The pentagonal icositetrahedron can be constructed from a snub cube without taking the dual. Square pyramids are added to the six square faces of the snub cube, and triangular pyramids are added to the eight triangular faces that do not share an edge with a square. The pyramid heights are adjusted to make them coplanar with the other 24 triangular faces of the snub cube. The result is the pentagonal icositetrahedron.
Cartesian coordinates
Denote the tribonacci constant by . (See snub cube for a geometric explanation of the tribonacci constant.) Then Cartesian coordinates for the 38 vertices of a pentagonal icositetrahedron centered at the origin, are as follows:
the 12 even permutations of (±1, ±(2t+1), ±t2) with an even number of minus signs
the 12 odd permutations of (±1, ±(2t+1), ±t2) with an odd number of minus signs
the 6 points (±t3, 0, 0), (0, ±t3, 0) and (0, 0, ±t3)
the 8 points (±t2, ±t2, ±t2)
The convex hulls for these vertices[4] scaled by result in a unit circumradius octahedron centered at the origin, a unit cube centered at the origin scaled to , and an irregular chiral snub cube scaled to , as visualized in the figure below:
Geometry
The pentagonal faces have four angles of and one angle of . The pentagon has three short edges of unit length each, and two long edges of length . The acute angle is between the two long edges. The dihedral angle equals .
If its dual snub cube has unit edge length, its surface area and volume are:[5]
Orthogonal projections
The pentagonal icositetrahedron has three symmetry positions, two centered on vertices, and one on midedge.
Orthogonal projections
Projective symmetry
[3]
[4]+
[2]
Image
Dual image
Variations
Isohedral variations with the same chiral octahedral symmetry can be constructed with pentagonal faces having 3 edge lengths.
This variation shown can be constructed by adding pyramids to 6 square faces and 8 triangular faces of a snub cube such that the new triangular faces with 3 coplanar triangles merged into identical pentagon faces.
Snub cube with augmented pyramids and merged faces
Pentagonal icositetrahedron
Net
Related polyhedra and tilings
Spherical pentagonal icositetrahedron
This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations (V3.3.3.3.n). (The sequence progresses into tilings the hyperbolic plane to any n.) These face-transitive figures have (n32) rotational symmetry.
^Koca, Mehmet; Ozdes Koca, Nazife; Koc, Ramazon (2010). "Catalan Solids Derived From 3D-Root Systems and Quaternions". Journal of Mathematical Physics. 51 (4). arXiv:0908.3272. doi:10.1063/1.3356985.
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN0-486-23729-X. (Section 3-9)
The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 287, pentagonal icosikaitetrahedron)
External links
– Interactive Polyhedron Model
March 04, 2023
pentagonal, icositetrahedron, click, rotating, models, type, catalanconway, notation, gccoxeter, diagramface, polygon, irregular, pentagonfaces, 24edges, 60vertices, 24face, configuration, 4dihedral, angle, symmetry, group, 432dual, polyhedron, snub, cubeprope. Pentagonal icositetrahedron Click ccw or cw for rotating models Type CatalanConway notation gCCoxeter diagramFace polygon irregular pentagonFaces 24Edges 60Vertices 38 6 8 24Face configuration V3 3 3 3 4Dihedral angle 136 18 33 Symmetry group O BC3 4 3 432Dual polyhedron snub cubeProperties convex face transitive chiralNetIn geometry a pentagonal icositetrahedron or pentagonal icosikaitetrahedron 1 is a Catalan solid which is the dual of the snub cube In crystallography it is also called a gyroid 2 3 A geometric construction of the Tribonacci constant AC with compass and marked ruler according to the method described by Xerardo Neira 3d model of a pentagonal icositetrahedron It has two distinct forms which are mirror images or enantiomorphs of each other Contents 1 Construction 2 Cartesian coordinates 3 Geometry 4 Orthogonal projections 4 1 Variations 5 Related polyhedra and tilings 6 References 7 External linksConstruction EditThe pentagonal icositetrahedron can be constructed from a snub cube without taking the dual Square pyramids are added to the six square faces of the snub cube and triangular pyramids are added to the eight triangular faces that do not share an edge with a square The pyramid heights are adjusted to make them coplanar with the other 24 triangular faces of the snub cube The result is the pentagonal icositetrahedron Cartesian coordinates EditDenote the tribonacci constant by t 1 839 286 755 21 displaystyle t approx 1 839 286 755 21 See snub cube for a geometric explanation of the tribonacci constant Then Cartesian coordinates for the 38 vertices of a pentagonal icositetrahedron centered at the origin are as follows the 12 even permutations of 1 2t 1 t2 with an even number of minus signs the 12 odd permutations of 1 2t 1 t2 with an odd number of minus signs the 6 points t3 0 0 0 t3 0 and 0 0 t3 the 8 points t2 t2 t2 The convex hulls for these vertices 4 scaled by t 3 displaystyle t 3 result in a unit circumradius octahedron centered at the origin a unit cube centered at the origin scaled to R 0 9416969935 displaystyle R approx 0 9416969935 and an irregular chiral snub cube scaled to R displaystyle R as visualized in the figure below Geometry EditThe pentagonal faces have four angles of arccos 1 t 2 114 812 074 477 90 displaystyle arccos 1 t 2 approx 114 812 074 477 90 circ and one angle of arccos 2 t 80 751 702 088 39 displaystyle arccos 2 t approx 80 751 702 088 39 circ The pentagon has three short edges of unit length each and two long edges of length t 1 2 1 419 643 377 607 08 displaystyle t 1 2 approx 1 419 643 377 607 08 The acute angle is between the two long edges The dihedral angle equals arccos 1 t 2 2 136 309 232 892 32 displaystyle arccos 1 t 2 2 approx 136 309 232 892 32 circ If its dual snub cube has unit edge length its surface area and volume are 5 A 3 22 5 t 1 4 t 3 19 299 94 V 11 t 4 2 20 t 37 7 4474 displaystyle begin aligned A amp 3 sqrt frac 22 5t 1 4t 3 amp amp approx 19 299 94 V amp sqrt frac 11 t 4 2 20t 37 amp amp approx 7 4474 end aligned Orthogonal projections EditThe pentagonal icositetrahedron has three symmetry positions two centered on vertices and one on midedge Orthogonal projections Projectivesymmetry 3 4 2 Image Dualimage Variations Edit Isohedral variations with the same chiral octahedral symmetry can be constructed with pentagonal faces having 3 edge lengths This variation shown can be constructed by adding pyramids to 6 square faces and 8 triangular faces of a snub cube such that the new triangular faces with 3 coplanar triangles merged into identical pentagon faces Snub cube with augmented pyramids and merged faces Pentagonal icositetrahedron NetRelated polyhedra and tilings Edit Spherical pentagonal icositetrahedron This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations V3 3 3 3 n The sequence progresses into tilings the hyperbolic plane to any n These face transitive figures have n32 rotational symmetry n32 symmetry mutations of snub tilings 3 3 3 3 n vteSymmetryn32 Spherical Euclidean Compact hyperbolic Paracomp 232 332 432 532 632 732 832 32Snubfigures Config 3 3 3 3 2 3 3 3 3 3 3 3 3 3 4 3 3 3 3 5 3 3 3 3 6 3 3 3 3 7 3 3 3 3 8 3 3 3 3 Gyrofigures Config V3 3 3 3 2 V3 3 3 3 3 V3 3 3 3 4 V3 3 3 3 5 V3 3 3 3 6 V3 3 3 3 7 V3 3 3 3 8 V3 3 3 3 The pentagonal icositetrahedron is second in a series of dual snub polyhedra and tilings with face configuration V3 3 4 3 n 4n2 symmetry mutations of snub tilings 3 3 4 3 n vteSymmetry4n2 Spherical Euclidean Compact hyperbolic Paracomp 242 342 442 542 642 742 842 42Snubfigures Config 3 3 4 3 2 3 3 4 3 3 3 3 4 3 4 3 3 4 3 5 3 3 4 3 6 3 3 4 3 7 3 3 4 3 8 3 3 4 3 Gyrofigures Config V3 3 4 3 2 V3 3 4 3 3 V3 3 4 3 4 V3 3 4 3 5 V3 3 4 3 6 V3 3 4 3 7 V3 3 4 3 8 V3 3 4 3 The pentagonal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron Uniform octahedral polyhedraSymmetry 4 3 432 4 3 432 1 4 3 3 3 332 3 4 3 2 4 3 t 4 3 r 4 3 r 31 1 t 3 4 t 31 1 3 4 31 1 rr 4 3 s2 3 4 tr 4 3 sr 4 3 h 4 3 3 3 h2 4 3 t 3 3 s 3 4 s 31 1 or or Duals to uniform polyhedraV43 V3 82 V 3 4 2 V4 62 V34 V3 43 V4 6 8 V34 4 V33 V3 62 V35 References Edit Conway Symmetries of things p 284 Promorphology of Crystals I Crystal Form Zones amp Habit Archived from the original on 2003 08 23 Koca Mehmet Ozdes Koca Nazife Koc Ramazon 2010 Catalan Solids Derived From 3D Root Systems and Quaternions Journal of Mathematical Physics 51 4 arXiv 0908 3272 doi 10 1063 1 3356985 Eric W Weisstein Pentagonal icositetrahedron Catalan solid at MathWorld Williams Robert 1979 The Geometrical Foundation of Natural Structure A Source Book of Design Dover Publications Inc ISBN 0 486 23729 X Section 3 9 Wenninger Magnus 1983 Dual Models Cambridge University Press doi 10 1017 CBO9780511569371 ISBN 978 0 521 54325 5 MR 0730208 The thirteen semiregular convex polyhedra and their duals Page 28 Pentagonal icositetrahedron The Symmetries of Things 2008 John H Conway Heidi Burgiel Chaim Goodman Strauss ISBN 978 1 56881 220 5 1 Chapter 21 Naming the Archimedean and Catalan polyhedra and tilings page 287 pentagonal icosikaitetrahedron External links EditPentagonal Icositetrahedron Interactive Polyhedron Model Retrieved from https en wikipedia org w index php title Pentagonal icositetrahedron amp oldid 1138825781, wikipedia, wiki, book, books, library,
