Three-torus

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The three-dimensional torus, or three-torus, is defined as the Cartesian product of three circles,

In contrast, the usual torus is the Cartesian product of two circles only.

The three-torus is a three-dimensional compact manifold with no boundary. It can be obtained by "gluing" the three pairs of opposite faces of a cube, where being "glued" can be intuitively understood to mean that when a particle moving in the interior of the cube reaches a point on a face, it goes through it and appears to come forth from the corresponding point on the opposite face. After gluing the first pair of opposite faces, the cube looks like a thick washer (annular cylinder), after gluing the second pair the flat faces of the washer it looks like two nested two-tori, the last gluing the inner nested torus to the outer nested torus is physically impossible in three-dimensional space so it has to happen in four dimensions.

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Klein bottle

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Surface (topology) Two-dimensional manifold

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.

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Torus Doughnut-shaped surface of revolution

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 mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to 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.

24-cell

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Real projective plane A compact non-orientable two-dimensional manifold

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R3 passing through the origin.

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Duoprism

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Clifford torus Four-dimensional geometrical object

In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S1
a
and S1
b
. It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S1
b
and S1
b
each exists in its own independent embedding space R2
a
and R2
b
, the resulting product space will be R4 rather than R3. The historically popular view that the cartesian product of two circles is an R3 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.

Two-dimensional space Geometric model of the planar projection of the physical universe

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Linear flow on the torus

In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus

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, radius r1 and a line segment of length 2r2:

Cubinder

In four-dimensional geometry, the cubinder is one way to generalise the 3D cylinder to 4D. Like the duocylinder and spherinder, it is also analogous to a cylinder in 3-space, which is the Cartesian product of a disk with a line segment.

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