jack

1. No-one says rolling dice has to result in a *face*. It's obviously more convenient, but if you roll your dice in a V-shaped valley, you'll end up with them landing on an edge. And you could get a corner by using a conical well. (Though you need to shake it to make sure it doesn't get stuck.) Then any polyhedron with 7n equivalent faces, or edges, or vertices would do the trick. Unfortunately, I don't think it helps, I think you need 7N faces or regular septagons to get 7N edges or vertices, which you can't have.

2. You can have it in six dimensions, though. A 6-dimensional tetrahedron has seven 5-d-tetrahedral faces.