The Steinmetz solid
The Steinmetz solid (mouhefanggai or bicylinder) is the solid common to two (or three) right circular cylinders of equal radius intersecting at right angle is called the Steinmetz solid where each of the curves of the intersection of two cylinders is an ellipse.
Mouhefanggai in Chinese language means “two square umbrellas”.Three intersecting cylinders are called a tricylinder. Half of a bicylinder is called a vault.The Steinmetz solid named after renowned mathematician Charles Proteus Steinmetz who solved the problem of determining the volume of the intersection
Charles Proteus Steinmetz ( April 9, 1865 – October 26, 1923) was a German-born American mathematician and electrical engineer and professor. He formulated mathematical theories for engineers. He made ground-breaking discoveries in the understanding of hysteresis that enabled engineers to design better electromagnetic equipment. He is a genius in both mathematics and electronics, and earned him the nicknames “Forger of Thunderbolts” and “The Wizard of Schenectady”. Steinmetz’s equation, Steinmetz solids, Steinmetz curves, and Steinmetz equivalent circuit are all named after him, are numerous honours and scholarships,
We use this principle to find the volume of a bicylinder also known as the method of indivisibles. This principle gives the idea that if two objects have the same cross-sectional areas at every height, then the two objects have the same volume.
Imagine two identical cylindrical pipes meeting at right angles. Now think about the shape of the intersection of both pipes. Early Chinese mathematicians called this shape as “ The mouhefanggai”. This can be translated as “two square umbrellas”.Using the cavalieri’s principle,The cross-section of the mouhefanggai is always a square that circumscribes a circle. If the radius of the circle r, we can conclude that the ratio of areas of each cross-section of the bicylinder to the sphere is 4r2:πr2, or 4:π. Now Cavalieri’s principle implies that the ratio of the volumes of the two objects is also 4:π.so a sphere has volume 4/3πr3, then the bicylinder has the volume 16/3r3.
Check your knowledge
When two (or three) right circular cylinders of equal radius intersecting at right angle , the shape formed is called the Steinmetz solid
We use Cavalieri’s principle to find the volume of steinmetz shapes which gives the idea that if two objects have the same cross-sectional areas at every height, then the two objects have the same volume.
Bicylinder(steinmetz /mouhefanggai ) generated by two cylinders with radius r has
Volume =(16/3)r³and the surface area is A=16 r²