Duocylinder

Last updated March 29, 2024

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 r₁ and r₂:

Geometry

Bounding 3-manifolds

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

x^{2}+y^{2}=r_{1}^{2},z^{2}+w^{2}\leq r_{2}^{2}

and

z^{2}+w^{2}=r_{2}^{2},x^{2}+y^{2}\leq r_{1}^{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 $zw$ -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, 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-dimensional 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 (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.

Related Research Articles

In mathematics, the Klein bottle is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the Klein bottle is a two-dimensional manifold on which one cannot define a normal vector at each point that varies continuously over the whole manifold. Other related non-orientable surfaces include the Möbius strip and the real projective plane. While a Möbius strip is a surface with a 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. Formally, 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 center 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 geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut.

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.

<span class="mw-page-title-main">Hyperboloid</span> Unbounded quadric surface

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 one-dimensional complex manifold.

The Method of Mechanical Theorems, also referred to as The Method, is one of the major surviving works of the ancient Greek polymath Archimedes. The Method takes the form of a letter from Archimedes to Eratosthenes, the chief librarian at the Library of Alexandria, and contains the first attested explicit use of indivisibles. The work was originally thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes Palimpsest. The palimpsest includes Archimedes' account of the "mechanical method", so called because it relies on the center of weights of figures (centroid) and the law of the lever, which were demonstrated by Archimedes in On the Equilibrium of Planes.

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.

Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.

<span class="mw-page-title-main">Villarceau circles</span> Intersection of a torus and a plane

In geometry, Villarceau circles are a pair of circles produced by cutting a torus obliquely through its center at a special angle.

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, a three-dimensional space is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region, a solid figure.

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.

<span class="mw-page-title-main">Clifford torus</span> Geometrical object in four-dimensional space

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 $y$ .

<span class="mw-page-title-main">Hypercone</span> 4-dimensional figure

In geometry, a hypercone is the figure in the 4-dimensional Euclidean space represented by the equation

In mathematics, 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the 3D Cartesian coordinates across the unit 2-sphere $. They are so named because the loci where one coordinate is constant form spheres tangent to the origin from one of six sides. This coordinate system exists independently from and has no relation to the 6-sphere. The three coordinates are$

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

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 r₁ and a line segment of length 2r₂:

<span class="mw-page-title-main">Intersection (geometry)</span> Shape formed from points common to other shapes

In geometry, an intersection is a point, line, or curve common to two or more objects. The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point or does not exist. Other types of geometric intersection include:

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 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

External links

(Wayback Machine copy)

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.