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
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\tim ...
of two
disks of respective radii ''r''
1 and ''r''
2:
:
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.
Geometry
Bounding 3-manifolds
The duocylinder is bounded by two mutually
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
3-
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s 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
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an in ...
'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
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
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon ...
, 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.
The ridge of the duocylinder may be thought of as the actual global shape of the screens of
video games such as
Asteroids
An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere.
...
, 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.
The duocylinder can be constructed from the
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
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
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an - ...
, but since the 3-sphere is smooth/regular you have to generalize the operation.
The
dihedral angle
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
between the two 3-d hypersurfaces on either side of the ridge is 90 degrees.
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.
Relation to other shapes
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
In geometry, a tesseract is the four-dimensional analogue 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 six square faces, the hypersurface of the tesseract consists of eig ...
(1-
ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used fo ...
× 1-ball × 1-ball × 1-ball) and the
hypersphere (4-
ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used fo ...
). They are:
* the cubinder (2-ball × 1-ball × 1-ball), whose surface consists of four cylindrical cells and one square torus.
* the
spherinder
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 (or solid 2-sphere) of radius ''r''1 and a line segment of length 2''r''2:
:D = \
Like th ...
(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.
See also
*
Clifford torus
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon ...
*
Duoprism
*
Flat torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
*
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz ...
*
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
References
* ''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 Explainedmdash;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
External links
Exploring Hyperspace with the Geometric Product(
Wayback Machine copy)
Four-dimensional geometry
Algebraic topology