New Roses: Simple Symmetric Venn Diagrams with 11 and 13 Curves
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Abstract
A symmetric \(n\)Venn diagram is one that is invariant under \(n\)fold rotation, up to a relabeling of curves. A simple \(n\)Venn diagram is an \(n\)Venn diagram in which at most two curves intersect at any point. In this paper, we introduce a new property of Venn diagrams called crosscut symmetry, which is related to dihedral symmetry. Utilizing a computer search restricted to diagrams with crosscut symmetry, we found many simple symmetric Venn diagrams with 11 curves. The question of the existence of a simple 11Venn diagram has been open since the 1960s. The technique used to find the 11Venn diagram is extended and a symmetric 13Venn diagram is also demonstrated.
Keywords
Venn diagram Crosscut symmetry Symmetric graphs Hypercube1 Introduction
Mathematically, an \(n\) Venn diagram is a collection of \(n\) simple closed (Jordan) curves in the plane with the following properties: (a) Each of the \(2^n\) different intersections of the open interiors or exteriors of all \(n\) curves is nonempty and connected; (b) there are only finitely many points where the curves intersect. If each of the intersections is of only two curves, then the diagram is said to be simple.
Venn diagrams can be viewed as plane graphs where the edges are those curve segments that lie between two adjacent intersection points. Furthermore, because the intersections of the interiors and exteriors in the definition are connected, they form the faces of this plane graph. The unbounded face that corresponds to the intersection of the exteriors of all curves is called the outermost face, and the face that corresponds to the intersection of the interiors of all curves is called the innermost face. A \(k\) face (\(0 \le k \le n\)) in an \(n\)Venn diagram is a face which is in the interior of exactly \(k\) curves. In a monotone Venn diagram, every \(k\)face is adjacent to at least one \((k1)\)face (if \(k>0\)) and it is also adjacent to at least one \((k+1)\)face (if \(k < n\)). Monotone Venn diagrams are precisely those that can be drawn with convex curves [1]. The diagrams under consideration in this paper are both monotone and simple.
An \(n\)Venn diagram is symmetric if it is left fixed (up to a relabeling of the curves) by a rotation of the plane by \(2\pi /n\) radians. Interest in symmetric Venn diagrams was initiated by Henderson [9] in a 1963 paper in which he showed that a symmetric \(n\)Venn diagram could not exist unless \(n\) is a prime number (see also [14]). It is easy to draw symmetric \(2\) and \(3\)Venn diagrams using circles as the curves, but it was not until 1975 that Grünbaum [6] published a simple symmetric \(5\)Venn diagram, one that could be drawn using ellipses. Some 20 years later, in 1992, simple symmetric \(7\)Venn diagrams were discovered independently by Grünbaum [7] and by Edwards [4]. The total number of nonisomorphic simple \(7\)Venn diagrams that are convexly drawable is also known [12].
The construction of a simple symmetric 11Venn diagram has eluded all previous efforts until now. We know of several futile efforts that involved either incorrect constructions, or unsuccessful computer searches. The main purpose of this paper is to demonstrate that there are simple symmetric \(11\) and \(13\)Venn diagrams.
It should be noted that if the diagrams are not constrained to be simple, then the problem has been solved. Hamburger [8] was the first to discover a (nonsimple) symmetric 11Venn diagram, and Griggs, Killian and Savage (GKS) [5] have shown how to construct symmetric, but highly nonsimple, \(n\)Venn diagrams for any prime \(n\). These constructions, in a sense, are maximally nonsimple since they involve points where all \(n\) curves intersect. Some progress toward “simplifying” the GKS construction is reported in [11], but their approach could never succeed in producing truly simple diagrams.
Part of the interest in Venn diagrams is due to the fact that their geometric dual graphs are planar spanning subgraphs of the hypercube; furthermore, if the Venn diagram is simple then the subgraph is maximal in the sense that every face is a quadrilateral. Symmetric drawings of Venn diagrams imply symmetric drawings of spanning subgraphs of the hypercube. Some recent work on finding symmetric structures embedded in the hypercube is reported in [10] and [3].
2 Crosscut Symmetry
We define a crosscut of a Venn diagram as a segment of a curve that extends from the innermost face to the outermost face and “cuts” (i.e., intersects) every other curve exactly once. Except for \(n=2\) and 3, where the symmetric 2Venn and 3Venn diagrams have 4 and 6 crosscuts, respectively. For \(n > 3\), a symmetric \(n\)Venn diagram either has \(n\) crosscuts or it has none. Referring to Fig. 1a, notice that each of the 7 curves has a single crosscut. Recall that in this paper all Venn diagrams are assumed to be simple and monotone.
Lemma 1
If \(n > 3\), then a symmetric \(n\)Venn diagram has at most one crosscut per curve.
Proof
A curve of a Venn diagram touches a face at most once [2]. In a monotone Venn diagram, every curve touches the outermost face and the innermost face exactly once [1]. Thus, a curve in any Venn diagram cannot have three or more crosscuts, because that curve would touch the outermost face (and the innermost face) at least twice. Now suppose that some curve \(C\) in an \(n\)Venn diagram \(V\) has two crosscuts. Then those crosscuts must start/finish at the same edge of \(C\) on the outer face, and finish/start at an edge of \(C\) on the innermost face of \(V\). Thus, curve \(C\) contains exactly \(2(n1)\) intersections with the other curves. If the Venn diagram \(V\) is symmetric, then there must be a total of \(n(n1)\) intersection points. On the other hand, a simple symmetric Venn diagram has exactly \(2^n2\) intersection points [13]. Since \(n(n1) = 2^n2\) has a solution for \(n = 1,2,3\), but not for \(n > 3\), the lemma is proved. \(\square \)
A clump in a Venn diagram is a collection of faces that is bounded by a simple closed path of edges. The size of a clump is the number of faces that it contains. Aside from the innermost face and the outermost face, a rotationally symmetric \(n\)Venn diagram can be partitioned into \(n\) congruent clumps, each of size \((2^n2)/n\); in this case, we call the clump a cluster—it is like a fundamental region for the rotation, but omitting the parts of the fundamental region corresponding to the full set and to the empty set. Referring again to Fig. 1a, a cluster has been shaded, and this cluster is redrawn in Fig. 1b. Notice that the cluster of Fig. 1b has a central shaded section, which has a reflective symmetry about the crosscut. The essential aspects of this reflective symmetry are embodied in the definition of crosscut symmetry.
Definition 1
Given a rotationally symmetric \(n\)Venn diagram \(V\), we label the curves as \(C_1, C_2, \ldots , C_n\) according to the clockwise order in which they touch the unbounded outermost face. Assume that the faces of \(V\) (except the innermost and outermost) can be partitioned into \(n\) clusters \(S_1, S_2, \ldots , S_n\) in such a way that each cluster \(S_k\) has the property that every curve intersects \(S_k\) in a segment and that \(S_k\) contains the crosscut for curve \(C_k\). Let \(L_{i,k}\) be the list of curves that we encounter as we follow \(C_i\) in the cluster \(S_k\) in clockwise order starting at the innermost face, and let \(\ell _{i,k}\) denote the length of \(L_{i,k}\). We say that \(V\) is crosscut symmetric if for every cluster \(S_k\), for any \(i \ne k\), the list \(L_{i,k}\) is palindromic; that is, for \(1 \le j \le \ell _{i,k}\), we have \(L_{i,k}[j] = L_{i,k}[\ell _{i,k}j+1]\).
Remark 1
If \(j,k\) is an adjacent pair in a crossing sequence \(\mathcal {C}\) and \(jk > 1\), then the sequence \(\mathcal {C'}\) obtained by replacing the pair \(j,k\) with the pair \(k,j\) is also a crossing sequence of the same diagram.
Theorem 1

\(\rho \) is \(1,3,2,5,4, \ldots , n2, n3\).

\(\delta \) is \(n1, n2, \ldots , 3, 2\).

\(\alpha \) and \(\alpha ^{r+}\) are two sequences of length \((2^{n1}(n1)^2)/n\) such that \(\alpha ^{r+}\) is obtained by reversing \(\alpha \) and adding \(1\) to each element; that is, \(\alpha ^{r+}[i]=\alpha [\alpha i+1]+1\).
Proof
Given a cluster \(S_i\) of a crosscut symmetric \(n\)Venn diagram where a segment of \(C_i\) is the crosscut, there are the same number of faces on both sides of the crosscut. Furthermore, let \(r\) be a \(k\)face in \(S_i\) that lies in the exterior of \(C_i\) and interior to the curves in some set \(\mathcal {K}\); then there is a corresponding \((k+1)\)face \(r'\) that is in the interior of \(C_i\) and also interior to the curves in \(\mathcal {K}\).
2.1 Simple Symmetric 11Venn Diagrams
By Theorem 1, if we have the subsequence \(\alpha \) of a crossing sequence, then we can construct the corresponding simple monotone crosscut symmetric \(n\)Venn diagram. Therefore, for small values of prime \(n\), an exhaustive search of \(\alpha \) sequences may give us possible crosscut symmetric \(n\)Venn diagrams. For example, for \(n=3\) and \(n=5\), the sequence \(\alpha \) is empty and the only possible cases are the three circles Venn diagram and Grünbaum 5ellipses. For \(n=7\), the valid cases of \(\alpha \) are \([3,2,4,3]\) and \([3,2,3,4]\). Our search algorithm is of the backtracking variety; for each possible case of \(\alpha \), we construct the crossing sequence \(S=\rho ,\alpha , \delta , \alpha ^{r+}\) checking along the way whether it currently satisfies the Venn diagram constraints, and then doing a final check of whether \(S\) represents a valid rotationally symmetric Venn diagram. Using this algorithm for \(n=11\), we found more than 200,000 simple monotone symmetric Venn diagrams, any one of which affirmatively settles the longstanding open question of the existence of a simple monotone symmetric 11Venn diagram.
3 Iterated Crosscut Symmetry
Definition 2
3.1 Simple Symmetric 13Venn Diagrams
It is natural to ask whether it is possible to find a simple symmetric 13Venn diagram that has iterated crosscut symmetry. The answer is yes; applying the backtracking search using \(\rho _\mathrm{E},\alpha _\mathrm{E},11,\delta _\mathrm{E}\) as a seed, we found more than 30,000 cases of simple symmetric 13Venn diagrams that also have iterated crosscut symmetry.
4 Final Thoughts
The \(\alpha \) sequence of the simple symmetric 13Venn diagram of Fig. 11
\(\begin{array}{ll} \alpha _{T}= &{} 3,2,4,\quad 3,5,4,3,2,4, \quad 3,5,4,6,5,4,3,5,\quad 4,6,5,7,6,5,4,3,2, \quad 3,4,\\ &{} 3,4,5,\quad 4,5,6,5,4,3, \quad 6,5,6,5,4,5,4,7,\quad 6,5,4,6,5,7,6,8,7, \quad 6,5,\\ &{} 4,3,7,\quad 8,6,7,5,6,7, \quad 8,5,6,5,6,7,6,7,\quad 4,5,6,7,6,5,6,5,4, \quad 5,4,\\ &{}9,8,7,\quad 6,5,4,3,2,3, \quad 4,3,4,5,4,5,6,5,\quad 4,3,5,4,6,5,4,5,6, \quad 7,6,\\ &{}5,4,5,\quad 6,5,6,7,6,5,\quad 6,7,6,7,8,7,6,5, \quad 4,3,5,4,6,5,7,6,5,\quad 4,6,\\ &{} 5,7,6,\quad 8,7,8,7,6,5,\quad 4,5,6,7,6,5,4,7, \quad 6,8,7,6,5,7,6,5,8,\quad 7,6,\\ &{}9,8,7,\quad 6,5,4,8,7,8,\quad 7,6,7,6,5,9,8,7, \quad 6,8,7,6,5,9,8,7,6,\quad 10,9,\\ &{}8,7,6,\quad 5,4,3,7,8,9,\quad 10,6,7,8,9,7,8,9,\,\,10,6,7,8,7,8,9,8,9, \,\,5,6,\\ &{}7,8,9,10,7,8,9,6,7,\,\,\quad 8,6,7,8,9,7,8,5, \quad 6,7,8,7,6,5,6,7,8,\quad 9,8,\\ &{}9,7,8,\quad 6,7,5,6,7,8, \quad 6,7,5,6,4,5,6,7, \quad 8,9,8,7,8,7,6,7,8,\quad 7,6,\\ &{}7,6,5, \quad 6,7,8,7,6,5, \quad 6,7,5,6,4,5,6,7, \quad 6,5,6,5,4,5,4\end{array}\) 
Conjecture 1
There is no simple Venn diagram with dihedral symmetry if \(n > 3\).
It is natural to wonder whether our method can be used to find a simple symmetric \(n\)Venn diagrams, for prime numbers \(n > 13\). The sizes of the \(\alpha \) sequences for \(n = 11,13,\) and \(17\) are 84,304, and 3,840, respectively. This tenfold increase in the size of the alpha sequence, and hence the depth of the backtracking tree, suggests that it is not surprising that our current methods fail for \(n = 17\).
One natural approach is to try to exploit further symmetries of the diagram. In the past, attention has focused on what is known as polar symmetry—in the cylindrical representation, this means that the diagram is not only rotationally symmetric but that it is also symmetric by reflection about a horizontal plane, followed by a rotation; an equivalent definition, which we use below is that there is an axis of rotation through the equator that leaves the diagram fixed. In the past, polar symmetry did not help in finding 11Venn diagrams (in fact, no polar symmetric Venn diagrams are yet known), but it is natural to wonder whether it might be fruitful to search for diagrams that are both polar symmetric and crosscut symmetric. However, the theorem proven below proves that there are no such diagrams for primes larger than 7.
Of the four faces incident to an intersection point in a simple Venn diagram, a pair of nonadjacent faces is in the interior of the same number of curves and the number of containing curves of the other two differs by two. Define a \(k\) point in a simple monotone Venn diagram to be an intersection point that is incident to two \(k\)faces (and thus also one \((k1)\)face and one \((k+1)\)face).
Lemma 2
Proof
Theorem 2
There is no monotone simple symmetric \(n\)Venn diagram with crosscut and polar symmetry for \(n > 7\).
Proof
Let \(V\) be a monotone simple symmetric \(n\)Venn diagram which has been drawn in the cylindrical representation with both polar and crosscut symmetry, and let \(S\) be a cluster of \(V\) with crosscut \(C\). Since the diagram is polar symmetric, \(S\) remains fixed under a rotation of \(\pi \) radians about some axis through the equator. Under the polar symmetry action, a crosscut must map to a crosscut, and since there are \(n\) crosscuts and \(n\) is odd, one crosscut must map to itself, and so one endpoint of this axis can be taken to be the central point, call it \(x\), of some crosscut (the other endpoint of the axis will then be midway between two crosscuts).
Notes
Acknowledgments
The authors are grateful to Mark Weston at the University of Victoria and to Rick Mabry at Louisiana State University in Shreveport for independently verifying that the symmetric 11Venn diagram illustrated in this paper is correct; thanks also to Wendy Myrvold for checking the correctness of the first 13Venn diagram that we found. Branko Grünbaum and Anthony Edwards are to be thanked for supplying us with historical background. Finally, we thank a referee for his/her thoughtful and useful suggestions on earlier versions of this paper.
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