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| In [[geometry]], the '''Steinmetz solid''' is the solid body generated by the intersection of two or three [[cylinder (geometry)|cylinders]] of equal radius at right angles. It is named after [[Charles Proteus Steinmetz]], though these solids were known long before Steinmetz studied them.
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| If two cylinders are intersected, the overlap is called a '''bicylinder''' or '''mouhefanggai''' ([[Chinese language|Chinese]] for two square umbrellas,<ref>http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=3736</ref> written in Chinese as 牟合方蓋). It can be seen topologically as a square [[hosohedron]]. If three cylinders are intersected, then the overlap is called a '''tricylinder'''.
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| == Bicylinder ==
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| [[Image:Bicylinder Steinmetz solid.gif|right|90px]]
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| === Volume ===
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| [[Archimedes]] and [[Zu Chongzhi]] calculated the volume of a bicylinder in which both cylinders have radius ''r''. It is
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| :<math>\frac{16}{3} r^3</math>
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| [[File:Sphere volume derivation using bicylinder.jpg|thumb|right|Zu Chongzhi's method (similar to [[Cavalieri's principle]]) for calculating a sphere's volume includes calculating the volume of a bicylinder.]] The volume of the two intersecting cylinders can be calculated by subtracting the volume of the overlap (or the bisector in this case) from the volume of the two cylinders added together.
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| Deriving the volume of a bicylinder (white) can be done by packing it in a cube (red). A plane (parallel with the cylinders' axes) intersecting the bicylinder forms a square and its intersection with the cube is a larger square. The difference between the areas of the two squares is the same as 4 small squares (blue). As the plane moves through the solids, these blue squares describe square pyramids with isosceles faces in the corners of the cube; the pyramids have their apexes at the midpoints of the four cube edges. Moving the plane through the whole bicylinder describes a total of 8 pyramids.
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| [[File:Bicylinder and cube sections related by pyramids.png|thumb|left|Relationship of the area of a bicylinder section with a cube section]]
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| The volume of the cube (red) minus the volume of the eight pyramids (blue) is the volume of the bicylinder (white). The [[Pyramid_(geometry)#Volume|volume of the 8 pyramids]] is: <math>\textstyle 8 \times \frac{1}{3} r^2 \times r = \frac{8}{3} r^3 </math>, and then we can calculate that the bicylinder volume is <math>\textstyle (2 r)^3 - \frac{8}{3} r^3 = \frac{16}{3} r^3</math>
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| === Surface Area ===
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| The surface area is 16''r''<sup>2</sup>. The ratio of <math>\tfrac{r}{3}</math> between surface area and volume holds more generally for a large family of shapes circumscribed around a sphere, including spheres themselves, cylinders, cubes, and both types of Steinmetz solid (Apostol and Mnatsakanian 2006).
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| The surface of the bicylinder consists of four cylindrical patches, separated by four curves each of which is half of an [[ellipse]]. The four patches and four separating curves all meet at two opposite vertices.
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| === Derived Solids ===
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| A bisected bicylinder is called a '''vault''',<ref>{{cite web | author= Weisstein, Eric W.| title= Steinmetz Solid| url= http://mathworld.wolfram.com/SteinmetzSolid.html| work= MathWorld—A Wolfram Web Resource| publisher= Wolfram Research, Inc.| date= c. 1999–2009| accessdate=2009-06-09}}</ref> and a [[groin vault]] in architecture has this shape.
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| {{br}}
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| == Tricylinder ==
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| [[Image:Tricylinder Steinmetz solid.gif|right|90px]]
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| The tricylinder has fourteen vertices connected by elliptical arcs in a pattern combinatorially equivalent to the [[rhombic dodecahedron]]. Its volume is
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| :<math>(16 - 8\sqrt{2}) r^3 \, </math>
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| and its surface area is
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| :<math>3(16 - 8\sqrt{2}) r^2. \, </math> | |
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| == References ==
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| {{reflist}}
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| == Bibliography ==
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| *{{cite journal
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| | doi = 10.2307/27641977
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| | author = Apostol, Tom M.; Mnatsakanian, Mamikon A.
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| | title = Solids circumscribing spheres
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| | journal = [[American Mathematical Monthly]]
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| | volume = 113
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| | year = 2006
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| | issue = 6
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| | pages = 521–540
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| | url = http://www.mamikon.com/USArticles/CircumSolids.pdf
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| | mr = 2231137
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| | jstor = 27641977}}
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| *{{cite journal
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| | author = Hogendijk, Jan P.
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| | title = The surface area of the bicylinder and Archimedes' Method
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| | journal = Historia Math.
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| | volume = 29
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| | year = 2002
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| | issue = 2
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| | pages = 199–203
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| | mr = 1896975
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| | doi = 10.1006/hmat.2002.2349}}
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| *{{cite journal
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| | author = Moore, M.
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| | title = Symmetrical intersections of right circular cylinders
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| | jstor = 3615957
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| | journal = [[The Mathematical Gazette]]
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| | volume = 58
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| | issue = 405
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| | pages = 181–185
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| | year = 1974
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| | doi = 10.2307/3615957}}
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| == External links ==
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| *[http://sketchup.google.com/3dwarehouse/details?mid=b170ed71b798d500a6b083aba0e2034a A 3D model of Steinmetz solid in Google 3D Warehouse]
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| *{{mathworld | urlname = SteinmetzSolid | title = Steinmetz Solid}}
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| *[http://local.wasp.uwa.edu.au/~pbourke/geometry/cylinders Intersecting cylinders] (Paul Bourke, 2003)
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| {{DEFAULTSORT:Steinmetz Solid}}
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| [[Category:Euclidean solid geometry]]
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