Duocylinder
From Wikipedia, the free encyclopedia
The duocylinder is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks of radius r:
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[edit] Geometry
[edit] Bounding 3-manifolds
The duocylinder is bounded by two mutually perpendicular 3-manifolds with torus-like surfaces, described by the equations:
and
The duocylinder is so-called because these two bounding 3-manifolds may be thought of as 3-dimensional cylinders 'bent around' in 4-dimensional space such that they form closed loops in the XY and ZW planes. The duocylinder has rotational symmetry in both of these planes.
[edit] The ridge
The ridge of the duocylinder (the 2-manifold that is the boundary between the two bounding tori) may be constructed as follows. Roll a 2-dimensional rectangle into a cylinder, so that its top and bottom edges meet. Then roll the cylinder in the plane perpendicular to the 3-dimensional hyperplane that the cylinder lies in, so that its two circular ends meet.
The resulting shape is known as the true Euclidean 2-torus, and is topologically equivalent to a 3-dimensional doughnut. However, unlike the latter, all parts of its surface are identically deformed. On the doughnut, the surface around the 'doughnut hole' is deformed differently from the surface outside.
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 3-dimensional space, because it requires two degrees of freedom in addition to its inherent 2-dimensional surface in order for both pairs of edges to be joined.
[edit] Projections
Parallel projections of the duocylinder into 3-dimensional space and its cross-sections with 3-dimensional space both form cylinders. Perspective projections of the duocylinder form torus-like shapes with the 'doughnut hole' filled in.
The duocylinder is the limiting shape of duoprisms as the number of sides in the constituent polygonal prisms approach infinity. As such, the duoprisms serve as good polytopic approximations of the duocylinder.
[edit] Nomenclature
The duocylinder is also known as the double cylinder.
[edit] See also
[edit] 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)
