Spherinder

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The spherinder can be seen as the volume between two parallel and equal solid 2-spheres (3-balls) in 4-dimensional space, here stereographically projected into 3D. Couronne solide.svg
The spherinder can be seen as the volume between two parallel and equal solid 2-spheres (3-balls) in 4-dimensional space, here stereographically projected into 3D.

In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3-ball (or solid 2-sphere), radius r1 and a line segment of length 2r2:

Contents

Like the duocylinder, it is also analogous to a cylinder in 3-space, which is the Cartesian product of a disk with a line segment.

It can be seen in 3-dimensional space by stereographic projection as two concentric spheres, in a similar way that a tesseract (cubic prism) can be projected as two concentric cubes, and how a circular cylinder can be projected into 2-dimensional space as two concentric circles.

Relation to other shapes

In 3-space, a cylinder can be considered intermediate between a cube and a sphere. In 4-space there are three intermediate forms between the tesseract and the hypersphere. Altogether, they are the:

These constructions correspond to the five partitions of 4, the number of dimensions.

If the two ends of a spherinder are connected together, or equivalently if a sphere is dragged around a circle perpendicular to its 3-space, it traces out a spheritorus. If the two ends of an uncapped spherinder are rolled inward, the resulting shape is a torisphere.

Spherindrical coordinate system

One can define a "spherindrical" coordinate system (r, θ, φ, w) where x, y, and z are the same as the spherical coordinates with an extra coordinate w. This is analogous to how cylindrical coordinates are defined: x and y being polar coordinates with an elevation coordinate z. Spherindrical coordinates can be converted to Cartesian coordinates using the formulas

where r is the radius, θ is the zenith angle, φ is the azimuthal angle, and w is the height. Cartesian coordinates can be converted to spherindrical coordinates using the formulas

The hypervolume element for spherindrical coordinates is which can be derived by computing the Jacobian.

Measurements

Hypervolume

Given a spherinder with a spherical base of radius r and a height h, the hypervolume of the spherinder is given by

Surface volume

The surface volume of a spherinder, like the surface area of a cylinder, is made up of three parts:


Therefore, the total surface volume is

Proof

The above formulas for hypervolume and surface volume can be proven using integration. The hypervolume of an arbitrary 4D region is given by the quadruple integral

The hypervolume of the spherinder can be integrated over spherindrical coordinates.

The related truncated icosidodecahedral prism is constructed from two truncated icosidodecahedra connected by prisms, shown here in stereographic projection with some prisms hidden. Truncated icosidodecahedral prism.png
The related truncated icosidodecahedral prism is constructed from two truncated icosidodecahedra connected by prisms, shown here in stereographic projection with some prisms hidden.

The spherinder is related to the uniform prismatic polychora, which are cartesian product of a regular or semiregular polyhedron and a line segment. There are eighteen convex uniform prisms based on the Platonic and Archimedean solids (tetrahedral prism, truncated tetrahedral prism, cubic prism, cuboctahedral prism, octahedral prism, rhombicuboctahedral prism, truncated cubic prism, truncated octahedral prism, truncated cuboctahedral prism, snub cubic prism, dodecahedral prism, icosidodecahedral prism, icosahedral prism, truncated dodecahedral prism, rhombicosidodecahedral prism, truncated icosahedral prism, truncated icosidodecahedral prism, snub dodecahedral prism), plus an infinite family based on antiprisms, and another infinite family of uniform duoprisms, which are products of two regular polygons.

See also

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