It is an open problem to characterize the cone of
f
f
-vectors of
4
4
-dimensional convex polytopes. The question whether the “fatness” of the
f
f
-vector of a
4
4
-polytope can be arbitrarily large is a key problem in this context. Here we construct a
2
2
-parameter family of
4
4
-dimensional polytopes
π
(
P
n
2
r
)
\pi (P^{2r}_n)
with extreme combinatorial structure. In this family, the “fatness” of the
f
f
-vector gets arbitrarily close to
9
9
; an analogous invariant of the flag vector, the “complexity,” gets arbitrarily close to
16
16
.
The polytopes are obtained from suitable deformed products of even polygons by a projection to
R
4
\mathbb {R}^4
.