All of the cubes in the image are the same cube, since light in the manifold wraps around into closed loops. The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, In contrast, the usual torus is the Cartesian product of only two circles.

2012年5月4日 · A 3-torus can be constructed by starting with a cube and then conceptually joining the top and bottom, the right and left, and the front and back. In such a space, an object which travels out to the left hand side of the cube instantly reappears at the right hand side (and similarly for the top-bottom and the front-back).

1 天前 · In one dimension, a line bends into circle, giving the 1-torus. In two dimensions, a rectangle wraps to a usual torus, also called the 2-torus. In three dimensions, the cube wraps to form a 3-manifold, or 3-torus. In each case, the -torus is an object that exists in dimension .

There is a three-dimensional analog of a 2-torus called (not surprisingly) a 3-torus. A 3-torus can be constructed from a cube with top and bottom faces glued, left and right glued, and front and back glued. The faces can only be physically glued in 4-space, so we will concentrate our thoughts on the abstractly glued cube, which is easier to ...

The cubes must be glued face-to-face, and the boundary of the resulting object should be topologically equivalent to an $n$-torus, by which I mean a genus-$n$ handlebody in $\mathbb{R}^3$ (as per Kevin Walker's terminological correction). For example, 8 cubes are needed to form a 1-toroid:

Each frame shows a stereographic projection into three dimensional space R 3 after rotating in four dimensional space (applying an orthogonal 4x4 matrix). Similarly as with the visualization of the three dimensional sphere, we draw a two parameter family of circles on the torus and encode one of the parameters with color.

当三种拓扑(ring、mesh 和 torus)具有相同数量的节点，且网络流量随机且均匀时(每一个节点向其余节点发送消息的概率均相同时)， ring结构的跳数比 mesh 结构和 torus结构都要多。

The simplest example of a finite 3-dimensional surface (generally called a 3-manifold) is the 3-torus. It can be constructed by starting with a cube (called the fundamental domain of the 3-torus) and then conceptually joining opposite faces.

When attached top to bottom and side to side, a four-by-four grid forms the polyhedral torus in the hypercube. An easier technique is to leave out certain of the 24 square faces.

Metatron’s cube is formed by connecting all the centers of the circles in the fruit of life. These form the Platonic solids. The torus is a manifestation of the seed of life that is rotated. As is spirals out it forms the Flower of Life. A rotating Flower of Life …