Steinmetz solid
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In geometry, the Steinmetz solid is the solid body generated by the intersection of two or three cylinders of equal radius at right angles. It is named after Charles Proteus Steinmetz, though these solids were known long before Steinmetz studied them.
If two cylinders are intersected, the overlap is called a bicylinder or mouhefanggai (Chinese for two square umbrellas,[1] written in Chinese as 牟合方蓋). If three cylinders are intersected, then the overlap is called a tricylinder.
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[edit] Bicylinder
Archimedes and Zu Chongzhi calculated the volume of a bicylinder in which both cylinders have radius r. It is
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.
The surface area is 16r2. The ratio of 
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).
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.
A bisected bicylinder is called a vault,[2] and a groin vault in architecture has this shape.
[edit] Tricylinder
The tricylinder has fourteen vertices connected by elliptical arcs in a pattern combinatorially equivalent to the rhombic dodecahedron. Its volume is
and its surface area is
[edit] In literature
In the 2003 novel The Curious Incident of the Dog in the Night-time, the main character talks about mental exercises he does to stay calm, including "visualizing the shape that is formed by the intersection of three rods."
[edit] References
- ^ http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=3736
- ^ Weisstein, Eric W. (c. 1999-2009). "Steinmetz Solid". MathWorld--A Wolfram Web Resource. Wolfram Research, Inc.. http://mathworld.wolfram.com/SteinmetzSolid.html. Retrieved on 2009-06-09.
[edit] Bibliography
- Apostol, Tom M.; Mnatsakanian, Mamikon A. (2006). "Solids circumscribing spheres". American Mathematical Monthly 113 (6): 521–540. MR2231137. http://www.mamikon.com/USArticles/CircumSolids.pdf.
- Hogendijk, Jan P. (2002). "The surface area of the bicylinder and Archimedes' Method". Historia Math. 29 (2): 199–203. doi:. MR1896975.
- Moore, M. (1974). "Symmetrical intersections of right circular cylinders". The Mathematical Gazette 58: 181–185. doi:.
[edit] External links
- Weistein, Eric W., "Steinmetz Solid" from MathWorld.
- Intersecting cylinders (Paul Bourke, 2003)
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