square, cube. 4D case known as tesseract.
Convex hull of all (Cartesian product power) D-tuples, e.g. in 3D:
( 1,  1,  1)
( 1,  1, -1)
( 1, -1,  1)
( 1, -1, -1)
(-1,  1,  1)
(-1,  1, -1)
(-1, -1,  1)
(-1, -1, -1)
From this we see that there are vertices.
Two vertices are linked iff they differ by a single number. So each vertex has D neighbors.
The non-regular version of the hypercube.

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  1. Classification of regular polytopes
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