Set of dual-uniform n-gonal trapezohedra | |
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![]() Example: dual-uniform pentagonal trapezohedron | |
Type | dual-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 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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]
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.
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.
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.
Trapezohedron type | Twisted trapezohedron | Unequal trapezohedron | Unequal twisted trapezohedron | |
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Symmetry group | D6, (662), [6,2]+ | C6v, (*66), [6] | C6, (66), [6]+ | |
Polyhedron image | ![]() | ![]() | ![]() | ![]() |
Net | ![]() | ![]() | ![]() | ![]() |
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.
Trapezohedron name | Digonal trapezohedron (Tetrahedron) | Trigonal trapezohedron | Tetragonal trapezohedron | Pentagonal trapezohedron | Hexagonal trapezohedron | Heptagonal trapezohedron | Octagonal trapezohedron | Decagonal trapezohedron | Dodecagonal trapezohedron | ... | Apeirogonal trapezohedron |
---|---|---|---|---|---|---|---|---|---|---|---|
Polyhedron image | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ... | |
Spherical tiling image | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | Plane tiling image | ![]() |
Face configuration | V2.3.3.3 | V3.3.3.3 | V4.3.3.3 | V5.3.3.3 | V6.3.3.3 | V7.3.3.3 | V8.3.3.3 | V10.3.3.3 | V12.3.3.3 | ... | V∞.3.3.3 |
Special cases:
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 p − q < 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 p − q = 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 .
5/2 | 5/3 | 7/2 | 7/3 | 7/4 | 8/3 | 8/5 | 9/2 | 9/4 | 9/5 |
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10/3 | 11/2 | 11/3 | 11/4 | 11/5 | 11/6 | 11/7 | 12/5 | 12/7 |
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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.
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.
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.
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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.
In geometry, a deltoidal icositetrahedron is a Catalan solid. Its dual polyhedron is the rhombicuboctahedron.
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.
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.
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.
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.
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.
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.
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.
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.
In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.
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.
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".
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.