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:
ball), whose hypersurface is a 3-sphere without any connecting boundaries.
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), consisting of
spherical coordinates with an extra coordinate w. This is analogous to how
cylindrical coordinates are defined: r and φ 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 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,