# Spherinder

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 radius r2:

$D = \{ (x,y,z,w) | x^2+y^2+z^2\leq r_1^2,\ w^2\leq r_2^2 \}$

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.

## 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 (1-ball × 1-ball × 1-ball × 1-ball) and the hypersphere (4-ball). They are the:

• cubinder (2-ball × 1-ball × 1-ball), whose surface consists of four cylindrical cells and one square torus.
• spherinder (3-ball × 1-ball), whose surface consists of three cells - two spheres, and the region in between.
• duocylinder (2-ball × 2-ball), whose surface consists of two toroidal cells.

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

## Related 4-polytopes

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

## References

• The Fourth Dimension Simply Explained, Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online: The Fourth Dimension Simply Explained—contains a description of duoprisms and duocylinders (double cylinders)
• The Visual Guide To Extra Dimensions: Visualizing The Fourth Dimension, Higher-Dimensional Polytopes, And Curved Hypersurfaces, Chris McMullen, 2008, ISBN 978-1438298924