The **duocylinder**, also called the **double cylinder** or the **bidisc**, is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks of respective radii *r*_{1} and *r*_{2}:

- Geometry
- Bounding 3-manifolds
- The ridge
- Projections
- Relation to other shapes
- See also
- References
- External links

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.

Its dual is a duospindle, constructed from two circles, one at the XY plane and the other in the ZW plane.

The duocylinder is bounded by two mutually perpendicular 3-manifolds with torus-like surfaces, respectively described by the formulae:

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) torus 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 (2D surface, embedded in 3D) doughnut, the surface around the 'doughnut hole' is deformed with negative curvature (like a saddle) while the surface outside is deformed with positive curvature (like a sphere).

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 ("directions") in addition to its inherent 2-dimensional surface in order for both pairs of edges to be joined.

The duocylinder can be constructed from the 3-sphere by "slicing" off the bulge of the 3-sphere on either side of the ridge. The analog of this on the 2-sphere is to draw minor latitude circles at ±45 degrees and slicing off the bulge between them, leaving a cylindrical wall, and slicing off the tops, leaving flat tops. This operation is equivalent to removing select vertices/pyramids from polytopes, but since the 3-sphere is smooth/regular you have to generalize the operation.

The dihedral angle between the two 3-d hypersurfaces on either side of the ridge is 90 degrees.

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. The duoprisms therefore serve as good polytopic approximations of the duocylinder.

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 × 1-ball × 1-ball × 1-ball) and the hypersphere (4-ball). They are:

- the cubinder
^{[ citation needed ]}(2-ball × 1-ball × 1-ball), whose surface consists of four cylindrical cells and one square torus. - the spherinder (3-ball × 1-ball), whose surface consists of three cells - two spheres, and the region in between.
- the
**duocylinder**(2-ball × 2-ball), whose surface consists of two toroidal cells.

The duocylinder is the only one of the above three that is regular. These constructions correspond to the five partitions of 4, the number of dimensions.

In topology, a branch of mathematics, the **Klein bottle** is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the real projective plane. While a Möbius strip is a surface with boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary.

A **sphere** is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance *r* from a given point in three-dimensional space. That given point is the centre of the sphere, and *r* is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

In the part of mathematics referred to as topology, a **surface** is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

In mathematics, an ** n-sphere** or a

In geometry, a **torus** is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.

In geometry, a **coordinate system** is a system that uses one or more numbers, or **coordinates**, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the *x*-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and *vice versa*; this is the basis of analytic geometry.

In mathematics, a **plane** is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point, a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of two-dimensional Euclidean geometry. Sometimes the word *plane* is used more generally to describe a two-dimensional surface, for example the hyperbolic plane and elliptic plane.

In mathematics, **homology** is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In geometry, a **hyperboloid of revolution**, sometimes called a **circular hyperboloid**, is the surface generated by rotating a hyperbola around one of its principal axes. A **hyperboloid** is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mathematics, particularly in complex analysis, a **Riemann surface** is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:

In geometry, a **hypersurface** is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension *n* − 1, which is embedded in an ambient space of dimension *n*, generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally, and sometimes globally.

In geometric topology, a branch of mathematics, a **Dehn twist** is a certain type of self-homeomorphism of a surface.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or *-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.

In geometry of 4 dimensions or higher, a **double prism** or **duoprism** is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (*n*+*m*)-polytope, where n and m are dimensions of 2 (polygon) or higher.

In geometry, a **three-dimensional space** is a mathematical structure in which three values (*coordinates*) are required to determine the position of a point. More specifically, *the three-dimensional space* is the Euclidean space of dimension three that models physical space.

The **Lambert azimuthal equal-area projection** is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the **Lambert zenithal equal-area projection**.

In geometric topology, the **Clifford torus** is the simplest and most symmetric flat embedding of the cartesian product of two circles *S*^{1}_{a} and *S*^{1}_{b}. It is named after William Kingdon Clifford. It resides in **R**^{4}, as opposed to in **R**^{3}. To see why **R**^{4} is necessary, note that if *S*^{1}_{a} and *S*^{1}_{b} each exists in its own independent embedding space **R**^{2}_{a} and **R**^{2}_{b}, the resulting product space will be **R**^{4} rather than **R**^{3}. The historically popular view that the cartesian product of two circles is an **R**^{3} torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis *z* available to it after the first circle consumes *x* and *y*.

In four-dimensional geometry, the **spherinder**, or **spherical cylinder** or **spherical prism**, is a geometric object, defined as the Cartesian product of a 3-ball of radius *r*_{1} and a line segment of length 2*r*_{2}:

*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)*The Visual Guide To Extra Dimensions: Visualizing The Fourth Dimension, Higher-Dimensional Polytopes, And Curved Hypersurfaces*, Chris McMullen, 2008, ISBN 978-1438298924

- Rotachora (4-dimensional objects with circular surfaces)
- Classification of rotatopes
- Diagrams of duocylinder projected into 3-dimensional space
- Exploring Hyperspace with the Geometric Product

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