Steinmetz solid

Steinmetz solid (intersection of two cylinders)

A 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 mathematician Charles Proteus Steinmetz^[1], who solved the geometric problem of determining the volume of the intersection, though these solids were known long before Steinmetz studied them.

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

Bicylinder

Animated depiction of a bicylinder

Volume

The bicylinder, known as a mouhefanggai in Chinese (for two square umbrellas,^[2] written as 牟合方蓋) is topologically equivalent to the square hosohedron. Archimedes and Zu Chongzhi calculated the volume of a bicylinder in which both cylinders have radius r. It is

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

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.

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.

Relationship of the area of a bicylinder section with a cube section

The volume of the cube (red) minus the volume of the eight pyramids (blue) is the volume of the bicylinder (white). The volume of the 8 pyramids is: $\textstyle 8\times {\frac {1}{3}}r^{2}\times r={\frac {8}{3}}r^{3}$ , and then we can calculate that the bicylinder volume is $\textstyle (2r)^{3}-{\frac {8}{3}}r^{3}={\frac {16}{3}}r^{3}$

Surface area

The surface area is 16r². The ratio of ${\tfrac {r}{3}}$ between volume and surface area 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.

Derived solids

A bisected bicylinder is called a vault,^[3] and a cloister vault in architecture has this shape.

Tricylinder

Animated depiction of a 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}.\,

More cylinders

With four cylinders, with axes connecting the vertices of a tetrahedron to the corresponding points on the other side of the solid, the volume is

12(2{\sqrt {2}}-{\sqrt {6}})r^{3}\,

With six cylinders, with axes parallel to the diagonals of the faces of a cube, the volume is:

{\frac {16}{3}}(3+2{\sqrt {3}}-4{\sqrt {2}})r^{3}\,

References

↑ Howard Eves, Slicing it thin, in: David Klarner, The mathematical Gardner, Wadsworth International 1981, S. 111
↑ http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=3736%5Bpermanent+dead+link%5D
↑ Weisstein, Eric W. (c. 1999–2009). "Steinmetz Solid". MathWorld—A Wolfram Web Resource. Wolfram Research, Inc. Retrieved 2009-06-09. Check date values in: |date= (help)

Bibliography

Apostol, Tom M.; Mnatsakanian, Mamikon A. (2006). "Solids circumscribing spheres" (PDF). American Mathematical Monthly. 113 (6): 521–540. doi:10.2307/27641977. JSTOR 27641977. MR 2231137. Archived from the original (PDF) on 2012-02-07. Retrieved 2007-03-25.
Jan Hogendijk (2002). "The surface area of the bicylinder and Archimedes' Method". Historia Math. 29 (2): 199–203. doi:10.1006/hmat.2002.2349. MR 1896975.
Moore, M. (1974). "Symmetrical intersections of right circular cylinders". The Mathematical Gazette. 58 (405): 181–185. doi:10.2307/3615957. JSTOR 3615957.

External links

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[1] Howard Eves, Slicing it thin, in: David Klarner, The mathematical Gardner, Wadsworth International 1981, S. 111

[2] ttp://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=3736%5Bpermanent+dead+link%5D

[3] Weisstein, Eric W. (c. 1999–2009). "Steinmetz Solid". MathWorld—A Wolfram Web Resource. Wolfram Research, Inc. Retrieved 2009-06-09. Check date values in: |date= (help)