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The duocylinder is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks of radius r: 2-dimensional renderings (ie. ...
In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes...
In geometry, a disk is the region in a plane contained inside of a circle. ...
Geometry
Bounding 3-manifolds The duocylinder is bounded by two mutually perpendicular 3-manifolds with torus-like surfaces, described by the equations: Perpendicular is a geometric term that may be used as a noun or adjective. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ...
A torus. ...
An open surface with X-, Y-, and Z-contours shown. ...
 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. In mathematics, a plane is the fundamental two-dimensional object. ...
Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. ...
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. In geometry, a rectangle is defined as a quadrilateral polygon in which all four angles are right angles. ...
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. Computer and video games A screenshot of Tetris for the Nintendo Game Boy A console game (better known as a video game) is a form of interactive multimedia used for entertainment, which consists of a moveable image displayed on a screen that is usually controlled and manipulated using a handheld...
Asteroids is a popular vector-based video arcade game released in 1979 by Atari. ...
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. A right circular cylinder In mathematics, a cylinder is a quadric, i. ...
A torus. ...
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. A duoprism is a 4-dimensional figure resulting from the Cartesian product of two polygons in the 2-dimensional Euclidean space. ...
In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...
Nomenclature The duocylinder is also known as the double cylinder.
See also A duoprism is a 4-dimensional figure resulting from the Cartesian product of two polygons in the 2-dimensional Euclidean space. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ...
A torus. ...
References - Rotachora (4-dimensional objects with circular surfaces)
- Classification of rotatopes
- The Fourth Dimension Simply Explained—contains a description of duoprisms and duocylinders (double cylinders)
- Diagrams of duocylinder projected into 3-dimensional space
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