The rhombic hexecontahedron can be dissected into 20 acute golden rhombohedra meeting at a central point. This gives the volume of a hexecontahedron of side length a to be and the area to be .
Construction
A rhombic hexecontahedron can be constructed from a rhombic triacontahedron.
A rhombic hexecontahedron can be constructed from a regular dodecahedron, by taking its vertices, its face centers and its edge centers and scaling them in or out from the body center to different extents. Thus, if the 20 vertices of a dodecahedron are pulled out to increase the circumradius by a factor of (ϕ+1)/2 ≈ 1.309, the 12 face centers are pushed in to decrease the inradius to (3-ϕ)/2 ≈ 0.691 of its original value, and the 30 edge centers are left unchanged, then a rhombic hexecontahedron is formed. (The circumradius is increased by 30.9% and the inradius is decreased by the same 30.9%.) Scaling the points by different amounts results in hexecontahedra with kite-shaped faces or other polyhedra.
Every golden rhombic face has a face center, a vertex, and two edge centers of the original dodecahedron, with the edge centers forming the short diagonal. Each edge center is connected to two vertices and two face centers. Each face center is connected to five edge centers, and each vertex is connected to three edge centers.
Stellation
The rhombic hexecontahedron is one of 227 self-supporting stellations of the rhombic triacontahedron. Its stellation diagram looks like this, with the original rhombic triacontahedron faces as the central rhombus.
In Brazilian culture, handcrafted rhombic hexecontahedra used to be made from colored fabric and cardboard, called giramundos ("world turners" in Portuguese) or happiness stars, sewn by mothers and given as wedding gifts to their daughters. The custom got lost with the urbanization of Brazil, though the technique for producing the handicrafts was still taught in Brazilian rural schools up until the first half of the twentieth century.[2]
The logo of the WolframAlpha website is a red rhombic hexecontahedron and was inspired by the logo of the related Mathematica software.[3]
rhombic, hexecontahedron, type, stellation, rhombic, triacontahedronvertices, edges, faces, golden, rhombisymmetry, properties, convex, zonohedronin, geometry, rhombic, hexecontahedron, stellation, rhombic, triacontahedron, nonconvex, with, golden, rhombic, fa. Rhombic hexecontahedronType Stellation of rhombic triacontahedronVertices 62 12 20 30 Edges 120 60 60 Faces 60 golden rhombiSymmetry Ih 5 3 532 Properties non convex zonohedronIn geometry a rhombic hexecontahedron is a stellation of the rhombic triacontahedron It is nonconvex with 60 golden rhombic faces with icosahedral symmetry It was described mathematically in 1940 by Helmut Unkelbach 1 3D model of a rhombic hexecontahedron It is topologically identical to the convex deltoidal hexecontahedron which has kite faces Contents 1 Dissection 2 Construction 3 Stellation 4 Related polyhedra 5 In popular culture 6 References 7 Bibliography 8 External linksDissection EditThe rhombic hexecontahedron can be dissected into 20 acute golden rhombohedra meeting at a central point This gives the volume of a hexecontahedron of side length a to be V 10 2 5 a 3 displaystyle V 10 2 sqrt 5 a 3 and the area to be A 24 5 a 2 displaystyle A 24 sqrt 5 a 2 Construction Edit A rhombic hexecontahedron can be constructed from a rhombic triacontahedron A rhombic hexecontahedron can be constructed from a regular dodecahedron by taking its vertices its face centers and its edge centers and scaling them in or out from the body center to different extents Thus if the 20 vertices of a dodecahedron are pulled out to increase the circumradius by a factor of ϕ 1 2 1 309 the 12 face centers are pushed in to decrease the inradius to 3 ϕ 2 0 691 of its original value and the 30 edge centers are left unchanged then a rhombic hexecontahedron is formed The circumradius is increased by 30 9 and the inradius is decreased by the same 30 9 Scaling the points by different amounts results in hexecontahedra with kite shaped faces or other polyhedra Every golden rhombic face has a face center a vertex and two edge centers of the original dodecahedron with the edge centers forming the short diagonal Each edge center is connected to two vertices and two face centers Each face center is connected to five edge centers and each vertex is connected to three edge centers Stellation EditThe rhombic hexecontahedron is one of 227 self supporting stellations of the rhombic triacontahedron Its stellation diagram looks like this with the original rhombic triacontahedron faces as the central rhombus Related polyhedra EditThe great rhombic triacontahedron contains the 30 larger intersecting rhombic faces In popular culture EditIn Brazilian culture handcrafted rhombic hexecontahedra used to be made from colored fabric and cardboard called giramundos world turners in Portuguese or happiness stars sewn by mothers and given as wedding gifts to their daughters The custom got lost with the urbanization of Brazil though the technique for producing the handicrafts was still taught in Brazilian rural schools up until the first half of the twentieth century 2 The logo of the WolframAlpha website is a red rhombic hexecontahedron and was inspired by the logo of the related Mathematica software 3 References Edit Grunbaum 1996b Artesanato se antecipou a descoberta de poliedro Handicraft anticipated the discovery of a polyhedron in Portuguese IMPA retrieved 2019 01 08 What s in the Logo That Which We Call a Rhombic Hexecontahedron Wolfram Alpha Blog Bibliography EditUnkelbach H Die kantensymmetrischen gleichkantigen Polyeder Deutsche Math 5 306 316 1940 Grunbaum B 1996a A New Rhombic Hexecontahedron Geombinatorics 15 18 Grunbaum B 1996b A New Rhombic Hexecontahedron Once More Geombinatorics 55 59 Grunbaum B 1997 Still More Rhombic Hexecontahedra Geombinatorics 140 142 Grunbaum B Parallelogram Faced Isohedra with Edges in Mirror Planes Discrete Math 221 93 100 2000 External links EditWeisstein Eric W Rhombic hexecontahedron MathWorld http www georgehart com virtual polyhedra zonohedra info html The Bilinski dodecahedron and assorted parallelohedra zonohedra monohedra isozonohedra and otherhedra Branko Grunbaum This polyhedron related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Rhombic hexecontahedron amp oldid 1120060586, wikipedia, wiki, book, books, library,
