Steinmetz solid
Encyclopedia
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.
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.
The surface area is 16r2. The ratio of
between surface area and volume 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.
A bisected bicylinder is called a vault, and a groin vault
in architecture has this shape.
The tricylinder has fourteen vertices connected by elliptical arcs in a pattern combinatorially equivalent to the rhombic dodecahedron
. Its volume is
and its surface area is
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the Steinmetz solid is the solid body generated by the intersection of two or three cylinders
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
of equal radius at right angles. It is named after Charles Proteus Steinmetz
Charles Proteus Steinmetz
Charles Proteus Steinmetz was a German-American mathematician and electrical engineer. He fostered the development of alternating current that made possible the expansion of the electric power industry in the United States, formulating mathematical theories for engineers...
, though these solids were known long before Steinmetz studied them.
If two cylinders are intersected, the overlap is called a bicylinder or mouhefanggai (Chinese
Chinese language
The Chinese language is a language or language family consisting of varieties which are mutually intelligible to varying degrees. Originally the indigenous languages spoken by the Han Chinese in China, it forms one of the branches of Sino-Tibetan family of languages...
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.
Bicylinder
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
and Zu Chongzhi
Zu Chongzhi
Zu Chongzhi , courtesy name Wenyuan , was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.-Life and works:...
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.
The surface area is 16r2. The ratio of
between surface area and volume 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
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
. The four patches and four separating curves all meet at two opposite vertices.
A bisected bicylinder is called a vault, and a groin vault
Groin vault
A groin vault or groined vault is produced by the intersection at right angles of two barrel vaults. The word groin refers to the edge between the intersecting vaults; cf. ribbed vault. Sometimes the arches of groin vaults are pointed instead of round...
in architecture has this shape.
Tricylinder
Rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...
. Its volume is
and its surface area is
External links
- A 3D model of Steinmetz solid in Google 3D Warehouse
- Intersecting cylinders (Paul Bourke, 2003)