These polytopes are built up by sum and product operations from cubes. They turn up as hard examples for incremental algorithms [11, 12] and as examples of zero-one polytopes with many facets [175]. The interesting thing about the latter result is that they turn out to have lots more facets if (any) one vertex is removed; this is the substance of [12].

faces>>size, vertex-degenerate, facet-degenerate, triangulation, incremental, zero-one

3d

<p> a=ceil(sqrt(d)) b=floor(d/a) c=d mod a

sumcube

c ? (8a)

^{b}*8c : (8a)^{b}

c ? V(sumcube(a))

^{b}x V(sumcube(c)) : V(sumcube(a))^{b}