In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the great duoantiprism is the only
uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
star-
duoantiprism
In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, whe ...
solution in
4-dimensional geometry. It has
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
or
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
, constructed from 10
pentagonal antiprism
In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for ...
s, 10
pentagrammic crossed-antiprism
In geometry, the pentagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams.
It differs from the pentagrammic antiprism by having oppo ...
s, and 50
tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
.
Its vertices are a subset of those of the
small stellated 120-cell
In geometry, the small stellated 120-cell or stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes.
Related polytopes
It has the same edge arrangement as the great gran ...
.
Construction
The great duoantiprism can be constructed from a nonuniform variant of the 10-10/3 duoprism (a duoprism of a
decagon
In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°.
A self-intersecting ''regular decagon'' i ...
and a
decagram) where the decagram's edge length is around 1.618 (
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
) times the edge length of the decagon via an
alternation process. The
decagonal prism
In geometry, the decagonal prism is the eighth in the infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra. The decagonal prism has 12 faces, 30 edges, a ...
s alternate into
pentagonal antiprism
In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for ...
s, the
decagrammic prisms alternate into
pentagrammic crossed-antiprism
In geometry, the pentagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams.
It differs from the pentagrammic antiprism by having oppo ...
s with new regular
tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
created at the deleted vertices. This is the only uniform solution for the p-q duoantiprism aside from the regular
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mi ...
(as a 2-2 duoantiprism).
Images
Other names
* ''Great duoantiprism'' (gudap) Jonathan Bowers
[http://www.polychora.com/12GudapsMovie.gif Animation of cross sections]
References
*''Regular Polytopes'',
H. S. M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, Dover Publications, Inc., 1973, New York, p. 124.
*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
**
N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
*
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4-polytopes