A Venn diagram is a schematic diagram used in logic and set theory to depict collections of sets and their relationships.
The Venn diagrams on two and three sets are illustrated above. The order-two diagram (left) consists of two intersecting circles, producing four regions
corresponding to membership in 
only, 
only, both 
and 
,
or neither. Here, 
denotes the intersection of sets 
and 
.
The order-three diagram (right) consists of three symmetrically placed mutually intersecting circles comprising
a total of eight regions. The regions
labeled 
,
, and 
consist of members which are only in one set
and no others, the three regions labelled 
, 
, and 
consist of members which are in two sets
but not the third, the region 
consists of members which are
simultaneously in all three, and the exterior region
represents membership in none of the sets.
In general, an order-
Venn diagram is a collection of 
Jordan curves in the plane,
any two of which intersect in finitely many points,
such that
1. The curves partition the plane into 
connected regions, and
2. Each subset 
of 
corresponds to a unique nonempty region formed by the
intersection of the interiors
of the curves in 
and the exteriors of the curves
not in 
(Ruskey and Weston 2005).
Since there are 
(the binomial coefficient) ways to pick 
members from a total of 
, the number of regions in an order-
Venn diagram is
 |
where the region outside the diagram is included in the count.
A Venn diagram is called simple if exactly two curves pass through each point of intersection.
It is called symmetric if its 
curves are successive images of
one curve under rotation through
radians
about a common point, giving 
-fold rotational symmetry
(Ruskey et al. 2006). Simplicity of the diagram is an additional condition
beyond simplicity of its individual Jordan curves.
For 
, a symmetric order-
Venn diagram exists iff 
is prime. Henderson (1963) gave
a proof of the necessity of primality, but his argument had a gap: it assumed without
justification that the regions other than the innermost
and outermost occur in orbits of size 
under rotation. Wagon and Webb
(2008) repaired this step using the Jordan curve
theorem. Griggs et al. (2004) proved the converse by constructing symmetric
Venn diagrams for every prime 
; their construction does not in general produce simple diagrams.
The region of intersection of the three circles 
in the order-three Venn diagram
in the special case of the center of each being located
at the intersection of the other two is a geometric
shape known as a Reuleaux triangle.
The left figure above shows an order-five Venn diagram due to Grünbaum (1975), while the sevenfold rosette in the middle is Ruskey's order-seven diagram called "Victoria" (Ruskey and Weston 2005). The right figure shows a non-simple symmetric order-11 Venn diagram drawn by Mark Weston (Ruskey et al. 2006, Fig. 2(c)).
Hamburger (2002) constructed the first symmetric order-11 Venn diagram, which was non-simple. The first simple symmetric order-11 Venn diagram was found by Mamakani and Ruskey (2012), who also constructed a simple symmetric order-13 diagram (Mamakani and Ruskey 2014). Dzoba (2026) constructed simple symmetric diagrams of orders 17 and 19, extending the previously known largest order of 13. His preprint supplies machine-checkable certificates for four order-17 and nine order-19 diagrams and reports formal verification in Lean 4 of one diagram of each order. According to Dzoba (2026), Claude (Anthropic) and Codex (OpenAI) developed the search and verification code under his direction.
In Season 4 episode "Power" of the television crime drama NUMB3RS, mathematical genius Charles Eppes constructs a Venn diagram to determine suspects who match a particular description and have a history of violence.
