In geometry, the snub square antiprism is the Johnson solid that can be constructed by snubbing the square antiprism. It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids, although it is a relative of the icosahedron that has fourfold symmetry instead of threefold.
The snub is the process of constructing polyhedra by cutting loose the edge's faces, twisting them, and then attaching equilateral triangles to their edges.[1] As the name suggested, the snub square antiprism is constructed by snubbing the square antiprism,[2] and this construction results in 24 equilateral triangles and 2 squares as its faces.[3] The Johnson solids are the convex polyhedra whose faces are regular, and the snub square antiprism is one of them, enumerated as , the 85th Johnson solid.[4]
Then, Cartesian coordinates of a snub square antiprism with edge length 2 are given by the union of the orbits of the points
under the action of the group generated by a rotation around the -axis by 90° and by a rotation by 180° around a straight line perpendicular to the -axis and making an angle of 22.5° with the -axis.[5] It has the three-dimensional symmetry of dihedral group of order 8.[2]
The surface area and volume of a snub square antiprism with edge length can be calculated as:[3]
Referencesedit
^Holme, Audun (2010). Geometry: Our Cultural Heritage. Springer. p. 99. doi:10.1007/978-3-642-14441-7. ISBN978-3-642-14441-7.
^ abBerman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
^Francis, Darryl (2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
^Timofeenko, A. V. (2009). "The non-Platonic and non-Archimedean noncomposite polyhedra". Journal of Mathematical Science. 162 (5): 725. doi:10.1007/s10958-009-9655-0. S2CID 120114341.
snub, square, antiprism, geometry, snub, square, antiprism, johnson, solid, that, constructed, snubbing, square, antiprism, elementary, johnson, solids, that, arise, from, paste, manipulations, platonic, archimedean, solids, although, relative, icosahedron, th. In geometry the snub square antiprism is the Johnson solid that can be constructed by snubbing the square antiprism It is one of the elementary Johnson solids that do not arise from cut and paste manipulations of the Platonic and Archimedean solids although it is a relative of the icosahedron that has fourfold symmetry instead of threefold Snub square antiprismTypeJohnsonJ84 J85 J86Faces24 triangles2 squaresEdges40Vertices16Vertex configuration8 35 8 34 4 displaystyle 8 times 3 5 8 times 3 4 times 4 Symmetry groupD4h displaystyle D 4h PropertiesconvexNet3D model of a snub square antiprismConstruction and properties editThe snub is the process of constructing polyhedra by cutting loose the edge s faces twisting them and then attaching equilateral triangles to their edges 1 As the name suggested the snub square antiprism is constructed by snubbing the square antiprism 2 and this construction results in 24 equilateral triangles and 2 squares as its faces 3 The Johnson solids are the convex polyhedra whose faces are regular and the snub square antiprism is one of them enumerated as J85 displaystyle J 85 nbsp the 85th Johnson solid 4 Let k 0 82354 displaystyle k approx 0 82354 nbsp be the positive root of the cubic polynomial9x3 33 5 2 x2 3 5 22 x 173 76 displaystyle 9x 3 3 sqrt 3 left 5 sqrt 2 right x 2 3 left 5 2 sqrt 2 right x 17 sqrt 3 7 sqrt 6 nbsp Furthermore let h 1 35374 displaystyle h approx 1 35374 nbsp be defined by h 2 8 23k 3 2 2 k243 3k2 displaystyle h frac sqrt 2 8 2 sqrt 3 k 3 left 2 sqrt 2 right k 2 4 sqrt 3 3k 2 nbsp Then Cartesian coordinates of a snub square antiprism with edge length 2 are given by the union of the orbits of the points 1 1 h 1 3k 0 h 3 3k2 displaystyle 1 1 h left 1 sqrt 3 k 0 h sqrt 3 3k 2 right nbsp under the action of the group generated by a rotation around the z displaystyle z nbsp axis by 90 and by a rotation by 180 around a straight line perpendicular to the z displaystyle z nbsp axis and making an angle of 22 5 with the x displaystyle x nbsp axis 5 It has the three dimensional symmetry of dihedral group D4h displaystyle D 4h nbsp of order 8 2 The surface area and volume of a snub square antiprism with edge length a displaystyle a nbsp can be calculated as 3 A 2 63 a2 12 392a2 V 3 602a3 displaystyle begin aligned A left 2 6 sqrt 3 right a 2 amp approx 12 392a 2 V amp approx 3 602a 3 end aligned nbsp References edit Holme Audun 2010 Geometry Our Cultural Heritage Springer p 99 doi 10 1007 978 3 642 14441 7 ISBN 978 3 642 14441 7 a b Johnson Norman W 1966 Convex polyhedra with regular faces Canadian Journal of Mathematics 18 169 200 doi 10 4153 cjm 1966 021 8 MR 0185507 Zbl 0132 14603 a b Berman Martin 1971 Regular faced convex polyhedra Journal of the Franklin Institute 291 5 329 352 doi 10 1016 0016 0032 71 90071 8 MR 0290245 Francis Darryl 2013 Johnson solids amp their acronyms Word Ways 46 3 177 Timofeenko A V 2009 The non Platonic and non Archimedean noncomposite polyhedra Journal of Mathematical Science 162 5 725 doi 10 1007 s10958 009 9655 0 S2CID 120114341 External links editWeisstein Eric W Snub square antiprism Johnson solid at MathWorld Retrieved from https en wikipedia org w index php title Snub square antiprism amp oldid 1207526979 Snub antiprisms, wikipedia, wiki, book, books, library,
