The motivation for some of the early studies of this structure was for its applications in the crystallography of crystal structures formed by rod-shaped molecules.[2]
Shrinking the square cross-sections of the prisms slightly causes the remaining space, consisting of the cubical voids, to become linked up into a single polyhedral set, bounded by axis-parallel faces. Polyhedra constructed in this way from finitely many prisms provide examples of axis-parallel polyhedra with vertices and faces that require pieces when subdivided into convex pieces;[4] they have been called Thurston polyhedra, after William Thurston,[5] who suggested using these shapes for this lower bound application.[4] Like the Schönhardt polyhedron, these polyhedra have no triangulation into tetrahedra unless additional vertices are introduced.[5]
Anduriel Widmark has used the tetrastix and hexastix structures as the basis for artworks made from glass rods, fused to form tangled knots.[6]
Related structuresedit
The space occupied by the union of the prisms can be divided into the prisms of the tetrastix structure in two different ways.[3] If the prisms are divided into unit cubes, offset by half a unit from the integer grid aligned with the prism sides, then these cubes together with the unit cube voids of the tetrastix structure form a tiling of space by cubes, combinatorially equivalent to the Weaire–Phelan structure for tiling space with unit volumes of low surface area. The tetrastix and Weaire–Phelan structures have the same group of symmetries.[7] Although this cube tiling includes some cubes (the ones filling the voids of the tetrastix) that do not meet face-to-face with any other cube, results of Oskar Perron on Keller's conjecture prove that (like the cubes within each prism of the tetrastix) every tiling of three-dimensional space by unit cubes must include an infinite column of cubes that all meet face-to-face.[8]
Similar constructions to the tetrastix are possible with triangular and hexagonal prisms, in four directions,[1] called by Conway et al. "tristix" and hexastix.[3]
See alsoedit
Mucube, a self-complementary structure formed by the union of three sets of axis-parallel infinite square prisms that intersect in cubes
^ abPaterson, Mike; Yao, F. Frances (1989), "Binary partitions with applications to hidden surface removal and solid modelling", in Mehlhorn, Kurt (ed.), Proceedings of the Fifth Annual Symposium on Computational Geometry, Saarbrücken, Germany, June 5-7, 1989(PDF), New York: ACM, pp. 23–32, doi:10.1145/73833.73836
^ abCarrigan, Braxton; Bezdek, András (2012), "Tiling polyhedra with tetrahedra" (PDF), Proceedings of the 24th Canadian Conference on Computational Geometry, CCCG 2012, Charlottetown, Prince Edward Island, Canada, August 8-10, 2012, pp. 217–222
^Widmark, Anduriel (April 2020), "Stixhexaknot: A symmetric cylinder arrangement of knotted glass", Journal of Mathematics and the Arts, 14 (1–2): 167–169, doi:10.1080/17513472.2020.1734517
^Perron, Oskar (1940), "Über lückenlose Ausfüllung des -dimensionalen Raumes durch kongruente Würfel", Mathematische Zeitschrift, 46: 1–26, doi:10.1007/BF01181421, MR 0003041, S2CID 186236462; —— (1940), "Über lückenlose Ausfüllung des -dimensionalen Raumes durch kongruente Würfel. II", Mathematische Zeitschrift, 46: 161–180, doi:10.1007/BF01181436, MR 0006068, S2CID 123877436
January 01, 1970
tetrastix, confused, with, tetrastick, geometry, possible, fill, volume, three, dimensional, euclidean, space, three, sets, infinitely, long, square, prisms, aligned, with, three, coordinate, axes, leaving, cubical, voids, john, horton, conway, heidi, burgiel,. Not to be confused with tetrastick In geometry it is possible to fill 3 4 of the volume of three dimensional Euclidean space by three sets of infinitely long square prisms aligned with the three coordinate axes leaving cubical voids 1 2 John Horton Conway Heidi Burgiel and Chaim Goodman Strauss have named this structure tetrastix 3 Tetrastix arrangement showing 6 sticks in each direction Contents 1 Applications 2 Related structures 3 See also 4 ReferencesApplications editThe motivation for some of the early studies of this structure was for its applications in the crystallography of crystal structures formed by rod shaped molecules 2 Shrinking the square cross sections of the prisms slightly causes the remaining space consisting of the cubical voids to become linked up into a single polyhedral set bounded by axis parallel faces Polyhedra constructed in this way from finitely many prisms provide examples of axis parallel polyhedra with n displaystyle n nbsp vertices and faces that require W n 3 2 textstyle Omega n 3 2 nbsp pieces when subdivided into convex pieces 4 they have been called Thurston polyhedra after William Thurston 5 who suggested using these shapes for this lower bound application 4 Like the Schonhardt polyhedron these polyhedra have no triangulation into tetrahedra unless additional vertices are introduced 5 Anduriel Widmark has used the tetrastix and hexastix structures as the basis for artworks made from glass rods fused to form tangled knots 6 Related structures editThe space occupied by the union of the prisms can be divided into the prisms of the tetrastix structure in two different ways 3 If the prisms are divided into unit cubes offset by half a unit from the integer grid aligned with the prism sides then these cubes together with the unit cube voids of the tetrastix structure form a tiling of space by cubes combinatorially equivalent to the Weaire Phelan structure for tiling space with unit volumes of low surface area The tetrastix and Weaire Phelan structures have the same group of symmetries 7 Although this cube tiling includes some cubes the ones filling the voids of the tetrastix that do not meet face to face with any other cube results of Oskar Perron on Keller s conjecture prove that like the cubes within each prism of the tetrastix every tiling of three dimensional space by unit cubes must include an infinite column of cubes that all meet face to face 8 Similar constructions to the tetrastix are possible with triangular and hexagonal prisms in four directions 1 called by Conway et al tristix and hexastix 3 See also editMucube a self complementary structure formed by the union of three sets of axis parallel infinite square prisms that intersect in cubes Blue phase mode LCD Burr puzzle HexastixReferences edit a b Holden Alan 1971 Shapes Space and Symmetry New York Columbia University Press p 161 reprinted by Dover 1991 a b O Keeffe M Andersson Sten November 1977 Rod packings and crystal chemistry Acta Crystallographica Section A 33 6 914 923 doi 10 1107 s0567739477002228 a b c Conway John H Burgiel Heidi Goodman Strauss Chaim 2008 Polystix The Symmetries of Things Wellesley Massachusetts A K Peters pp 346 348 ISBN 978 1 56881 220 5 MR 2410150 a b Paterson Mike Yao F Frances 1989 Binary partitions with applications to hidden surface removal and solid modelling in Mehlhorn Kurt ed Proceedings of the Fifth Annual Symposium on Computational Geometry Saarbrucken Germany June 5 7 1989 PDF New York ACM pp 23 32 doi 10 1145 73833 73836 a b Carrigan Braxton Bezdek Andras 2012 Tiling polyhedra with tetrahedra PDF Proceedings of the 24th Canadian Conference on Computational Geometry CCCG 2012 Charlottetown Prince Edward Island Canada August 8 10 2012 pp 217 222 Widmark Anduriel April 2020 Stixhexaknot A symmetric cylinder arrangement of knotted glass Journal of Mathematics and the Arts 14 1 2 167 169 doi 10 1080 17513472 2020 1734517 Conway Burgiel amp Goodman Strauss 2008 Understanding the Irish Bubbles p 351 Perron Oskar 1940 Uber luckenlose Ausfullung des n displaystyle n nbsp dimensionalen Raumes durch kongruente Wurfel Mathematische Zeitschrift 46 1 26 doi 10 1007 BF01181421 MR 0003041 S2CID 186236462 1940 Uber luckenlose Ausfullung des n displaystyle n nbsp dimensionalen Raumes durch kongruente Wurfel II Mathematische Zeitschrift 46 161 180 doi 10 1007 BF01181436 MR 0006068 S2CID 123877436 Retrieved from https en wikipedia org w index php title Tetrastix amp oldid 1109391724, wikipedia, wiki, book, books, library,
