6-cube 6-cube In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol {4,3}, being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the "4-cube") with "hex" for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets. It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex,
Prince Rupert's cube for the largest square that lies within a unit cube. More generally, show how to find the largest rectangle of a given aspect ratio that lies within a unit cube. As they show, the optimal rectangle must always pass through the center of the cube, with its vertices on edges of the cube. Based on this, they show, depending on the desired aspect ratio, that the optimal rectangle must either lie on a plane that cuts diagonally through four corners of the cube, or it must be formed by an isosceles right triangle on one corner of the cube and