A parallelotope is the generalization of a parallelogram to 
dimensions.
Moving a point 
along a line from an initial point to a final point traces
out a line segment 
. When 
is translated from an initial
position to a final position, it traces out a parallelogram 
. When 
is translated, it traces
out a parallelepiped 
. Continuing this construction gives an 
-dimensional parallelotope 
. The number 
of 
-dimensional faces
of 
is
 |
where 
is a binomial coefficient and 
, 1, ..., 
(Coxeter 1973). These are also the coefficients of 
.
Grozev and Nikolov (2026) proved that a parallelotope 
containing one facet
of the unit hypercube ![[0,1]^n](/images/equations/Parallelotope/Inline18.svg)
and a vertex on the
opposite facet satisfies 
. Thus its integer translates
cover the entire space. In three dimensions, the same
conclusion holds for every parallelepiped containing
any five vertices of the unit cube.
The covering result yields the sharp pyramid volume
bound 
for a parallelotope-based pyramid contained in ![[0,1]^n](/images/equations/Parallelotope/Inline21.svg)
(Grozev and Nikolov 2026).
