Duocylinder

The duocylinder is a geometric object embedded in 4-Dimensional Euclidean space, defined as the Cartesian product of two circles of radius r :

:D = { (x,y,z,w) | x^2+y^2leq r^2, z^2+w^2leq r^2 }

It is bounded by two mutually perpendicular 3-manifolds with torus-like surfaces, described by the equations:

:x^2 + y^2 = r^2, z^2 + w^2 leq r^2 and :z^2 + w^2 = r^2, x^2 + y^2 leq r^2

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 Plane_(geometry). The duocylinder has rotational symmetry in both of these planes.

In order to get the wireframe , so to speak, of this object, we can start with a two dimensional rectangle. Join the top and bottom edges, so that it forms a cylinder. Then join the other two edges by bending it around through 4-dimensional space. This then is 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. This 2 dimensional manifold, thus has been called the true Euclidean 2-torus . Though this surface is topologically equivalent to a doughnut ; the familiar doughnut shape can exist in 3-dimensional space because it has been curved into a spherical geometry on the outside, and a hyperbolic geometry around the hole. This both stretches and crunches the surface in different parts. This is because once a flat rectangle is curled into a cylinder, it is no longer flat in three dimensions, and cannot be bent any further without distorting it. However, the duocylinder is not distorted in this manner at all since it is curved in the fourth dimension, where the cylinder is still flat and can be bent without distortion of its surface.

Parallel projections of the duocylinder into 3-dimensional space and its cross-sections with 3-dimensional space both form Cylinder_(geometry). 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 Polytope approximations of the duocylinder.

=Nomenclature=

The duocylinder is also known as the double cylinder.

=See also=

*Duoprism *Manifold *3-Torus

=References=

*[http://tetraspace.alkaline.org/shapes/rotachora.htm Rotachora (4-dimensional objects with circular surfaces)] *[http://tetraspace.alkaline.org/shapes/rotatopeclass.htm Classification of rotatopes] *[http://etext.lib.virginia.edu/etcbin/toccer-new2id=ManFour.sgm&images=images/modeng&data=/texts/english/modeng/parsed&tag=public&part=all The Fourth Dimension Simply Explained]—contains a description of duoprisms and duocylinders (double cylinders) *[http://eusebeia.dyndns.org/~hsteoh/4d/duocylinder.html Diagrams of duocylinder projected into 3-dimensional space]