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| [[Image:Umbilic Torus.png|thumb|right|Computer representation of an Umbilic Torus]]
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| [[File:Petrie Plaza.jpg|thumb|upright|"Eternity" by [[John Robinson (sculptor)|John Robinson]]]]
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| The '''umbilic torus''' or '''umbilic braclet''' is a single-edged 3-dimensional shape. The lone edge goes three times around the ring before returning to the starting point. The shape also has a single external face. A [[cross section (geometry)|cross section]] of the surface forms a [[Deltoid curve|deltoid]].
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| The umbilic torus occurs in the mathematical subject of [[singularity theory]], in particular in the classification of [[umbilical point]]s which are determined by real [[cubic form]]s <math>a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3</math>. The equivalence classes of such cubics form a three dimensional real projective space and the subset of parabolic forms define a surface – the umbilic torus. [[Christopher Zeeman]] named this set the umbilic braclet in 1976.<ref name=port>{{Citation|first=Ian R.|last=Porteous|title=Geometric Differentiation, For the Intelligence of Curves and Surfaces|ISBN=978-0-521-00264-6|pp=350
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| |date=2001|publisher=Cambridge University Press| edition=2nd}}</ref>
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| The torus is defined by the following set of [[parametric equations]].<ref>Larson, Roland E., et al. ''Calculus''. Ed. Charles Hartford. 6th ed. Boston: Houghton Mifflin Company, 1998.</ref>
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| :<math>x = \sin u \left(7+\cos\left({u \over 3} - 2v\right) + 2\cos\left({u \over3} + v\right)\right) </math>
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| :<math>y = \cos u \left(7 + \cos\left({u \over 3} - 2v\right) + 2\cos\left({u \over 3} + v\right)\right)</math>
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| :<math>z = \sin\left({u \over 3} - 2v\right) + 2\sin \left({u \over 3} + v\right)</math>
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| :::<math>\mbox{for }-\pi \le u \le \pi,\quad -\pi \le v \le \pi \,</math>
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| [[John Robinson (sculptor)|John Robinson]] created a sculpture ''Eternity'' based on the shape in 1989, this had a triangular cross-section rather than a deltoid of a true Umbilic bracelet. This appeared on the cover of Geometric Differentiation by [[Ian R. Porteous]].<ref name="port"/>
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| [[Helaman Ferguson]] has created a 27-inch (69 centimeters) bronze sculpture, '''Umbilic Torus''', and it is his most widely known piece of art. In 2010, it was announced that [[James Harris Simons|Jim Simons]] had commissioned an Umbilic Torus sculpture to be constructed outside the Math and Physics buildings at [[Stony Brook University]], in proximity to the [[Simons Center for Geometry and Physics]]. The torus is made out of cast bronze, and is mounted on a stainless steel column. The total weight of the sculpture is 65 tonnes, and has a height of {{convert|28|ft|m}}. The torus has a diameter of {{convert|24|ft|m}}, the same diameter as the granite base. Various mathematical formulas defining the torus are inscribed on the base. Installation was completed in September, 2012. <ref>Helaman Ferguson, "Two Theorems, Two Sculptures, Two Posters", American Mathematical Monthly, Volume 97, Number 7,August-September 1990, pages 589-610.</ref>
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| == See also ==
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| *[[Torus]]
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| *[[Möbius strip]]
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| *[[Mathematics and art]]
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| == References ==
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| <references/>
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| ==External links==
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| * [http://helasculpt.com/gallery/?album=3&gallery=66 Umbilic Torus on Ferguson site]
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| * [http://www.popmath.org.uk/sculpmath/pagesm/fibundle.html Discussion of Robinson's Eternity]
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| [[Category:Sculptures]]
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| [[Category:Mathematics and culture]]
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