Steinmetz solid

From Wikipedia, the free encyclopedia

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, then it 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 it is called a tricylinder.

1 Bicylinder
2 Tricylinder
3 References
4 External links

[edit] 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 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 groin vault in architecture has the shape of a bisected bicylinder.

[edit] Tricylinder

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

$(16 - \sqrt{128}) r^3$

and its surface area is

$3(16 - \sqrt{128}) r^2.$

[edit] References

Apostol, Tom M.; Mnatsakanian, Mamikon A. (2006). "Solids circumscribing spheres". American Mathematical Monthly 113 (6): 521–540. MR 2231137.

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. MR 1896975.