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[[File:Couronne solide.svg|thumb|The '''spherinder''' can be seen as the volume between two parallel and equal solid 2-[[sphere]]s (3-balls) in 4-dimensional space, here stereographically projected into 3D.]]
In [[Four-dimensional space|four-dimensional geometry]], the '''spherinder''', or '''spherical cylinder''' or '''spherical prism''', is a geometric object, defined as the [[Cartesian product]] of a 3-[[ball (mathematics)|ball]] (or solid 2-[[sphere]]), radius ''r''<sub>1</sub> and a [[line segment]] of radius ''r''<sub>2</sub>:

:<math>D = \{ (x,y,z,w) | x^2+y^2+z^2\leq r_1^2,\ w^2\leq r_2^2 \}</math>

Like the [[duocylinder]], it is also analogous to a [[Cylinder (geometry)|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 (mathematics)|ball]] × 1-ball × 1-ball × 1-ball) and the [[hypersphere]] (4-[[ball (mathematics)|ball]]).  They are the [[cubinder]] (2-ball × 1-ball × 1-ball or [[circle]]×[[square]]), the [[duocylinder]] (2-ball × 2-ball) and the '''spherinder''' (3-ball × 1-ball).  These constructions correspond to the five [[Partition (number theory)|partition]]s of 4, the number of dimensions.

== Related 4-polytopes ==
[[File:Truncated icosidodecahedral prism.png|thumb|The related [[truncated icosidodecahedral prism]] is constructed from two [[icosidodecahedron|icosidodecahedra]] connected by [[prism (geometry)|prisms]], shown here in [[stereographic projection]] with some prisms hidden.]]
It is related to the [[Uniform_polychoron#Prismatic_uniform_polychora|uniform prismatic polychora]], which are [[cartesian product]] of a regular or semiregular [[polyhedron]] and a [[line segment]]. For example: [[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]], and the [[snub dodecahedral prism]].

==See also==
*[[Clifford torus]]

==References==
*''The Fourth Dimension Simply Explained'', Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online: [http://etext.lib.virginia.edu/etcbin/toccer-new2?id=ManFour.sgm&images=images/modeng&data=/texts/english/modeng/parsed&tag=public&part=all The Fourth Dimension Simply Explained]&mdash;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

[[Category:Four-dimensional geometry]]
[[Category:Geometric objects]]
[[Category:Spherical geometry]]


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