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A simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. So, and correct me if I'm wrong, you can say a simplex integer lattice is a generalization of a lattice built from the end points of simplexes in arbitrary dimensions.

But what about the geometric shape square in 2D and cube in 3D? Is there a word for generalizing them in arbitrary dimensions? What if you use them in a lattice? Simpletic honeycomb comes to mind, but obviously doesn't fit.

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    $\begingroup$ The word you are looking for is hypercube (en.wikipedia.org/wiki/Hypercube) $\endgroup$
    – parsiad
    Mar 27, 2017 at 15:36
  • $\begingroup$ @parsiad So it is! Thank you. $\endgroup$
    – Mossmyr
    Mar 27, 2017 at 15:53

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