In geometry, the truncated dodecadodecahedron (or stellatruncated dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U59. It is given a Schläfli symbolt0,1,2{5⁄3,5}. It has 54 faces (30 squares, 12 decagons, and 12 decagrams), 180 edges, and 120 vertices.[1] The central region of the polyhedron is connected to the exterior via 20 small triangular holes.
3D model of a truncated dodecadodecahedron
The name truncated dodecadodecahedron is somewhat misleading: truncation of the dodecadodecahedron would produce rectangular faces rather than squares, and the pentagram faces of the dodecadodecahedron would turn into truncated pentagrams rather than decagrams. However, it is the quasitruncation of the dodecadodecahedron, as defined by Coxeter, Longuet-Higgins & Miller (1954).[2] For this reason, it is also known as the quasitruncated dodecadodecahedron.[3] Coxeter et al. credit its discovery to a paper published in 1881 by Austrian mathematician Johann Pitsch.[4]
Cartesian coordinates for the vertices of a truncated dodecadodecahedron are all the triples of numbers obtained by circular shifts and sign changes from the following points (where is the golden ratio):
Each of these five points has eight possible sign patterns and three possible circular shifts, giving a total of 120 different points.
As a Cayley graphedit
The truncated dodecadodecahedron forms a Cayley graph for the symmetric group on five elements, as generated by two group members: one that swaps the first two elements of a five-tuple, and one that performs a circular shift operation on the last four elements. That is, the 120 vertices of the polyhedron may be placed in one-to-one correspondence with the 5! permutations on five elements, in such a way that the three neighbors of each vertex are the three permutations formed from it by swapping the first two elements or circularly shifting (in either direction) the last four elements.[5]
^Coxeter, H. S. M.; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246 (916): 401–450, Bibcode:1954RSPTA.246..401C, doi:10.1098/rsta.1954.0003, JSTOR 91532, MR 0062446. See especially the description as a quasitruncation on p. 411 and the photograph of a model of its skeleton in Fig. 114, Plate IV.
^Wenninger writes "quasitruncated dodecahedron", but this appears to be a mistake. Wenninger, Magnus J. (1971), "98 Quasitruncated dodecahedron", Polyhedron Models, Cambridge University Press, pp. 152–153.
^Pitsch, Johann (1881), "Über halbreguläre Sternpolyeder", Zeitschrift für das Realschulwesen, 6: 9–24, 72–89, 216. According to Coxeter, Longuet-Higgins & Miller (1954), the truncated dodecadodecahedron appears as no. XII on p.86.
^Eppstein, David (2009), "The topology of bendless three-dimensional orthogonal graph drawing", in Tollis, Ioannis G.; Patrignani, Marizio (eds.), Graph Drawing, Lecture Notes in Computer Science, vol. 5417, Heraklion, Crete: Springer-Verlag, pp. 78–89, arXiv:0709.4087, doi:10.1007/978-3-642-00219-9_9, ISBN978-3-642-00218-2.
truncated, dodecadodecahedron, type, uniform, star, polyhedron, elements, 180v, faces, sides, coxeter, diagram, wythoff, symbol, symmetry, group, index, references, dual, polyhedron, medial, disdyakis, triacontahedron, vertex, figure, bowers, acronym, quitdid,. Truncated dodecadodecahedron Type Uniform star polyhedron Elements F 54 E 180V 120 x 6 Faces by sides 30 4 12 10 12 10 3 Coxeter diagram Wythoff symbol 2 5 5 3 Symmetry group Ih 5 3 532 Index references U59 C75 W98 Dual polyhedron Medial disdyakis triacontahedron Vertex figure 4 10 9 10 3 Bowers acronym Quitdid In geometry the truncated dodecadodecahedron or stellatruncated dodecadodecahedron is a nonconvex uniform polyhedron indexed as U59 It is given a Schlafli symbol t0 1 2 5 3 5 It has 54 faces 30 squares 12 decagons and 12 decagrams 180 edges and 120 vertices 1 The central region of the polyhedron is connected to the exterior via 20 small triangular holes 3D model of a truncated dodecadodecahedron The name truncated dodecadodecahedron is somewhat misleading truncation of the dodecadodecahedron would produce rectangular faces rather than squares and the pentagram faces of the dodecadodecahedron would turn into truncated pentagrams rather than decagrams However it is the quasitruncation of the dodecadodecahedron as defined by Coxeter Longuet Higgins amp Miller 1954 2 For this reason it is also known as the quasitruncated dodecadodecahedron 3 Coxeter et al credit its discovery to a paper published in 1881 by Austrian mathematician Johann Pitsch 4 Contents 1 Cartesian coordinates 2 As a Cayley graph 3 Related polyhedra 3 1 Medial disdyakis triacontahedron 4 See also 5 References 6 External linksCartesian coordinates editCartesian coordinates for the vertices of a truncated dodecadodecahedron are all the triples of numbers obtained by circular shifts and sign changes from the following points where f 1 5 2 displaystyle varphi tfrac 1 sqrt 5 2 nbsp is the golden ratio 1 1 3 1 f 1 f 2 2 f f 2 f f 2 f 2 1 f 2 2 5 1 5 displaystyle begin array lcr Bigl 1 amp 1 amp 3 Bigr Bigl frac 1 varphi amp frac 1 varphi 2 amp 2 varphi Bigr Bigl varphi amp frac 2 varphi amp varphi 2 Bigr Bigl varphi 2 amp frac 1 varphi 2 amp 2 Bigr Bigl sqrt 5 amp 1 amp sqrt 5 Bigr end array nbsp Each of these five points has eight possible sign patterns and three possible circular shifts giving a total of 120 different points As a Cayley graph editThe truncated dodecadodecahedron forms a Cayley graph for the symmetric group on five elements as generated by two group members one that swaps the first two elements of a five tuple and one that performs a circular shift operation on the last four elements That is the 120 vertices of the polyhedron may be placed in one to one correspondence with the 5 permutations on five elements in such a way that the three neighbors of each vertex are the three permutations formed from it by swapping the first two elements or circularly shifting in either direction the last four elements 5 Related polyhedra editMedial disdyakis triacontahedron edit Medial disdyakis triacontahedron nbsp Type Star polyhedron Face nbsp Elements F 120 E 180V 54 x 6 Symmetry group Ih 5 3 532 Index references DU59 dual polyhedron Truncated dodecadodecahedron nbsp 3D model of a medial disdyakis triacontahedron The medial disdyakis triacontahedron is a nonconvex isohedral polyhedron It is the dual of the uniform truncated dodecadodecahedron See also editList of uniform polyhedraReferences edit Maeder Roman 59 truncated dodecadodecahedron MathConsult Coxeter H S M Longuet Higgins M S Miller J C P 1954 Uniform polyhedra Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical Sciences 246 916 401 450 Bibcode 1954RSPTA 246 401C doi 10 1098 rsta 1954 0003 JSTOR 91532 MR 0062446 See especially the description as a quasitruncation on p 411 and the photograph of a model of its skeleton in Fig 114 Plate IV Wenninger writes quasitruncated dodecahedron but this appears to be a mistake Wenninger Magnus J 1971 98 Quasitruncated dodecahedron Polyhedron Models Cambridge University Press pp 152 153 Pitsch Johann 1881 Uber halbregulare Sternpolyeder Zeitschrift fur das Realschulwesen 6 9 24 72 89 216 According to Coxeter Longuet Higgins amp Miller 1954 the truncated dodecadodecahedron appears as no XII on p 86 Eppstein David 2009 The topology of bendless three dimensional orthogonal graph drawing in Tollis Ioannis G Patrignani Marizio eds Graph Drawing Lecture Notes in Computer Science vol 5417 Heraklion Crete Springer Verlag pp 78 89 arXiv 0709 4087 doi 10 1007 978 3 642 00219 9 9 ISBN 978 3 642 00218 2 Wenninger Magnus 1983 Dual Models Cambridge University Press doi 10 1017 CBO9780511569371 ISBN 978 0 521 54325 5 MR 0730208External links editWeisstein Eric W Truncated dodecadodecahedron MathWorld Weisstein Eric W Medial disdyakis triacontahedron MathWorld Retrieved from https en wikipedia org w index php title Truncated dodecadodecahedron amp oldid 1185183862, wikipedia, wiki, book, books, library,
