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, 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.

The bicylinder 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

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.

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: ${\displaystyle \textstyle 8\times {\frac {1}{3}}r^{2}\times r={\frac {8}{3}}r^{3}}$, and then we can calculate that the bicylinder volume is ${\displaystyle \textstyle (2r)^{3}-{\frac {8}{3}}r^{3}={\frac {16}{3}}r^{3}}$

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