Trapezohedron

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Set of dual-uniform n-gonal trapezohedra
Pentagonal trapezohedron.svg
Example: dual-uniform pentagonal trapezohedron
Typedual-uniform in the sense of dual-semiregular polyhedron
Faces 2n congruent kites
Edges 4n
Vertices 2n + 2
Vertex configuration V3.3.3.n
Schläfli symbol { } ⨁ {n} [1]
Conway notation dAn
Coxeter diagram CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 2x.pngCDel n.pngCDel node.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel n.pngCDel node fh.png
Symmetry group Dnd, [2+,2n], (2*n), order 4n
Rotation group Dn, [2,n]+, (22n), order 2n
Dual polyhedron (convex) uniform n-gonal antiprism
Properties convex, face-transitive, regular vertices [2]

In geometry, an n-gonaltrapezohedron, antidipyramid, antibipyramid, or 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 (also called trapezoids, or deltoids). [3]

Contents

The n-gon part of the name does not refer to faces here, but to two arrangements of each n vertices around an axis of n-fold symmetry. The dual n-gonal antiprism has two actual n-gon faces.

An n-gonal trapezohedron can be dissected into two equal n-gonal pyramids and an n-gonal antiprism.

Terminology

These figures, sometimes called deltohedra, must not be confused with deltahedra, whose faces are equilateral triangles.

A twisted trigonal trapezohedron (with six twisted trapezoidal faces) and a twisted tetragonal trapezohedron (with eight twisted trapezoidal faces) exist as crystals; in crystallography (describing the crystal habits of minerals), they are just called trigonal trapezohedron and tetragonal trapezohedron. They have no plane of symmetry, and no center of inversion symmetry; [4] , [5] but they have a center of symmetry: the intersection point of their symmetry axes. The trigonal trapezohedron has one 3-fold symmetry axis, perpendicular to three 2-fold symmetry axes. [4] The tetragonal trapezohedron has one 4-fold symmetry axis, perpendicular to four 2-fold symmetry axes.

Also in crystallography, the word trapezohedron is often used for the polyhedron with 24 trapezoidal faces properly known as a (deltoidal) icositetrahedron , [6] which has eighteen order-4 vertices and eight order-3 vertices. This is not to be confused with the dodecagonal trapezohedron, which also has 24 trapezoidal faces, but two order-12 apices (i.e. poles) and two rings of twelve order-3 vertices each.

Still in crystallography, the polyhedron with 12 trapezoidal faces known as a deltoid dodecahedron, [7] has six order-4 vertices and eight order-3 vertices (the rhombic dodecahedron is a special case). This is not to be confused with the hexagonal trapezohedron , which also has 12 trapezoidal faces, [8] but two order-6 apices (i.e. poles) and two rings of six order-3 vertices each.

Symmetry

The symmetry group of an n-gonal trapezohedron is Dnd, of order 4n, except in the case of n = 3: a cube has the larger symmetry group Od of order 48 = 4×(4×3), which has four versions of D3d as subgroups.

The rotation group of an n-trapezohedron is Dn, of order 2n, except in the case of n = 3: a cube has the larger rotation group O of order 24 = 4×(2×3), which has four versions of D3 as subgroups.

One degree of freedom within symmetry from Dnd (order 4n) to Dn (order 2n) changes the congruent kites into congruent quadrilaterals with three edge lengths, called twisted kites, and the n-trapezohedron is called a twisted trapezohedron. (In the limit, one edge of each quadrilateral goes to zero length, and the n-trapezohedron becomes an n-bipyramid.)

If the kites surrounding the two peaks are not twisted but are of two different shapes, the n-trapezohedron can only have Cnv (cyclic with vertical mirrors) symmetry, order 2n, and is called an unequal or asymmetric trapezohedron. Its dual is an unequal n-antiprism , with the top and bottom polygons of different radii.

If the kites are twisted and are of two different shapes, the n-trapezohedron can only have Cn (cyclic) symmetry, order n, and is called an unequal twisted trapezohedron.

Example: variations with hexagonal trapezohedra (n = 6)
Trapezohedron typeTwisted trapezohedronUnequal trapezohedronUnequal twisted trapezohedron
Symmetry group D6, (662), [6,2]+C6v, (*66), [6]C6, (66), [6]+
Polyhedron image 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

Forms

An n-trapezohedron has 2n quadrilateral faces, with 2n+2 vertices. Two apices are on the polar axis, and the other vertices are in two regular n-gonal rings of vertices.

Family of n-gonal trapezohedra
Trapezohedron nameDigonal trapezohedron
(Tetrahedron)
Trigonal trapezohedron Tetragonal trapezohedron Pentagonal trapezohedron Hexagonal trapezohedron Heptagonal trapezohedron Octagonal trapezohedron Decagonal trapezohedron Dodecagonal trapezohedron ... Apeirogonal trapezohedron
Polyhedron image Digonal trapezohedron.png TrigonalTrapezohedron.svg Tetragonal trapezohedron.png Pentagonal trapezohedron.svg Hexagonal trapezohedron.png Heptagonal trapezohedron.png Octagonal trapezohedron.png Decagonal trapezohedron.png Dodecagonal trapezohedron.png ...
Spherical tiling image Spherical digonal antiprism.png Spherical trigonal trapezohedron.png Spherical tetragonal trapezohedron.png Spherical pentagonal trapezohedron.png Spherical hexagonal trapezohedron.png Spherical heptagonal trapezohedron.png Spherical octagonal trapezohedron.png Spherical decagonal trapezohedron.png Spherical dodecagonal trapezohedron.png Plane tiling image Apeirogonal trapezohedron.svg
Face configuration V2.3.3.3V3.3.3.3V4.3.3.3V5.3.3.3V6.3.3.3V7.3.3.3V8.3.3.3V10.3.3.3V12.3.3.3...V∞.3.3.3

Special cases:

Examples

Star trapezohedron

A face-transitive star p/q-trapezohedron is defined by a regular zig-zag skew star 2p/q-gon base, two symmetric apices with no degree of freedom right above and right below the base, and kite faces connecting each pair of adjacent base edges to one apex.

Such a star p/q-trapezohedron is a self-intersecting, crossed, or non-convex form. It exists for any regular zig-zag skew star 2p/q-gon base; but if p/q < 3/2, then pq < q/2, so the dual star antiprism (of the star trapezohedron) cannot be uniform (i.e. cannot have equal edge lengths); and if p/q = 3/2, then pq = q/2, so the dual star antiprism must be flat, thus degenerate, to be uniform.

A dual-uniform star p/q-trapezohedron has Coxeter-Dynkin diagram CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel p.pngCDel rat.pngCDel q.pngCDel node fh.png.

Dual-uniform star p/q-trapezohedra up to p = 12
5/25/37/27/37/48/38/59/29/49/5
5-2 deltohedron.png 5-3 deltohedron.png 7-2 deltohedron.png 7-3 deltohedron.png 7-4 deltohedron.png 8-3 deltohedron.png 8-5 deltohedron.png 9-2 deltohedron.png 9-4 deltohedron.png 9-5 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 5.pngCDel rat.pngCDel 2x.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 5.pngCDel rat.pngCDel 3x.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 7.pngCDel rat.pngCDel 2x.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 7.pngCDel rat.pngCDel 3x.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 7.pngCDel rat.pngCDel 4.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 8.pngCDel rat.pngCDel 3x.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 8.pngCDel rat.pngCDel 5.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 9.pngCDel rat.pngCDel 2x.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 9.pngCDel rat.pngCDel 4.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 9.pngCDel rat.pngCDel 5.pngCDel node fh.png
10/311/211/311/411/511/611/712/512/7
10-3 deltohedron.png 11-2 deltohedron.png 11-3 deltohedron.png 11-4 deltohedron.png 11-5 deltohedron.png 11-6 deltohedron.png 11-7 deltohedron.png 12-5 deltohedron.png 12-7 deltohedron.png
CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 10.pngCDel rat.pngCDel 3x.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 2x.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 3x.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 4.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 5.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 6.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 7.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 12.pngCDel rat.pngCDel 5.pngCDel node fh.pngCDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 12.pngCDel rat.pngCDel 7.pngCDel node fh.png

See also

Related Research Articles

Antiprism Polyhedron with parallel bases connected by triangles

In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An.

Bipyramid Polyhedron formed by joining mirroring pyramids base-to-base

A (symmetric) n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices.

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.

Octahedron Polyhedron with 8 triangular faces

In geometry, an octahedron 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 each vertex.

Prism (geometry) Solid with parallel bases connected by parallelograms

In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.

Rhombic dodecahedron 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.

Deltoidal icositetrahedron Catalan solid with 24 faces

In geometry, a deltoidal icositetrahedron is a Catalan solid. Its dual polyhedron is the rhombicuboctahedron.

Deltoidal hexecontahedron

In geometry, a deltoidal hexecontahedron is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, an Archimedean solid. It is one of six Catalan solids to not have a Hamiltonian path among its vertices.

Snub disphenoid 84th Johnson solid (12 triangular faces)

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 vertices have four faces and others have five. It is a dodecahedron, one of the eight deltahedra, and is the 84th Johnson solid. It can be thought of as a square antiprism where both squares are replaced with two equilateral triangles.

Hexagonal bipyramid Polyhedron; 2 hexagonal pyramids joined base-to-base

A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles.

Uniform polyhedron Isogonal polyhedron with regular faces

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

Trigonal trapezohedron Polyhedron with 6 congruent rhombus faces

In geometry, a trigonal trapezohedron is a rhombohedron in which, additionally, all six faces are congruent. Alternative names for the same shape are the trigonal deltohedron or isohedral rhombohedron. Some sources just call them rhombohedra.

Hexagonal trapezohedron Polyhedron made of 12 congruent kites

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.

Octagonal trapezohedron

In geometry, a octagonal trapezohedron' or deltohedron is the sixth in an infinite series trapezohedra which are dual polyhedron to the antiprisms. It has sixteen faces which are congruent kites.

Decagonal trapezohedron

In geometry, a decagonal trapezohedron is the eighth in an infinite series of face-uniform polyhedra which are dual polyhedra to the antiprisms. It has twenty faces which are congruent kites.

In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces. The precise number of near misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons. Some high symmetry near-misses are also symmetrohedra with some perfect regular polygon faces.

Compound of two tetrahedra Polyhedral compound

In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.

Diminished trapezohedron Polyhedron made by truncating one end of a trapezohedron

In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular n-gonal base face, n triangle faces around the base, and n kites meeting on top. The kites can also be replaced by rhombi with specific proportions.

Icosahedron Polyhedron with 20 faces

In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and from Ancient Greek ἕδρα (hédra) ' seat'. The plural can be either "icosahedra" or "icosahedrons".

Heptagonal trapezohedron

In geometry, a heptagonal trapezohedron or deltohedron is the fifth in an infinite series of trapezohedra which are dual polyhedron to the antiprisms. It has 14 faces which are congruent kites.

References

  1. N.W. Johnson: Geometries and Transformations, (2018) ISBN   978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3c
  2. "duality". maths.ac-noumea.nc. Retrieved 2020-10-19.
  3. Spencer 1911, p. 575, or p. 597 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, TETRAHEDRAL CLASS, footnote: « [Deltoid]: From the Greek letter δ, Δ; in general, a triangular-shaped object; also an alternative name for a trapezoid ». Remark: a twisted kite can look like and even be a trapezoid.
  4. 1 2 Spencer 1911, p. 581, or p. 603 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, Rhombohedral Division, TRAPEZOHEDRAL CLASS, FIG. 74.
  5. Spencer 1911, p. 577, or p. 599 on Wikisource, CRYSTALLOGRAPHY, 2. TETRAGONAL SYSTEM, TRAPEZOHEDRAL CLASS.
  6. Spencer 1911, p. 574, or p. 596 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, HOLOSYMMETRIC CLASS, FIG. 17.
  7. Spencer 1911, p. 575, or p. 597 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, TETRAHEDRAL CLASS, FIG. 27.
  8. Spencer 1911, p. 582, or p. 604 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, Hexagonal Division, TRAPEZOHEDRAL CLASS.
  9. Trigonal-trapezohedric Class, 3 2 and Hexagonal-trapezohedric Class, 6 2 2