Duocylinder

The duocylinder, or double cylinder, is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks of radii r¹ and r²:

It is analogous to a cylinder in 3-space, which is the Cartesian product of a disk with a line segment. But unlike the cylinder, both hypersurfaces (of a regular duocylinder) are congruent.

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.

A regular duocylinder consists of two congruent cells, one square flat torus face (the ridge), zero edges, and zero vertices.

The ridge of the duocylinder is the 2-manifold that is the boundary between the two bounding (solid) tori cells. 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 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 topologically equivalent to a Euclidean 2-torus (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.

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