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Finite cubes and finite affine spaces

We begin by considering certain analogues, in finite 3-dimensional affine spaces, of the ordinary cube. From the properties of these structures (as geometric lattices having point sets which are threefold Cartesian products) we derive the notion of an (abstract) finite cube. By studying the additional properties (including the properties of certain groups associated with the cube) of members of a suitably restricted class of embedded finite cubes, we arrive at the notion of a perfect finite cube. Our main theorem asserts that every perfect finite cube can be embedded in a finite 3-dimensional affine space.