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Steinmetz Solid


The Steinmetz solid is the solid common to two (or three) right circular cylinders of equal radii intersecting at right angles is called the Steinmetz solid. Two cylinders intersecting at right angles are called a bicylinder or mouhefanggai (Chinese for "two square umbrellas"), and three intersecting cylinders a tricylinder. Half of a bicylinder is called a vault.

Closed forms exist for the volume of a cylinder-cylinder intersection where the cylinders are of different radii and intersect centrally at arbitrary angle beta (Hubbell 1965).

SteinmetzCylinders2
SteinmetzSolid2

For two cylinders of radius r oriented long the z- and x-axes gives the equations

 x^2+y^2=r^2
(1)
 y^2+z^2=r^2
(2)

which can be solved for x and y gives the parametric equations of the edges of the solid,

x=+/-z
(3)
y=+/-sqrt(r^2-z^2).
(4)

The surface area can be found as intxds, where

ds=sqrt(1+((dy)/(dz))^2)dz
(5)
=r/(sqrt(r^2-z^2))dz.
(6)

Taking the range of integration as a quarter or one face and then multiplying by 16 gives

 S_2=16int_0^r(rz)/(sqrt(r^2-z^2))dz=16r^2.
(7)

The volume common to two cylinders was known to Archimedes (Heath 1953, Gardner 1962) and the Chinese mathematician Tsu Ch'ung-Chih (Kiang 1972), and does not require calculus to derive. Using calculus provides a simple derivation, however. Noting that the solid has a square cross section of side-half-length sqrt(r^2-z^2), the volume is given by

 V_2(r,r)=int_(-r)^r(2sqrt(r^2-z^2))^2dz=(16)/3r^3
(8)

(Moore 1974). The volume can also be found using cylindrical algebraic decomposition, which reduces the inequalities

 {x^2+y^2<1; -L<z<L; y^2+z^2<1; -L<x<L
(9)

to

 {-1<x<1; -sqrt(1-x^2)<y<sqrt(1-x^2); -sqrt(1-y^2)<z<sqrt(1-y^2),
(10)

giving the integral

 V_2(1,1)=int_(-1)^1int_(-sqrt(1-x^2))^(sqrt(1-x^2))int_(-sqrt(1-y^2))^(sqrt(1-y^2))dzdydx=(16)/3.
(11)

If the two right cylinders are of different radii a and b with a>b, then the volume common to them is

 V_2(a,b)=8/3a[(a^2+b^2)E(k)-(a^2-b^2)K(k)],
(12)

where K(k) is the complete elliptic integral of the first kind, E(k) is the complete elliptic integral of the second kind, and k=b/a is the elliptic modulus.

Each of the curves of intersection of two cylinders of radii a and b is sometimes known as a Steinmetz curve.

The volume common to two elliptic cylinders

 (x^2)/(a^2)+(z^2)/(c^2)=1    (y^2)/(b^2)+(z^2)/(c^('2))=1
(13)

with c<c^' is

 V_2(a,c;b,c^')=(8ab)/(3c)[(c^('2)+c^2)E(k)-(c^('2)-c^2)K(k)],
(14)

where k=c/c^' (Bowman 1961, p. 34).

SteinmetzCylinders3
SteinmetzSolid3
SteinmetzSolid3Exploded

For three cylinders of radii r intersecting at right angles, The resulting solid has 12 curved faces. If tangent planes are drawn where the faces meet, the result is a rhombic dodecahedron (Wells 1991). The volume of intersection can be computed in a number of different ways,

V_3(r,r,r)=16r^3int_0^(pi/4)int_0^1ssqrt(1-s^2cos^2t)dsdt
(15)
=(sqrt(2)r)^3+6int_(r/sqrt(2))^r(2sqrt(r^2-z^2))^2dz
(16)
=8(2-sqrt(2))r^3
(17)

(Moore 1974). According to the protagonist Christopher in the novel The Curious Incident of the Dog in the Night-Time, "...People go on holidays to see new things and relax, but it wouldn't make me relaxed and you can see new things by looking at earth under a microscope or drawing the shape of the solid made when 3 circular rods of equal thickness intersect at right angles" (Haddon 2003, p. 178), which is of course precisely the Steinmetz solid formed by three symmetrically placed cylinders.

SteinmetzTetrahedra

Four cylinders can also be placed with axes along the lines joining the vertices of a tetrahedron with the centers on the opposite sides. The resulting solid of intersection has volume

 V_4=12(2sqrt(2)-sqrt(6))
(18)

and 24 curved faces analogous to a cube-octahedron compound (Moore 1974, Wells 1991).

Steinmetz6

Six cylinders can be placed with axes parallel to the face diagonals of a cube. The resulting solid of intersection has volume

 V_4=(16)/3(3+2sqrt(3)-4sqrt(2))
(19)

and 36 curved faces, 24 of which are kite-shaped and 12 of which are rhombic (Moore 1974).


See also

Cylinder, Cylinder-Cylinder Intersection, Elliptic Cylinder, Reuleaux Tetrahedron, Rhombic Dodecahedron, Right Angle, Steinmetz Curve, Trihyperboloid, Vault

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References

Angell, I. O. and Moore, M. "Symmetrical Intersections of Cylinders." Acta Cryst. Sect. A 43, 244-250, 1987.Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, p. 871, 1984.Baumann, E. "Intersection of Cylinders." http://www.baumanneduard.ch/all_e.htm.Bowman, F. Introduction to Elliptic Functions, with Applications. New York: Dover, 1961.Gardner, M. "Mathematical Games: Some Puzzles Based on Checkerboards." Sci. Amer. 207, 151-164, Nov. 1962.Gardner, M. The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: Chicago University Press, pp. 183-185, 1991.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.Haddon, M. The Curious Incident of the Dog in the Night-Time. New York: Vintage, 2003.Heath, T. L. The Method of Archimedes. New York: Dover, 1953.Hubbell, J. H. "Common Volume of Two Intersecting Cylinders." J. Research National Bureau of Standards--C. Engineering and Instrumentation 69C, 139-143, April-June 1965.Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 128, 1948.Kiang, T. "An Old Chinese Way of Finding the Volume of a Sphere." Math. Gaz. 56, 88-91, 1972.Moore, M. "Symmetrical Intersections of Right Circular Cylinders." Math. Gaz. 58, 181-185, 1974.Trott, M. The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, pp. 32-34, 2006. http://www.mathematicaguidebooks.org/.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 118-119, 1991.Wells, D. G. #555 in The Penguin Book of Curious and Interesting Puzzles. London: Penguin Books, 1992.

Cite this as:

Weisstein, Eric W. "Steinmetz Solid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SteinmetzSolid.html

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