08-01-2026, 09:28 PM
Steinmetz solid
Summary
A Steinmetz solid is a three-dimensional geometric shape formed by the intersection of two or more cylinders of equal radius whose axes meet at right angles. The most common case, called a bicylinder, is created by two perpendicular cylinders and has the remarkably simple volume $(V=\frac{16}{3}r^3)$ and surface area $(A=16r^2)$.
A tricylinder, formed by three mutually perpendicular cylinders, has a more intricate shape and a volume of $(V=8(2-\sqrt{2})r^3)$. Although the solid is named after the American-German mathematician and engineer Charles Proteus Steinmetz,
similar results had already been discovered by ancient mathematicians such as Archimedes, Zu Chongzhi, and Piero della Francesca. Steinmetz solids are celebrated in geometry for their elegant symmetry, surprising mathematical properties, and applications in engineering, architecture, and computer graphics.
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