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| [[Image:Polyhedral schlegel diagrams.svg|thumb|Examples colored by the number of sides on each face. Yellow [[triangle]]s, red [[Square (geometry)|squares]], and green [[pentagon]]s.]]
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| [[Image:Hypercube.svg|thumb|A [[tesseract]] projected into 3-space as a Schlegel diagram. There are 8 cubic cells visible, one in the center, one below each of the six exterior faces, and the last one is ''inside-out'' representing the space outside the cubic boundary.]] | |
| In [[geometry]], a '''Schlegel diagram''' is a projection of a [[polytope]] from <math>R^d</math> into <math>R^{d-1}</math> through a point beyond one of its facets. The resulting entity is a [[polytopal subdivision]] of the facet in <math>R^{d-1}</math> that is combinatorially equivalent to the original polytope. In 1886 [[Victor Schlegel]] introduced this tool for studying combinatorial and topological properties of polytopes. In dimensions 3 and 4, a Schlegel diagram is a projection of a [[polyhedron]] into a plane figure and a projection of a [[polychoron]] to [[3-space]], respectively. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes.
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| ==Construction== | |
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| A Schlegel diagram can be constructed by a [[perspective projection]] viewed from a point outside of the polytope, above the center of a [[Facet (mathematics)|facet]]. All vertices and edges of the polytope are projected onto a hyperplane of that facet. If the polytope is convex, a point near the facet will exist which maps the facet outside, and all other facets inside, so no edges need to cross in the projection.
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| == Examples ==
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| {| class=wikitable width=400
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| ![[Dodecahedron]]
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| ![[120-cell|Dodecaplex]]
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| |-
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| |valign=top|[[Image:Dodecahedron schlegel diagram.png|200px]]<BR>12 pentagon faces in the plane
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| |valign=top|[[Image:Schlegel wireframe 120-cell.png|200px]]<BR>120 dodecahedral cells in 3-space
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| |}
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| == See also ==
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| * [[Net (polyhedron)]] – A different approach for visualization by lowering the dimension of a [[polytope]] is to build a net, disconnecting [[facet]]s, and ''unfolding'' until the facets can exist on a single [[hyperplane]]. This maintains the geometric scale and shape, but makes the topological connections harder to see.
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| == References ==
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| * [[Victor Schlegel]] (1883) ''Theorie der homogen zusammengesetzten Raumgebilde'', Nova Acta, Ksl. Leop.-Carol. Deutsche Akademie der Naturforscher, Band XLIV, Nr. 4, Druck von E. Blochmann & Sohn in Dresden. [http://www.citr.auckland.ac.nz/dgt/Publications.php?id=544]
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| * Victor Schlegel (1886) ''Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper'', Waren.
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| * [[Coxeter|Coxeter, H.S.M.]]; ''[[Regular Polytopes (book)|Regular Polytopes]]'', (Methuen and Co., 1948). (p. 242)
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| ** ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
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| * {{Citation | last=Grünbaum | first=Branko | authorlink=Branko Grünbaum| title = Convex polytopes | location=New York & London | publisher=[[Springer-Verlag]] | year=2003 | isbn=0-387-00424-6 | edition=2nd | editor1-first=Volker | editor1-last=Kaibel | editor2-first=Victor | editor2-last=Klee | editor2-link=Victor Klee | editor3-first=Günter M. | editor3-last =Ziegler | editor3-link = Günter M. Ziegler}}.
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| == External links ==
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| {{Commons category|Schlegel diagrams}}
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| * {{mathworld | urlname = SchlegelGraph | title = Schlegel graph}}
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| ** {{mathworld | urlname = Skeleton | title = Skeleton}}
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| * [http://www.georgehart.com/hyperspace/hart-120-cell.html George W. Hart: 4D Polytope Projection Models by 3D Printing]
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| * [http://www.nrich.maths.org/public/viewer.php?obj_id=897 Nrich maths – for the teenager. Also useful for teachers.]
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| [[Category:Polytopes]]
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