Small complex icosidodecahedron
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In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. The faces in it are considered as two overlapping edges as topological polyhedron.
A small complex icosidodecahedron can be constructed from a number of different vertex figures.
A very similar figure emerges as a geometrical truncation of the great stellated dodecahedron, where the pentagram faces become doubly-wound pentagons ({5/2} --> {10/2}), making the internal pentagonal planes, and the three meeting at each vertex become triangles, making the external triangular planes.
As a compound
The small complex icosidodecahedron can be seen as a compound of the icosahedron {3,5} and the great dodecahedron {5,5/2} where all vertices are precise and edges coincide. The small complex icosidodecahedron resembles an icosahedron, because the great dodecahedron is completely contained inside the icosahedron.
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Its two-dimensional analogue would be the compound of a regular pentagon, {5}, representing the icosahedron as the n-dimensional pentagonal polytope, and regular pentagram, {5/2}, as the n-dimensional star. These shapes would share vertices, similarly to how its 3D equivalent shares edges.
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See also
References
- {{Citation | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Longuet-Higgins | first2=M. S. | last3=Miller | first3=J. C. P. | title=Uniform polyhedra | jstor=91532 | mr=0062446 | year=1954 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=246 | issue=916 | pages=401–450 | doi=10.1098/rsta.1954.0003| bibcode=1954RSPTA.246..401C | s2cid=202575183 }} (Table 6, degenerate cases)
- {{mathworld | urlname = SmallComplexIcosidodecahedron| title = Small complex icosidodecahedron}}
- {{KlitzingPolytopes|polyhedra-neu.htm|3D uniform polyhedra|x3/2o5o5*a - cid}}