Hexagonal antiprism: Difference between revisions

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Uniform hexagonal antiprism

Type	Prismatic uniform polyhedron
Elements	F = 14, E = 24 V = 12 (χ = 2)
Faces by sides	12{3}+2{6}
Schläfli symbol	s{2,12} sr{2,6}
Wythoff symbol	\| 2 2 6
Coxeter diagram
Symmetry group	D_6d, [2⁺,12], (2*6), order 24
Rotation group	D₆, [6,2]⁺, (622), order 12
References	U_77(d)
Dual	Hexagonal trapezohedron
Properties	convex
Vertex figure 3.3.3.6

In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.

Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.

In the case of a regular n-sided base, one usually considers the case where its copy is twisted by an angle $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠180°/n⁠$ . Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it has the two $n$ -gonal bases and, connecting those bases, $2 n$ isosceles triangles.

If faces are all regular, it is a semiregular polyhedron.

Crossed antiprism

A crossed hexagonal antiprism is a star polyhedron, topologically identical to the convex hexagonal antiprism with the same vertex arrangement, but it can't be made uniform; the sides are isosceles triangles. Its vertex configuration is 3.3/2.3.6, with one triangle retrograde. It has D_6d symmetry, order 24.

Related polyhedra

The hexagonal faces can be replaced by coplanar triangles, leading to a nonconvex polyhedron with 24 equilateral triangles.

Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622)							[6,2]⁺, (622)	[6,2⁺], (2*3)


{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

V6²	V12²	V6²	V4.4.6	V2⁶	V4.4.6	V4.4.12	V3.3.3.6	V3.3.3.3

Family of uniform n-gonal antiprisms
v
t
e
Antiprism name	Digonal antiprism	(Trigonal) Triangular antiprism	(Tetragonal) Square antiprism	Pentagonal antiprism	Hexagonal antiprism	Heptagonal antiprism	...	Apeirogonal antiprism
Polyhedron image							...
Spherical tiling image							Plane tiling image
Vertex config.	2.3.3.3	3.3.3.3	4.3.3.3	5.3.3.3	6.3.3.3	7.3.3.3	...	∞.3.3.3

External links

Weisstein, Eric W. "Antiprism". MathWorld.
Hexagonal Antiprism: Interactive Polyhedron model
Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
- VRML model Archived 2007-02-07 at the Wayback Machine
polyhedronisme A6

@@ Line 1: / Line 1: @@
+{{short description|Antiprism with 6-sided caps}}
 {{Prism polyhedra db|Prism polyhedron stat table|AP6}}
-In [[geometry]], the '''hexagonal antiprism''' is the 4th in an infinite set of [[antiprisms]] formed by an even-numbered sequence of triangle sides closed by two polygon caps.
+In [[geometry]], the '''hexagonal antiprism''' is the 4th in an infinite set of [[antiprisms]] formed by an even-numbered sequence of triangle sides closed by two [[polygon]] caps.
-Antiprisms are similar to [[prism (geometry)|prism]]s except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.
+Antiprisms are similar to [[prism (geometry)|prism]]s except the bases are twisted relative to each other, and that the side faces are triangles, rather than [[quadrilateral]]s.
-In the case of a regular 6-sided base, one usually considers the case where its copy is twisted by an angle 180°/''n''. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a '''right antiprism'''. As faces, it has the two [[n-gon|''n''-gonal]] bases and, connecting those bases, 2''n'' isosceles triangles.
+In the case of a regular ''n''-sided base, one usually considers the case where its copy is twisted by an angle {{math|{{sfrac|180°|''n''}}}}. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a '''right antiprism'''. As faces, it has the two {{nowrap|[[n-gon|{{mvar|n}}-gonal]]}} bases and, connecting those bases, {{math|2''n''}} [[isosceles triangle]]s.
 If faces are all regular, it is a [[semiregular polyhedron]].
@@ Line 24: / Line 26: @@
 * [https://web.archive.org/web/20080616071808/http://polyhedra.org/poly/show/29/hexagonal_antiprism Hexagonal Antiprism: Interactive Polyhedron model]
 * [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] www.georgehart.com: The Encyclopedia of Polyhedra
-** [[VRML]] [http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_antiprism.wrl model]
+** [[VRML]] [http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_antiprism.wrl model] {{Webarchive|url=https://web.archive.org/web/20070207112806/http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_antiprism.wrl |date=2007-02-07 }}
+* [http://levskaya.github.io/polyhedronisme/?recipe=C100A6 polyhedronisme] A6
-** [http://www.georgehart.com/virtual-polyhedra/conway_notation.html Conway Notation for Polyhedra] Try: "A6"
 [[Category:Prismatoid polyhedra]]
 {{Polyhedron navigator}}
-{{Polyhedron-stub}}