedron2.png Unequal hexagonal trapezohedron.png Unequal twisted hexagonal trapezohedron.png
Net Twisted hexagonal trapezohedron net.png Twisted hexagonal trapezohedron2 net.png Unequal hexagonal trapezohedron net.png Unequal twisted hexagonal trapezohedron net.png

Spherical tiling

The hexagonal trapezohedron also exists as a spherical tiling, with 2 vertices on the poles, and alternating vertices equally spaced above and below the equator.

Spherical hexagonal trapezohedron.svg
Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622)[6,2]+, (622)[6,2+], (2*3)
Hexagonal dihedron.png Dodecagonal dihedron.png Hexagonal dihedron.png Spherical hexagonal prism.svg Spherical hexagonal hosohedron.svg Spherical truncated trigonal prism.png Spherical dodecagonal prism2.png Spherical hexagonal antiprism.svg Spherical trigonal antiprism.svg
CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel node.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.png
{6,2} t{6,2} r{6,2} t{2,6} {2,6} rr{6,2} tr{6,2} sr{6,2} s{2,6}
Duals to uniforms
Spherical hexagonal hosohedron.svg Spherical dodecagonal hosohedron.svg Spherical hexagonal hosohedron.svg Spherical hexagonal bipyramid.svg Hexagonal dihedron.png Spherical hexagonal bipyramid.svg Spherical dodecagonal bipyramid.svg Spherical hexagonal trapezohedron.svg Spherical trigonal trapezohedron.svg
V62 V122 V62 V4.4.6 V26 V4.4.6 V4.4.12 V3.3.3.6 V3.3.3.3
Family of n-gonal trapezohedra
Trapezohedron nameDigonal trapezohedron
(Tetrahedron)
Trigonal trapezohedron Tetragonal trapezohedron Pentagonal trapezohedron Hexagonal trapezohedron ... Apeirogonal trapezohedron
Polyhedron image Digonal trapezohedron.png TrigonalTrapezohedron.svg Tetragonal trapezohedron.png Pentagonal trapezohedron.svg Hexagonal trapezohedron.png ...
Spherical tiling image Spherical digonal antiprism.svg Spherical trigonal trapezohedron.svg Spherical tetragonal trapezohedron.svg Spherical pentagonal trapezohedron.svg Spherical hexagonal trapezohedron.svg Plane tiling image Apeirogonal trapezohedron.svg
Face configuration V2.3.3.3V3.3.3.3V4.3.3.3V5.3.3.3V6.3.3.3...V∞.3.3.3

Related Research Articles

<span class="mw-page-title-main">Archimedean solid</span> Polyhedra in which all vertices are the same

In geometry, an Archimedean solid is one of 13 convex polyhedra whose faces are regular polygons and whose vertices are all symmetric to each other. They were first enumerated by Archimedes. They belong to the class of convex uniform polyhedra, the convex polyhedra with regular faces and symmetric vertices, which is divided into the Archimedean solids, the five Platonic solids, and the two infinite families of prisms and antiprisms. The pseudorhombicuboctahedron is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not vertex-transitive. An even larger class than the convex uniform polyhedra is the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.

In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two pyramids together base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise specified the base vertices are usually coplanar and a bipyramid is usually symmetric, meaning the two pyramids are mirror images across their common base plane. When each apex of the bipyramid is on a line perpendicular to the base and passing through its center, it is a right bipyramid; otherwise it is oblique. When the base is a regular polygon, the bipyramid is also called regular.

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

In geometry, an octahedron is a polyhedron with eight faces. An octahedron can be considered as a square bipyramid. When the edges of a square bipyramid are all equal in length, it produces a regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. It is also an example of a deltahedron. An octahedron is the three-dimensional case of the more general concept of a cross polytope.

<span class="mw-page-title-main">Hexagon</span> Shape with six sides

In geometry, a hexagon is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

<span class="mw-page-title-main">Kite (geometry)</span> Quadrilateral symmetric across a diagonal

In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.

<span class="mw-page-title-main">Rhombic dodecahedron</span> Catalan solid with 12 faces

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

<span class="mw-page-title-main">Trapezohedron</span> Polyhedron made of congruent kites arranged radially

In geometry, an n-gonaltrapezohedron, n-trapezohedron, n-antidipyramid, n-antibipyramid, or n-deltohedron, is the dual polyhedron of an n-gonal antiprism. The 2n faces of an n-trapezohedron are congruent and symmetrically staggered; they are called twisted kites. With a higher symmetry, its 2n faces are kites.

<span class="mw-page-title-main">Deltoidal icositetrahedron</span> Catalan solid with 24 kite faces

In geometry, the deltoidal icositetrahedron is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icositetrahedron, whose dual is the (uniform) rhombicuboctahedron, is tightly related to the pseudo-deltoidal icositetrahedron, whose dual is the pseudorhombicuboctahedron; but the actual and pseudo-d.i. are not to be confused with each other.

In geometry, the term semiregular polyhedron is used variously by different authors.

<span class="mw-page-title-main">Uniform polyhedron</span> Isogonal polyhedron with regular faces

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular, quasi-regular, or semi-regular. The faces and vertices don't need to be convex, so many of the uniform polyhedra are also star polyhedra.

<span class="mw-page-title-main">Hexagonal tiling</span> Regular tiling of a two-dimensional space

In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} .

<span class="mw-page-title-main">Trigonal trapezohedron</span> Polyhedron with 6 congruent rhombus faces

In geometry, a trigonal trapezohedron is a polyhedron with six congruent quadrilateral faces, which may be scalene or rhomboid. The variety with rhombus-shaped faces faces is a rhombohedron. An alternative name for the same shape is the trigonal deltohedron.

<span class="mw-page-title-main">Tetragonal trapezohedron</span> Trapezohedron with eight faces

In geometry, a tetragonal trapezohedron, or deltohedron, is the second in an infinite series of trapezohedra, which are dual to the antiprisms. It has eight faces, which are congruent kites, and is dual to the square antiprism.

<span class="mw-page-title-main">Pentagonal trapezohedron</span>

In geometry, a pentagonal trapezohedron is the third in an infinite series of face-transitive polyhedra which are dual polyhedra to the antiprisms. It has ten faces which are congruent kites.

<span class="mw-page-title-main">Truncated 5-cell</span>

In geometry, a truncated 5-cell is a uniform 4-polytope formed as the truncation of the regular 5-cell.

<span class="mw-page-title-main">Isohedral figure</span> ≥2-dimensional tessellation or ≥3-dimensional polytope with identical faces

In geometry, a tessellation of dimension 2 or higher, or a polytope of dimension 3 or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any two faces A and B, there must be a symmetry of the entire figure by translations, rotations, and/or reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.

In geometry, a polytope 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 two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.

<span class="mw-page-title-main">Tetradecahedron</span> Polyhedron with 14 faces

A tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces.

<span class="mw-page-title-main">Chamfer (geometry)</span> Geometric operation which truncates the edges of polyhedra

In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion: it moves the faces apart (outward), and adds a new face between each two adjacent faces; but contrary to expansion, it maintains the original vertices. For a polyhedron, this operation adds a new hexagonal face in place of each original edge.

References

  1. McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243–256, doi:10.2307/3619822, JSTOR   3619822 .
  2. 3 2 and Hexagonal-trapezohedric Class, 6 2 2
  3. Hahn, Theo, ed. (2005). International tables for crystallography (5th ed.). Dordrecht, Netherlands: Published for the International Union of Crystallography by Springer. ISBN   978-0-7923-6590-7.
  4. "Crystallography: The Hexagonal System". www.mindat.org. Retrieved 6 January 2023.