Tesseract
The Tesserakt [ tɛsərakt ] is a generalization of the classic cube four dimensions. It also speaks of a four-dimensional hypercube. The tesseract is to the cube as the cube behaves squared. He has 16 vertices, 32 edges of equal length, 24 square faces, and is bounded by 8 cubical cells. These cells are referred to as boundary cube of the tesseract. In every corner of 4 edges, 6 faces and 4 cells meet each perpendicular to one another.
The images in this article must be understood as images of Tesserakten under parallel projections. Down in the first picture you can see a blue and a yellow cube, which are connected by six other rhombohedral distorted cube boundary. In the three-dimensional network of the tesseract (left in the first picture), all eight boundary cubes are folded into three-dimensional space, like the faces of a three-dimensional cube can be unfolded into a net of six squares. There are 261 different ways to develop a tesseract.
The following image is a network of the tesseract is shown on the left, and bottom right a two-dimensional parallel projection of the tesseract.
When his eight opposite limiting cubes two stapled together in a tesseract, creating a 4- torus.
Projections in 2 dimensions
The construction of a hypercube can be thought of as follows:
- One-dimensional: Two points A and B can be connected to a line, it is a new line AB.
- Two-dimensional: Two parallel lines AB and CD can be connected to a square, with the corners ABCD.
- Three-dimensional: Two parallel squares ABCD and EFGH can be connected to a cube, with the corners ABCDEFGH.
- Four-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to a hypercube, with the corners ABCDEFGHIJKLMNOP.
Although this is difficult to imagine, but it is possible to project tesseracts in three - or two-dimensional spaces. Also, projections into the second dimension revealing if you rearranges the projected vertices. In this method, one can obtain images that no longer reflect the spatial relationships within the tesseract, but showing the connecting structure of the corners, as the following examples:
A tesseract is in principle formed by two connected cubes. The scheme is the construction of a cube from two squares similar to: Set side by side two copies of the lower dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length. Eight cubes that are interconnected.
Tesseracts are also two-part graph, just like lines, squares and cubes.
Projections in three dimensions
First, the cell - to-parallel projection of the tesseract in the three-dimensional space has a cubical casing. The next and the most distant areas are projected on the die, and the remaining cells 6 are projected on the square surface of the cube.
First, the area - to-parallel projection of the tesseract in 3 -dimensional space has a rectangular envelope. Two pairs of projecting the upper and lower halves of the casing 4, and the remaining cells are projected on the side faces.
First, the edge - to-parallel projection of the tesseract in the three-dimensional space has a cover in the form of a hexagonal prism. Six cells are projected onto rhombic prisms which are arranged in the hexagonal prism are designed to the same way as the surfaces of a cube in 3D, a hexagonal shell in the corner -first projection. The remaining two cells are projected onto the base of the prism.
The corner -first parallel projection of the tesseract into three-dimensional space has a rhombic dodekaederförmige envelope.