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| [[Image:ReuleauxTetrahedron Animation.gif|frame|right|Animation of a Reuleaux tetrahedron, showing also the tetrahedron from which it is formed.]] | | Greetings! I am [http://www.articlestunner.com/cures-for-the-yeast-infection-suggestions-to-use-now/ at home std test] Marvella and I feel [http://Www.Patient.Co.uk/health/Anogenital-Warts comfortable] when individuals use the complete name. She is a librarian but she's always wanted her personal company. North Dakota is her birth location but she will have to transfer [https://grouponsa.zendesk.com/entries/96807503-Items-To-Know-While-Confronting-Candida at home std testing] 1 working day or another. He is std home test truly fond of performing ceramics but he is struggling to find time for it.<br><br>My web page at home std testing ([http://tcciaarusha-bic.info/groups/simple-stuff-you-could-do-to-remove-candida/ click for source]) |
| [[Image:Reuleaux-tetrahedron-intersection.png|thumb|Four spheres intersect to form a Reuleaux tetrahedron.]]
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| The '''Reuleaux tetrahedron''' is the intersection of four [[sphere]]s of [[radius]] ''s'' centered at the [[Vertex (geometry)|vertices]] of a regular [[tetrahedron]] with side length ''s''. The sphere through each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron. The Reuleaux tetrahedron has the same face structure as a regular tetrahedron, but with curved faces: four vertices, and four curved faces, connected by six circular-arc edges.
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| This shape is defined and named by analogy to the [[Reuleaux triangle]], a two-dimensional [[curve of constant width]]. One can find repeated claims in the mathematical literature that the Reuleaux tetrahedron is analogously a [[surface of constant width]], but it is not true: the two midpoints of opposite edge arcs are separated by a larger distance,
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| :<math>\left(\sqrt3 - \frac{\sqrt2}2 \right) \cdot s\approx 1.0249s.</math>
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| The volume of a Reuleaux tetrahedron is <ref name=Weisstein>{{citation
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| | author = Weisstein, Eric W
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| | authorlink = Eric W. Weisstein
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| | title = Reuleaux Tetrahedron
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| | publisher = MathWorld–A Wolfram Web Resource
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| | year = 2008
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| | url = http://mathworld.wolfram.com/ReuleauxTetrahedron.html}}</ref>
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| :<math>\frac{s^3}{12}(3\sqrt2 - 49\pi + 162\tan^{-1}\sqrt2)\approx 0.422s^3</math>
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| == Meissner bodies ==
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| <!-- linked from [[Meissner body]], etc. -->
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| Meissner and Schilling<ref name=Meissner>{{citation
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| | last1 = Meissner | first1 = Ernst | last2 = Schilling | first2 = Friedrich
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| | title = Drei Gipsmodelle von Flächen konstanter Breite
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| | journal = Z. Math. Phys.
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| | volume = 60
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| | year = 1912
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| | pages = 92–94}}</ref> showed how to modify the Reuleaux tetrahedron to form a [[surface of constant width]], by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. According to which three edge arcs are replaced (three that have a common vertex or three that form a triangle) there result two noncongruent shapes that are sometimes called '''Meissner bodies''' or '''Meissner tetrahedra''' (for interactive pictures and films see Weber).<ref name=weber>{{cite web
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| | author = Weber, Christof
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| | year = 2009
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| | url = http://www.swisseduc.ch/mathematik/geometrie/gleichdick/docs/meissner_en.pdf
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| | title = What does this solid have to do with a ball?}}
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| </ref> Bonnesen and Fenchel<ref name=bonnesen>{{citation
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| | last1 = Bonnesen | first1 = Tommy
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| | authorlink2 = Werner Fenchel | last2 = Fenchel | first2 = Werner
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| | title = Theorie der konvexen Körper
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| | publisher = Springer-Verlag
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| | year = 1934
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| | pages = 127–139}}</ref> conjectured that Meissner tetrahedra are the minimum-volume three-dimensional shapes of constant width, a conjecture which is still open.<ref>{{citation
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| | last1 = Kawohl | first1 = Bernd | last2 = Weber | first2 = Christof
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| | title = Meissner's Mysterious Bodies
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| | journal = [[Mathematical Intelligencer]]
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| | volume = 33
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| | issue = 3
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| | year = 2011
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| | pages = 94–101
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| | doi = 10.1007/s00283-011-9239-y
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| | url = http://www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/pub100.pdf}}</ref> In connection with this problem, Campi, Colesanti and Gronchi<ref name=campi>{{citation
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| | last1 = Campi | first1 = Stefano
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| | last2 = Colesanti | first2 = Andrea
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| | last3 = Gronchi | first3 = Paolo
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| | contribution = Minimum problems for volumes of convex bodies
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| | title = Partial Differential Equations and Applications: Collected Papers in Honor of Carlo Pucci
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| | publisher = Lecture Notes in Pure and Applied Mathematics, no. 177, Marcel Dekker
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| | year = 1996
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| | pages = 43–55}}</ref> showed that the minimum volume surface of revolution with constant width is the surface of revolution of a Reuleaux triangle through one of its symmetry axes.
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| == Etymology ==
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| The term derives from [[Franz Reuleaux]], a 19th-century German engineer who did pioneering work on ways that machines translate one type of motion into another.
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| == References ==
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| {{reflist}}
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| == External links ==
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| * {{cite web
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| | author = Lachand-Robert, Thomas; Oudet, Édouard
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| | title = Spheroforms
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| | url = http://www.lama.univ-savoie.fr/~lachand/Spheroforms.html}}
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| * {{cite web
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| | author = Weber, Christof
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| | title = Bodies of Constant Width
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| | url = http://www.swisseduc.ch/mathematik/geometrie/gleichdick/index-en.html}} There are also films and even [http://www.swisseduc.ch/mathematik/geometrie/gleichdick/meissner-en.html interactive pictures] of both Meissner bodies.
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| [[Category:Euclidean solid geometry]]
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| [[Category:Geometric shapes]]
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