Duocylinder
Encyclopedia
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, defined as the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of two disk
Disk (mathematics)
In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...
s of radius r:
It is analogous to a cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
in 3-space, which is the Cartesian product of a disk with a line segment
Line segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
.
Bounding 3-manifolds
The duocylinder is bounded by two mutually perpendicularPerpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
3-manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s with torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
-like surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
s, described by the equations:
and
The duocylinder is so called because these two bounding 3-manifolds may be thought of as 3-dimensional cylinders
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
'bent around' in 4-dimensional space such that they form closed loops in the XY and ZW planes. The duocylinder has rotational symmetry
Rotational symmetry
Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...
in both of these planes.
The ridge
The ridge of the duocylinder is the 2-manifold that is the boundary between the two bounding tori. It is in the shape of a Clifford torusClifford torus
In geometric topology, the Clifford torus is a special kind of torus sitting inside R4. Alternatively, it can be seen as a torus sitting inside C2 since C2 is topologically the same space as R4...
, which is the Cartesian product of two circles. Intuitively, it may be constructed as follows: Roll a 2-dimensional rectangle
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square 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
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
(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.
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.
Projections
Parallel projections of the duocylinder into 3-dimensional space and its cross-sections with 3-dimensional space both form cylindersCylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
. Perspective projections of the duocylinder form torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
-like shapes with the 'doughnut hole' filled in.
Relation to other shapes
The duocylinder is the limiting shape of duoprismDuoprism
In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...
s as the number of sides in the constituent polygonal prisms approach infinity. The duoprisms therefore serve as good polytopic
Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
approximations of the duocylinder.
In 3-space, a cylinder can be considered intermediate between a cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
and a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
. In 4-space there are three intermediate forms between the tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...
(1-ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
× 1-ball × 1-ball × 1-ball) and the hypersphere
Hypersphere
In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...
(4-ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
). They are the cubinder (2-ball × 1-ball × 1-ball), the duocylinder (2-ball × 2-ball) and the spherinder (3-ball × 1-ball). These constructions correspond to the five partition
Partition (number theory)
In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...
s of 4, the number of dimensions.