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| In [[geometry]], the '''Hill tetrahedra''' are a family of [[Space-filling polyhedron|space-filling]] [[tetrahedron|tetrahedra]]. They were discovered in 1896 by [[Micaiah John Muller Hill|M. J. M. Hill]], a professor of [[mathematics]] at the [[University College London]], who showed that they are [[Hilbert's third problem|scissor-congruent]] to a [[cube]].
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| == Construction ==
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| For every <math>\alpha \in (0,2\pi/3)</math>, let <math>v_1,v_2,v_3 \in \Bbb R^3</math>
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| be three unit vectors with angle <math>\alpha</math> between every two of them.
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| Define the ''Hill tetrahedron'' <math>Q(\alpha)</math> as follows:
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| :<math> Q(\alpha) \, = \, \{c_1 v_1+c_2 v_2+c_3 v_3 \mid
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| 0 \le c_1 \le c_2 \le c_3 \le 1\}.
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| </math>
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| A special case <math>Q=Q(\pi/2)</math> is the tetrahedron having all sides right triangles with sides 1, <math>\sqrt{2}</math> and <math>\sqrt{3}</math>. [[Ludwig Schläfli]] studied <math>Q</math> as a special case of the [[Schläfli orthoscheme|orthoscheme]], and [[H. S. M. Coxeter]] called it the characteristic tetrahedron of the cubic spacefilling.
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| == Properties ==
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| * A cube can be tiled with 6 copies of <math>Q</math>.
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| * Every <math> Q(\alpha)</math> can be [[Dissection (geometry)|dissected]] into three polytopes which can be reassembled into a [[prism (geometry)|prism]].
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| == Generalizations ==
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| In 1951 [[Hugo Hadwiger]] found the following ''n''-dimensional generalization of Hill tetrahedra:
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| :<math> Q(w) \, = \, \{c_1 v_1+\cdots +c_n v_n \mid | |
| 0 \le c_1 \le \cdots \le c_n \le 1\},
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| </math>
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| where vectors <math>v_1,\ldots,v_n</math> satisfy <math>(v_i,v_j) = w</math> for all <math>1\le i< j\le n</math>, and where <math>-1/(n-1)< w < 1</math>. Hadwiger showed that all such [[simplex|simplices]] are scissor congruent to a [[hypercube]].
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| == See also==
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| * [[Schläfli orthoscheme]]
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| == References ==
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| * M. J. M. Hill, Determination of the volumes of certain species of tetrahedra without employment of the method of limits, ''Proc. London Math. Soc.'', 27 (1895–1896), 39–53.
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| * [[Hugo Hadwiger|H. Hadwiger]], Hillsche Hypertetraeder, ''Gazeta Matemática (Lisboa)'', 12 (No. 50, 1951), 47–48.
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| * [[H.S.M. Coxeter]], [http://matwbn.icm.edu.pl/ksiazki/aa/aa18/aa18132.pdf Frieze patterns], ''Acta Arithmetica'' '''18''' (1971), 297–310.
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| * E. Hertel, Zwei Kennzeichnungen der Hillschen Tetraeder, ''J. Geom.'' 71 (2001), no. 1–2, 68–77.
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| * Greg N. Frederickson, ''Dissections: Plane and Fancy'', Cambridge University Press, 2003.
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| * [[Neil Sloane|N.J.A. Sloane]], V.A. Vaishampayan, ''[http://arxiv.org/pdf/0710.3857 Generalizations of Schobi’s Tetrahedral Dissection]'', [[arXiv]]:0710.3857.
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| == External links ==
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| * [http://demonstrations.wolfram.com/ThreePieceDissectionOfAHillTetrahedronIntoATriangularPrism Three piece dissection of a Hill tetrahedron into a triangular prism]
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| [[Category:Polyhedra]]
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| [[Category:Space-filling polyhedra]]
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