 Duocylinder

The duocylinder, or double cylinder, is a geometric object embedded in 4dimensional Euclidean space, defined as the Cartesian product of two disks of radius r:
It is analogous to a cylinder in 3space, which is the Cartesian product of a disk with a line segment.
Contents
Geometry
Bounding 3manifolds
The duocylinder is bounded by two mutually perpendicular 3manifolds with toruslike surfaces, described by the equations:
and
The duocylinder is so called because these two bounding 3manifolds may be thought of as 3dimensional cylinders 'bent around' in 4dimensional space such that they form closed loops in the XY and ZW planes. The duocylinder has rotational symmetry in both of these planes.
The ridge
The ridge of the duocylinder is the 2manifold that is the boundary between the two bounding tori. It is in the shape of a Clifford torus, which is the Cartesian product of two circles. Intuitively, it may be constructed as follows: Roll a 2dimensional rectangle into a cylinder, so that its top and bottom edges meet. Then roll the cylinder in the plane perpendicular to the 3dimensional hyperplane that the cylinder lies in, so that its two circular ends meet.
The resulting shape is topologically equivalent to a Euclidean 2torus (a doughnut shape). However, unlike the latter, all parts of its surface are identically deformed. On the doughnut, the surface around the 'doughnut hole' is deformed with negative curvature while the surface outside is deformed with positive curvature.
The ridge of the duocylinder may be thought of as the actual global shape of the screens of video games such as Asteroids, where going off the edge of one side of the screen leads to the other side. It cannot be embedded without distortion in 3dimensional space, because it requires two degrees of freedom in addition to its inherent 2dimensional surface in order for both pairs of edges to be joined.
Projections
Parallel projections of the duocylinder into 3dimensional space and its crosssections with 3dimensional space both form cylinders. Perspective projections of the duocylinder form toruslike shapes with the 'doughnut hole' filled in.
Relation to other shapes
The duocylinder is the limiting shape of duoprisms as the number of sides in the constituent polygonal prisms approach infinity. The duoprisms therefore serve as good polytopic approximations of the duocylinder.
In 3space, a cylinder can be considered intermediate between a cube and a sphere. In 4space there are three intermediate forms between the tesseract (1ball × 1ball × 1ball × 1ball) and the hypersphere (4ball). They are the cubinder (2ball × 1ball × 1ball), the duocylinder (2ball × 2ball) and the spherinder (3ball × 1ball). These constructions correspond to the five partitions of 4, the number of dimensions.
See also
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)
External links
Categories: Fourdimensional geometry
 Algebraic topology
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