We wrote about the basic properties of Venn diagrams in an earlier post. Now we take a deeper look. John Venn, a logician and philosopher, born in Hull, Yorkshire in 1834, introduced the diagrams in a paper in 1880 and in his book Symbolic Logic, published one year later. The diagrams were used long before Venn’s paper, but he formalized and popularized them. He used them as logical diagrams: the interior of each set means the truth of a particular proposition. Unions and intersections of sets correspond to the logical operators OR and AND.
We can define a Venn diagram in a strictly mathematical way, but we can also describe it in an intuitive way. It is a partitioning of the plane by a set of n simple closed curves. We assume that each two curves intersect exactly twice and no more than two meet at any point. Thus, we rule out many “awkward” possibilities like these:
Since each point not on any of the curves is either in the interior or exterior of each, there are 2^{n} possibilities. We assume that the n curves divide the plane into 2^{n} regions (each simply connected). For example, the familiar threeset Venn diagram gives us 2^{3} = 8 regions.
We can specify a region by a binary number with n bits, each bit indicating whether the point is inside (1) or outside (0) a particular curve (membership or otherwise of a particular set); see figure at head of post.
Higher order
We can try to extend the pictures of Venn diagrams to four or more sets. Trying the pattern of four circles in a symmetric arrangement (as in fig below) looks promising but, when we count the regions, there are only 14. There should be 2^{4} = 16 regions; something is missing. We cannot make a Venn4 diagram using only circles.
A pattern using ellipses was found by Venn himself and is shown below. It is not the only possible solution.
There are other possibilities, and a method of constructing Venn diagrams of arbitrary order is presented in Ruskey and Weston (Ref below).
Graphs
A graph is a pair (V, E), where V is a set of vertices and E is a set of edges each linking two vertices. There is a graph associated with every Venn diagram:

Each intersection corresponds to a vertex.

Each segment between intersections corrresponds to an edge.
If we consider Venn2 there are two intersections and four curve segments between them. So the graph of Venn2 is just a pair of vertices connected by four edges:
The dual graph of a Venn diagram is formed as follows:

Place a vertex within each interior region and one in the exterior region.

Place an edge between two vertices if their regions share a common boundary.
For Venn2, the dual comprises four points, each linked to two neighbours. It corresponds to the boundary of a square: vertices are at the corners and the sides are the graph edges.
Now we consider the threeset diagram Venn3. There are six intersections, linked by twelve segments with eight regions (seven interior and the exterior region). Thus, the graph has six vertices linked by twelve edges. It can be represented by a polyhedron with eight faces, a octahedron. The number of faces follows from Euler’s formula
V – E + F = 2
In the plane, it has the form of a triangle in a circle. It can also be represented by a hexagram/hexagon pair sharing the same six vertices:
The dual graph has a vertex in each of eight regions, with twelve edges linking them. It also corresponds to a polyhedron, a cube, the dual of an octahedron. In the plane, it can be depicted as a pair of squares with corresponding corners linked:
Sources
Ruskey and Weston: A Survey of Venn Diagrams. http://www.combinatorics.org/files/Surveys/ds5/VennEJC.html
Venn, John, 1880: On the diagrammatic and mechanical representation of propositions and reasonings. London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 9 (1880), 118.
Venn, John, 1880: Symbolic Logic. MacMillan, London 1881, 2nd ed., (1894).