  Shape complexity is a hard-to-quantify quality, mainly due to its relative
nature. Biased by Euclidean thinking, circles are commonly considered as the
simplest. However, their constructions as digital images are only
approximations to the ideal form. Consequently, complexity orders computed in
reference to circle are unstable. Unlike circles which lose their circleness in
digital images, squares retain their qualities. Hence, we consider squares
(hypercubes in $\mathbb Z^n$) to be the simplest shapes relative to which
complexity orders are constructed. Using the connection between $L^\infty$ norm
and squares we effectively encode squareness-adapted simplification through
which we obtain multi-scale complexity measure, where scale determines the
level of interest to the boundary. The emergent scale above which the effect of
a boundary feature (appendage) disappears is related to the ratio of the
contacting width of the appendage to that of the main body. We discuss what
zero complexity implies in terms of information repetition and constructibility
and what kind of shapes in addition to squares have zero complexity.
