Steinmetz solid

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). It can be seen topologically as a square hosohedron. If three cylinders are intersected, then the overlap is called a tricylinder.

Bicylinder

Archimedes and Zu Chongzhi calculated the volume of a bicylinder in which both cylinders have radius r. It is

$\frac{16}{3} r^3.$

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 $\tfrac{r}{3}$ 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.

Tricylinder

The tricylinder has fourteen vertices connected by elliptical arcs in a pattern combinatorially equivalent to the rhombic dodecahedron. Its volume is

$(16 - 8\sqrt{2}) r^3 \,$

and its surface area is

$3(16 - 8\sqrt{2}) r^2. \,$

References

1. ^ http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=3736
2. ^ Weisstein, Eric W. (c. 1999–2009). "Steinmetz Solid". MathWorld—A Wolfram Web Resource. Wolfram Research, Inc.. http://mathworld.wolfram.com/SteinmetzSolid.html. Retrieved 2009-06-09. 

Bibliography

• Apostol, Tom M.; Mnatsakanian, Mamikon A. (2006). "Solids circumscribing spheres". American Mathematical Monthly 113 (6): 521–540. doi:10.2307/27641977. JSTOR 27641977. 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:10.1006/hmat.2002.2349. MR1896975. 
• Moore, M. (1974). "Symmetrical intersections of right circular cylinders". The Mathematical Gazette 58 (405): 181–185. doi:10.2307/3615957. JSTOR 3615957. 

External links

• A 3D model of Steinmetz solid in Google 3D Warehouse
• Weisstein, Eric W., "Steinmetz Solid" from MathWorld.
• Intersecting cylinders (Paul Bourke, 2003)