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| {{Infobox polyhedron
| | The name of the writer is Numbers. For years I've been operating as a payroll clerk. For many years he's been living in North Dakota and his family members loves it. The favorite pastime for my kids and me is to perform baseball and I'm trying to make it a profession.<br><br>My homepage ... std home test ([http://www.dhspoteh.co.kr/xe/index.php?document_srl=331302&mid=Sports her comment is here]) |
| |image=Szilassi polyhedron.svg
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| |type=[[Toroidal polyhedron]]
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| |faces=7 [[hexagon]]s
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| |edges=21
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| |vertices=14
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| |euler=0 (Genus 1)
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| |symmetry=C<sub>1</sub>, [ ]<sup>+</sup>, (11)
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| |vertex_config=6.6.6
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| |dual=[[Császár polyhedron]]
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| |properties=Nonconvex
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| }}
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| The '''Szilassi polyhedron''' is a nonconvex [[polyhedron]], topologically a [[torus]], with seven [[hexagon|hexagonal]] faces. | |
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| Each face of this polyhedron shares an edge with each other face. As a result, it requires seven colours to colour each adjacent face, providing the lower bound for the [[seven colour theorem]]. It has an [[Rotational symmetry|axis of 180-degree symmetry]]; three pairs of faces are congruent leaving one unpaired hexagon that has the same rotational symmetry as the polyhedron. The 14 vertices and 21 edges of the Szilassi polyhedron form an embedding of the [[Heawood graph]] onto the surface of a torus.
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| [[File:Szilassi polyhedron.gif|thumb|left]]
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| The [[tetrahedron]] and the Szilassi polyhedron are the only two known polyhedra in which each face shares an edge with each other face. If a polyhedron with ''f'' faces is embedded onto a surface with ''h'' holes, in such a way that each face shares an edge with each other face, it follows by some manipulation of the [[Euler characteristic]] that
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| :<math>h = \frac{(f - 4)(f - 3)}{12}.</math>
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| This equation is satisfied for the tetrahedron with ''h'' = 0 and ''f'' = 4, and for the Szilassi polyhedron with ''h'' = 1 and ''f'' = 7. The next possible solution, ''h'' = 6 and ''f'' = 12, would correspond to a polyhedron with 44 vertices and 66 edges, but it is not known whether such a polyhedron exists. More generally this equation can be satisfied precisely when ''f'' is congruent to 0, 3, 4, or 7 modulo 12.
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| The Szilassi polyhedron is named after Hungarian mathematician Lajos Szilassi, who discovered it in 1977. The [[dual polyhedron|dual]] to the Szilassi polyhedron, the [[Császár polyhedron]], was discovered earlier by {{harvs|first=Ákos|last=Császár|authorlink=Ákos Császár|year=1949|txt}}; it has seven vertices, 21 edges connecting every pair of vertices, and 14 triangular faces. Like the Szilassi polyhedron, the Császár polyhedron has the topology of a torus.
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| ==References==
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| *{{citation
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| | last = Császár | first = Ákos | author-link = Ákos Császár
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| | journal = Acta Sci. Math. Szeged
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| | pages = 140–142
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| | title = A polyhedron without diagonals
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| | volume = 13
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| | year = 1949}}.
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| *{{citation
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| | doi = 10.1038/scientificamerican1178-22
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| | last = Gardner | first = Martin | author-link = Martin Gardner
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| | contribution = In Which a Mathematical Aesthetic is Applied to Modern Minimal Art
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| | journal = [[Scientific American]]
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| | pages = 22–32
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| | title = Mathematical Games
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| | issue = 5
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| | volume = 239
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| | year = 1978}}.
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| *{{citation
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| | last1 = Jungerman | first1 = M.
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| | last2 = Ringel | first2 = Gerhard | author2-link = Gerhard Ringel
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| | doi = 10.1007/BF02414187
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| | issue = 1–2
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| | journal = Acta Mathematica
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| | pages = 121–154
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| | title = Minimal triangulations on orientable surfaces
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| | volume = 145
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| | year = 1980}}.
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| *{{citation
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| | last = Peterson | first = Ivars | author-link = Ivars Peterson
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| | publisher = [[Mathematical Association of America]]
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| | contribution = A polyhedron with a hole
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| | title = MathTrek
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| | url = http://www.maa.org/mathland/mathtrek_01_22_07.html
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| | year = 2007}}.
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| *{{citation
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| | last = Szilassi | first = Lajos
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| | journal = Structural Topology
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| | pages = 69–80
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| | title = Regular toroids
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| | url = http://haydn.upc.es/people/ros/StructuralTopology/ST13/st13-06-a3-ocr.pdf
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| | volume = 13
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| | year = 1986}}.
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| ==External links==
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| *{{citation
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| | last = Ace | first = Tom
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| | title = The Szilassi polyhedron
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| | url = http://www.minortriad.com/szilassi.html}}.
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| *{{MathWorld | urlname=SzilassiPolyhedron | title=Szilassi Polyhedron}}
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| * [http://cutoutfoldup.com/patterns/0927_a4.pdf Szilassi Polyhedron] - Papercraft model at [http://cutoutfoldup.com CutOutFoldUp.com]
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| [[Category:Nonconvex polyhedra]]
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| [[Category:Toroidal polyhedra]]
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The name of the writer is Numbers. For years I've been operating as a payroll clerk. For many years he's been living in North Dakota and his family members loves it. The favorite pastime for my kids and me is to perform baseball and I'm trying to make it a profession.
My homepage ... std home test (her comment is here)