A combinatorial cube with perpendicular facets
A combinatorial cube is a polytope whose face lattice is isomorphic to the face lattice of the standard cube. In other words, the vertices have been moved without adding or removing any faces.
For d > 2, there exist combinatorial cubes similar to this one. For each pair of opposite facets, the respective supporting hyperplanes are perpendicular rather than parallel!
To see this, click on the circles to the left of Equations 1 and 2 to observe the supporting planes. Then try Equations 3 and 4, and then 5 and 6.
[Construction by my collaborator Joseph Doolittle.]
A combinatorial cube is a polytope whose face lattice is isomorphic to the face lattice of the standard cube. In other words, the vertices have been moved without adding or removing any faces.
For d > 2, there exist combinatorial cubes similar to this one. For each pair of opposite facets, the respective supporting hyperplanes are perpendicular. Try toggling the six equations two at a time (eq1&eq2, eq3&eq4, eq5&eq6) to see this property in action. [Construction by Joseph Doolittle.]