RotationĪn axis of rotation in 3D is a fixed line. The larger outer cube is one of the cells of the 4-cube. Projected image of a 4-cube by means of center projection. To display the 4-cube in 3D, central projection from 4D to 3D is analogous to central projection from 3D to 2D the function is the natural extension of see Figure 2.įigure 2. m files containing information for these polytopes are provided at : the positions of the vertices, vertex indices for the proper faces and which faces are neighbors. The 24 squares of the 4-cube are described in terms of their vertex indices.īesides the 4-cube, there are five other regular polytopes in four dimensions. The 16 vertices of a 4-cube can be defined as lists of length four of all possible combinations of In particular, no interior point of a cell as a 3D object is in the interior of the hypercube all the points of a cell are on the boundary of the 4-cube. No point of a proper face is strictly in the interior of the 4-cube that is, a hypersphere at such a point contains points inside and points outside the 4-cube. The proper faces of the 4-cube are its vertices, edges, squares and cells.Įach point of a proper face is on the 3D hypersurface of the 4-cube. The eight cubes are called cells, which are like the six square faces of a 3D cube. Overall, the 4D Rubik puzzle is a 4-cube (or 4D hypercube or tesseract), with vertices, edges, squares, eight cubes and one 4-cube. A cube and its image under a central projection. Five of the faces overlap with a sixth face, the price to pay for the loss of one dimension.įigure 1. Choose to obtain the projection shown on the right inįigure 1. The 3D cube can be represented in a 2D plane using central projection, defined by taking the intersection of the plane with the line joining the two points and. More generally, the number of -cubes (points, segments, squares, …) in an -cube,, is. Going up a dimension doubles the number of vertices. The 3D hypercube is a cube (or 3-cube), with eight vertices, edges, six square faces and one volume. The 2D hypercube (or 2-cube) is a square, with four vertices, four edges and one face (the square including its interior). The 1D hypercube (or 1-cube) is a segment, with two vertices and one edge. The zero-dimensional hypercube (or 0-cube) is a point, with one vertex. To understand the 4D hypercube, it helps to first see how its lower-dimensional analogs relate to each other. Basic Concepts Hypercubes in Low Dimensions Starting from a random coloring of the 4-cube, the goal of the puzzle is to return to the initial coloring of the 3-faces. Projecting the whole 4D configuration to 3D exhibits Rubik ’s 4-cube as a four-dimensional extension of Rubik ’s cube. The set can be rotated around the normal (a plane) through the center of. Each 3-face of determines a set of 27 4-subcubes that have a cube in the same hyperplane as. Rubik ’s 4-cube (or 4D hypercube) consists of 81 unit 4-subcubes, each containing eight 3D subcubes. The set can be rotated around the normal through the center of. Each face of determines a set of nine subcubes that have a face in the same plane as. The well-known three-dimensional Rubik ’s cube consists of 27 unit subcubes. This article constructs the basic concepts of the puzzle and implements it in a program. Rubik ’s cube has a natural extension to four-dimensional space.
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