In this publication, as well as some of Hinton’s later work, the word was occasionally spelled “tessaract”. The tesseract can be constructed in a number of ways. Each pair of non-parallel hyperplanes intersects to form 24 square read four views on free will pdf in a tesseract. Three cubes and three squares intersect at each edge.

There are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices. Two points A and B can be connected to a line, giving a new line segment AB. Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD. Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH. Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP. It is possible to project tesseracts into three- and two-dimensional spaces, similarly to projecting a cube into two-dimensional space.

Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices. A tesseract is in principle obtained by combining two cubes. The scheme is similar to the construction of a cube from two squares: juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length. 4 and there are many different paths to allow weight balancing. The nearest and farthest cells are projected onto the cube, and the remaining 6 cells are projected onto the 6 square faces of the cube. Two pairs of cells project to the upper and lower halves of this envelope, and the 4 remaining cells project to the side faces.

Six cells project onto rhombic prisms, which are laid out in the hexagonal prism in a way analogous to how the faces of the 3D cube project onto 6 rhombs in a hexagonal envelope under vertex-first projection. The two remaining cells project onto the prism bases. Two vertices of the tesseract are projected to the origin. This projection is also the one with maximal volume. The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.