  Venn diagrams are a graphical way to represent a set system. Each of the n
sets is represented by a simple closed curve. The n curves subdivide the plane
into 2^n open connected regions, each of which represents the intersection of
its containing curves' sets. For example, two overlapping circles can divide
the plane into 4 regions representing {}, A, B, and A intersect B. Three
overlapping circles can also be used to represent the 2^3 ways in which 3 sets
can intersect.
  One of the primary questions related to Venn diagrams concerns which shapes
can be used for the curves. The previous examples used 2 and 3 circles, but a
4-set Venn diagram cannot be represented by 4 circles; instead, ellipses must
be used.
  In this paper, we consider Venn diagrams whose curves are the outlines of
polyominoes. In particular, we give examples of Venn diagrams where the curves
are rotations and translations of a single polyomino, so-called congruent
polyVenn diagrams. We also consider the problem of area-minimization (relative
to a scaling factor) and present examples of Venn polyominoes which minimize
area according to various constraints.
  At present, these examples do not generalize and so we develop an algorithm
that comes close to minimizing the area. The algorithm is simple and utilizes
symmetric chain decompositions of the Boolean lattice. We also provide
asymptotic results that relate the area required by the algorithm's diagrams to
the theoretical minimum area. We conclude by presenting some open problems
related to Venn polyominoes and other shape-constrained Venn diagrams.

Combinatorics;Geometric Topology