This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron form an embedding of the complete graphK7 onto a topological torus. Of the 35 possible triangles from vertices of the polyhedron, only 14 are faces.
Interactive orthographic projection of a Csaszar polyhedron. In the SVG image, move the mouse left and right to rotate the model.
The tetrahedron and the Császár polyhedron are the only two known polyhedra (having a manifold boundary) without any diagonals: every two vertices of the polygon are connected by an edge, so there is no line segment between two vertices that does not lie on the polyhedron boundary. That is, the vertices and edges of the Császár polyhedron form a complete graph.
The combinatorial description of this polyhedron has been described earlier by Möbius.[1] Three additional different polyhedra of this type can be found in a paper by Bokowski, J. and Eggert, A.[2]
If the boundary of a polyhedron with v vertices forms a surface with h holes, in such a way that every pair of vertices is connected by an edge, it follows by some manipulation of the Euler characteristic that
This equation is satisfied for the tetrahedron with h = 0 and v = 4, and for the Császár polyhedron with h = 1 and v = 7. The next possible solution, h = 6 and v = 12, would correspond to a polyhedron with 44 faces and 66 edges, but it is not realizable as a polyhedron. It is not known whether such a polyhedron exists with a higher genus (Ziegler 2008).
More generally, this equation can be satisfied only when v is congruent to 0, 3, 4, or 7 modulo 12 (Lutz 2001).
History and related polyhedra
The Császár polyhedron is named after Hungarian topologist Ákos Császár, who discovered it in 1949. The dual to the Császár polyhedron, the Szilassi polyhedron, was discovered later, in 1977, by Lajos Szilassi; it has 14 vertices, 21 edges, and seven hexagonal faces, each sharing an edge with every other face. Like the Császár polyhedron, the Szilassi polyhedron has the topology of a torus.
There are other known polyhedra such as the Schönhardt polyhedron for which there are no interior diagonals (that is, all diagonals are outside the polyhedron) as well as non-manifold surfaces with no diagonals (Szabó 1984, 2009).
References
^Möbius, A.F. (1967) [1886]. "Mittheilungen aus Möbius' Nachlass: I". In Klein, Felix (ed.). Zur Theorie der Polyëder und der Elementarverwandtschaft. Gesammelte Werke. Vol. II. p. 552. OCLC 904788205.
^Bokowski, J.; Eggert, A. (1991). "All Realizations of Möbius' Torus with 7 Vertices". Structural Topology. 17: 59–76. CiteSeerX10.1.1.970.6870. hdl:2099/1067.
Császár, A. (1949), (PDF), Acta Sci. Math. Szeged, 13: 140–142, archived from the original (PDF) on 2017-09-18.
Gardner, Martin (1988), Time Travel and Other Mathematical Bewilderments, W. H. Freeman and Company, pp. 139–152, Bibcode:1988ttom.book.....G, ISBN0-7167-1924-X
Gardner, Martin (1992), Fractal Music, Hypercards and More: Mathematical Recreations from Scientific American, W. H. Freeman and Company, pp. 118–120, ISBN0-7167-2188-0
Lutz, Frank H. (2001), "Császár's Torus", Electronic Geometry Models: 2001.02.069.
Szabó, Sándor (1984), "Polyhedra without diagonals", Periodica Mathematica Hungarica, 15 (1): 41–49, doi:10.1007/BF02109370, S2CID 189834222.
Szabó, Sándor (2009), "Polyhedra without diagonals II", Periodica Mathematica Hungarica, 58 (2): 181–187, doi:10.1007/s10998-009-10181-x, S2CID 45731540.
Ziegler, Günter M. (2008), "Polyhedral Surfaces of High Genus", in Bobenko, A. I.; Schröder, P.; Sullivan, J. M.; Ziegler, G. M. (eds.), Discrete Differential Geometry, Oberwolfach Seminars, vol. 38, Springer-Verlag, pp. 191–213, arXiv:math.MG/0412093, doi:10.1007/978-3-7643-8621-4_10, ISBN978-3-7643-8620-7, S2CID 15911143.
Császár’s polyhedron in virtual reality in NeoTrie VR.
January 11, 2023
császár, polyhedron, geometry, hungarian, ʃaːsaːr, nonconvex, toroidal, polyhedron, with, triangular, faces, source, source, source, source, source, source, source, source, source, source, animation, being, rotated, unfoldedtypetoroidal, polyhedronfaces14, tri. In geometry the Csaszar polyhedron Hungarian ˈt ʃaːsaːr is a nonconvex toroidal polyhedron with 14 triangular faces Csaszar polyhedron source source source source source source source source source source An animation of the Csaszar polyhedron being rotated and unfoldedTypeToroidal polyhedronFaces14 trianglesEdges21Vertices7Euler char 0 Genus 1 Vertex configuration3 3 3 3 3 3Symmetry groupC1 11 Dual polyhedronSzilassi polyhedronPropertiesNon convexThis polyhedron has no diagonals every pair of vertices is connected by an edge The seven vertices and 21 edges of the Csaszar polyhedron form an embedding of the complete graph K7 onto a topological torus Of the 35 possible triangles from vertices of the polyhedron only 14 are faces Contents 1 Complete graph 2 History and related polyhedra 3 References 4 External linksComplete graph Edit STL 3D model of a Csaszar polyhedron Interactive orthographic projection of a Csaszar polyhedron In the SVG image move the mouse left and right to rotate the model The tetrahedron and the Csaszar polyhedron are the only two known polyhedra having a manifold boundary without any diagonals every two vertices of the polygon are connected by an edge so there is no line segment between two vertices that does not lie on the polyhedron boundary That is the vertices and edges of the Csaszar polyhedron form a complete graph The combinatorial description of this polyhedron has been described earlier by Mobius 1 Three additional different polyhedra of this type can be found in a paper by Bokowski J and Eggert A 2 If the boundary of a polyhedron with v vertices forms a surface with h holes in such a way that every pair of vertices is connected by an edge it follows by some manipulation of the Euler characteristic thath v 3 v 4 12 displaystyle h frac v 3 v 4 12 This equation is satisfied for the tetrahedron with h 0 and v 4 and for the Csaszar polyhedron with h 1 and v 7 The next possible solution h 6 and v 12 would correspond to a polyhedron with 44 faces and 66 edges but it is not realizable as a polyhedron It is not known whether such a polyhedron exists with a higher genus Ziegler 2008 More generally this equation can be satisfied only when v is congruent to 0 3 4 or 7 modulo 12 Lutz 2001 History and related polyhedra EditThe Csaszar polyhedron is named after Hungarian topologist Akos Csaszar who discovered it in 1949 The dual to the Csaszar polyhedron the Szilassi polyhedron was discovered later in 1977 by Lajos Szilassi it has 14 vertices 21 edges and seven hexagonal faces each sharing an edge with every other face Like the Csaszar polyhedron the Szilassi polyhedron has the topology of a torus There are other known polyhedra such as the Schonhardt polyhedron for which there are no interior diagonals that is all diagonals are outside the polyhedron as well as non manifold surfaces with no diagonals Szabo 1984 2009 References Edit Mobius A F 1967 1886 Mittheilungen aus Mobius Nachlass I In Klein Felix ed Zur Theorie der Polyeder und der Elementarverwandtschaft Gesammelte Werke Vol II p 552 OCLC 904788205 Bokowski J Eggert A 1991 All Realizations of Mobius Torus with 7 Vertices Structural Topology 17 59 76 CiteSeerX 10 1 1 970 6870 hdl 2099 1067 Csaszar A 1949 A polyhedron without diagonals PDF Acta Sci Math Szeged 13 140 142 archived from the original PDF on 2017 09 18 Gardner Martin 1988 Time Travel and Other Mathematical Bewilderments W H Freeman and Company pp 139 152 Bibcode 1988ttom book G ISBN 0 7167 1924 X Gardner Martin 1992 Fractal Music Hypercards and More Mathematical Recreations from Scientific American W H Freeman and Company pp 118 120 ISBN 0 7167 2188 0 Lutz Frank H 2001 Csaszar s Torus Electronic Geometry Models 2001 02 069 Szabo Sandor 1984 Polyhedra without diagonals Periodica Mathematica Hungarica 15 1 41 49 doi 10 1007 BF02109370 S2CID 189834222 Szabo Sandor 2009 Polyhedra without diagonals II Periodica Mathematica Hungarica 58 2 181 187 doi 10 1007 s10998 009 10181 x S2CID 45731540 Ziegler Gunter M 2008 Polyhedral Surfaces of High Genus in Bobenko A I Schroder P Sullivan J M Ziegler G M eds Discrete Differential Geometry Oberwolfach Seminars vol 38 Springer Verlag pp 191 213 arXiv math MG 0412093 doi 10 1007 978 3 7643 8621 4 10 ISBN 978 3 7643 8620 7 S2CID 15911143 External links EditWeisstein Eric W Csaszar Polyhedron MathWorld Csaszar s polyhedron in virtual reality in NeoTrie VR Retrieved from https en wikipedia org w index php title Csaszar polyhedron amp oldid 1113730873, wikipedia, wiki, book, books, library,
