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The ''deltoidal hexecontahedron'' can be constructed from either the regular icosahedron or regular dodecahedron by adding vertices mid-edge, and mid-face, and creating new edges from each edge center to the face centers. Conway polyhedron notation would give these as oI, and oD, ortho-icosahedron, and ortho-dodecahedron. These geometric variations exist as a continuum along one degree of freedom.
:
When projected onto a sphere (see right), it can be seen that the edges make up the edges of an icosahedron and dodecahedron arranged in their dual positions.
This tiling is topologically related as a part of sequence of deltoidal polyhedra with face figure (V3.4.''n''.4), and continues as tilings of the hyperbolic plane. These face-transitive figures have (*''n''32) reflectional symmetry.
(Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal hexecontahedron) * http://mathworld.wolfram.com/ArchimedeanDualGraph.html
Deltoidal Hexecontahedron (Trapezoidal Hexecontrahedron)
��Interactive Polyhedron Model
Example in real life
��A ball almost 4 meters in diameter, from ripstop nylon, and inflated by the wind. It bounces around on the ground so that kids can play with it at kite festivals. Catalan solids {{Polyhedron-stub
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 ...
, a deltoidal hexecontahedron (also sometimes called a ''trapezoidal hexecontahedron'', a ''strombic hexecontahedron'', or a ''tetragonal hexacontahedron'') is a Catalan solid which is the dual polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the othe ...
of the rhombicosidodecahedron
In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.
It has 20 regular triangular faces, 30 square faces, 12 regular ...
, an Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
. It is one of six Catalan solids to not have a Hamiltonian path among its vertices.
It is topologically identical to the nonconvex rhombic hexecontahedron.
Lengths and angles
The 60 faces are deltoids orkites
A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the face ...
. The short and long edges of each kite are in the ratio 1: ≈ 1:1.539344663...
The angle between two short edges in a single face is arccos()≈118.2686774705°. The opposite angle, between long edges, is arccos()≈67.783011547435° . The other two angles of each face, between a short and a long edge each, are both equal to arccos()≈86.97415549104°.
The dihedral angle between any pair of adjacent faces is arccos()≈154.12136312578°.
Topology
Topologically, the ''deltoidal hexecontahedron'' is identical to the nonconvex rhombic hexecontahedron. The deltoidal hexecontahedron can be derived from adodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentag ...
(or icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetric ...
) by pushing the face centers, edge centers and vertices out to different radii from the body center. The radii are chosen so that the resulting shape has planar kite faces each such that vertices go to degree-3 corners, faces to degree-five corners, and edge centers to degree-four points.
Orthogonal projections
The ''deltoidal hexecontahedron'' has 3 symmetry positions located on the 3 types of vertices:Variations
The ''deltoidal hexecontahedron'' can be constructed from either the regular icosahedron or regular dodecahedron by adding vertices mid-edge, and mid-face, and creating new edges from each edge center to the face centers. Conway polyhedron notation would give these as oI, and oD, ortho-icosahedron, and ortho-dodecahedron. These geometric variations exist as a continuum along one degree of freedom.
:
Related polyhedra and tilings
When projected onto a sphere (see right), it can be seen that the edges make up the edges of an icosahedron and dodecahedron arranged in their dual positions.
This tiling is topologically related as a part of sequence of deltoidal polyhedra with face figure (V3.4.''n''.4), and continues as tilings of the hyperbolic plane. These face-transitive figures have (*''n''32) reflectional symmetry.
See also
* Deltoidal icositetrahedronReferences
* (Section 3-9) * ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,(Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal hexecontahedron) * http://mathworld.wolfram.com/ArchimedeanDualGraph.html
External links
*Deltoidal Hexecontahedron (Trapezoidal Hexecontrahedron)
��Interactive Polyhedron Model
Example in real life
��A ball almost 4 meters in diameter, from ripstop nylon, and inflated by the wind. It bounces around on the ground so that kids can play with it at kite festivals. Catalan solids {{Polyhedron-stub