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New Roses: Simple Symmetric Venn Diagrams with 11 and 13 Curves
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 11-Venn diagram has been open since the 1960s. The technique used to find the 11-Venn diagram is extended and a symmetric 13-Venn diagram is also demonstrated.
... Given the intricate nature of presenting a holistic picture of the interplay between antecedents and outcomes in a single diagram, the proposed research model used a Venn diagram, adapted from Mamakani and Ruskey's (2014) work, to depict the study's two main components: the causal factors (i.e., 10 constructs) and the desired outcome (i.e., overall satisfaction and destination behavioral loyalty). The study aimed to examine the causal factors' impact on this outcome at three levels: (a) overall satisfaction, (b) destination behavioral loyalty, and (c) the combination of overall satisfaction and destination behavioral loyalty (See Figure 1). ...
Prior studies have identified the complexity involved in explaining the relationship between wellness tourism attributes and tourist behavior. This study investigated the configurational effects of 10 constructs in predicting the ambiguous area between overall satisfaction and destination behavioral loyalty. To achieve this, the study used a fuzzy set qualitative comparative analysis (fsQCA) and necessary condition analysis (NCA). The results of the fsQCA generated a range of models that effectively predicted tourists’ high or low levels of overall satisfaction and behavioral loyalty. Furthermore, the NCA results identified perceived quality as an essential construct for achieving the combination of overall satisfaction and behavioral loyalty. The study enhances our current knowledge by broadening our comprehension of the intricate aspects of tourists’ decision-making processes and the formation of their behaviors. Additionally, it provides a roadmap for the industry to effectively address challenges in managing wellness tourism resources.
... According to the binomial theorem, we have The term 1 n n−1 k + (−1) k+1 assumes only positive integer values for and odd prime n. A proof can be found in Mamakani [2]. The OEIS reference for these values is A219539 [3]. ...
This paper presents a concise and elementary proof of Fermat's Last Theorem, which
asserts that the Diophantine equation x^n +y^n =z^n has no non-trivial solutions for
any integer n greater than 2. The proof outlined in this work relies solely on elementary
mathematical methods, providing a clear and accessible demonstration of the
theorem's validity. By employing fundamental mathematical principles, we illuminate
the impossibility of non-trivial solutions, thereby contributing to the comprehension of this f
amous conjecture.
... This influences the insertion strategy so that CO sequences are preferred that have fewer children while still maximizing monotone faces. It is still an open problem to find simple symmetric Venn diagrams for any prime number of sets, and the largest to be produced is a 13-Venn diagram [27]. ...
Creating comprehensible visualizations of highly overlapping set-typed data is a challenging task due to its complexity. To facilitate insights into set connectivity and to leverage semantic relations between intersections, we propose a fast two-step layout technique for Euler diagrams that are both well-matched and well-formed. Our method conforms to established form guidelines for Euler diagrams regarding semantics, aesthetics, and readability. First, we establish an initial ordering of the data, which we then use to incrementally create a planar, connected, and monotone dual graph representation. In the next step, the graph is transformed into a circular layout that maintains the semantics and yields simple Euler diagrams with smooth curves. When the data cannot be represented by simple diagrams, our algorithm always falls back to a solution that is not well-formed but still well-matched, whereas previous methods often fail to produce expected results. We show the usefulness of our method for visualizing set-typed data using examples from text analysis and infographics. Furthermore, we discuss the characteristics of our approach and evaluate our method against state-of-the-art methods.
... Do simple, symmetric Venn diagrams with n curves exist for every prime number n? See Ruskey, Savage, and Wagon for a 2006 survey[21]. More recently, Mamakani and Ruskey gave the first examples of simple, symmetric Venn diagrams with n = 11 and n = 13 curves[18]. ...
This survey article collects a few of my favorite open problems of Branko Gr\"unbaum.
... The term 1 n n−1 k + (−1) k+1 assumes only positive integer values for and odd prime n. A proof can be found in Mamakani [3]. The OEIS reference for these values is A219539 [5]. ...
Fermat's Last Theorem states that the Diophantine equation X^n + Y^n = Z^n has no non-trivial solution for any n greater than 2. In this paper we give an approach to a brief and simple proof of the theorem using only elementary methods.
... is the number of q-points on the left side of a crosscut of simple symmetric p-Venn diagram [12]. This integer sequence is known as A219539 sequence in [13]. ...
We consider some class of the sums which naturally include the sums of powers of integers. We suggest a number of conjectures concerning a representation of these sums.
Creating comprehensible visualizations of highly overlapping set-typed data is a challenging task due to its complexity. To facilitate insights into set connectivity and to leverage semantic relations between intersections, we propose a fast two-step layout technique for Euler diagrams that are both well-matched and well-formed. Our method conforms to established form guidelines for Euler diagrams regarding semantics, aesthetics, and readability. First, we establish an initial ordering of the data, which we then use to incrementally create a planar, connected, and monotone dual graph representation. In the next step, the graph is transformed into a circular layout that maintains the semantics and yields simple Euler diagrams with smooth curves. When the data cannot be represented by simple diagrams, our algorithm always falls back to a solution that is not well-formed but still well-matched, whereas previous methods often fail to produce expected results. We show the usefulness of our method for visualizing set-typed data using examples from text analysis and infographics. Furthermore, we discuss the characteristics of our approach and evaluate our method against state-of-the-art methods.
We show a Brunnian link whose minimal projection is a simple symmetric Venn diagram of order 7, solving a problem that first appeared in 1998. The Venn diagram itself was discovered by Branko Grünbaum in 1992.
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums are made.
Using graph theory, we develop procedures for the construction of Venn diagrams. This allows us to determine the number of Venn diagrams on three sets, and to address further questions on enumeration of Venn diagrams. In so doing, we obtain examples of Venn diagrams which yield answers to several problems and conjectures of Grünbaum.
Using graph theory, we develop procedures for the construction of Venn diagrams. This allows us to determine the number of Venn diagrams on three sets, and to address further questions on enumeration of Venn diagrams. In so doing, we obtain examples of Venn diagrams which yield answers to several problems and conjectures of Grünbaum.
Canfield and Mason have conjectured that for all subgroups G of the automorphism group of the Boolean lattice B n (which can be regarded as the symmetric group S n ), the quotient order B n /G is a symmetric chain order. We provide a straightforward proof of a generalization of a result of K. K. Jordan: namely, B n /G is an SCO whenever G is generated by powers of disjoint cycles. In addition, the Boolean lattice B n can be replaced by any product of finite chains. The symmetric chain decompositions of Greene and Kleitman provide the basis for partitions of these quotients.
In this paper we are concerned with producing exhaustive lists of simple monotone Venn diagrams that have some symmetry (non-trivial isometry) when drawn on the sphere. A diagram is simple if at most two curves intersect at any point, and it is monotone if it has some embedding on the plane in which all curves are convex. We show that there are 23 such 7-Venn diagrams with a 7-fold rotational symmetry about the polar axis, and that 6 of these have an additional 2-fold rotational symmetry about an equatorial axis. In the case of simple monotone 6-Venn diagrams, we show that there are 39 020 non-isomorphic planar diagrams in total, and that 375 of them have a 2-fold symmetry by rotation about an equatorial axis, and amongst these we determine all those that have a richer isometry group on the sphere. Additionally, 270 of the 6-Venn diagrams also have the 2-fold symmetry induced by reflection about the center of the sphere.Since such exhaustive searches are prone to error, we have implemented the search in a couple of ways, and with independent programs. These distinct algorithms are described. We also prove that the Grünbaum encoding can be used to efficiently identify any monotone Venn diagram.
We discuss the proof of the theorem of D. W. Henderson [Am. Math. Mon. 70, No. 4, 424–426 (1963)] which states that symmetric n-Venn diagrams only exist when n is prime, pointing out what appears to be the omission in the literature of part of the argument, and providing a proof of the missing step.
A FISC or family of intersecting simple closed curves is a collection of simpleclosed curves in the plane with the properties that there is some open region commonto the interiors of all the curves, and that every two curves intersect in finitely manypoints.Let F be a FISC. Intersections of the curves represent the vertices of a plane graph,G(F), whose edges are the curve arcs between vertices. The directed dual of G(F),denoted~D(F), is the dual graph of G(F), but with edges...
A class of completely symmetric simple Venn diagrams for seven sets is constructed and displayed.
A Venn diagram is simple if at most two curves intersect at any given point. A recent paper of Griggs, Killian, and Savage (Elec. J. Combinatorics, Research Pa- per 2, 2004) shows how to construct rotationally symmetric Venn diagrams for any prime number of curves. However, the resulting diagrams contain only n bn=2c inter- section points, whereas a simple Venn diagram contains 2n 2 intersection points. We show how to modify their construction to give symmetric Venn diagrams with asymptotically at least 2n 1 intersection points, whence the name \half-simple."
In this paper we create and implement a method to construct a non-simple, symmetric
11-Venn diagram. By doing this we answer a question of Grunbaum. Daniel Kleitman's
mathematics seems to be remote from this area, but in fact, his results inspired this
solution, which holds promise of settling the general case for all prime numbers, and
the promise to nd simple doilies with 11 and more curves as well.
Let Nn denote the quotient poset of the Boolean lattice, Bn, under the relation equivalence under rotation. Griggs, Killian, and Savage proved that Np is a symmetric chain order for prime p. In this paper, we settle the question posed in that paper, namely whether Nn is a symmetric chain order for all n. This paper provides an algorithm that produces a symmetric chain decomposition (or SCD). We accomplish this by modifying bracketing from Greene and Kleitman. This allows us to take appropriate “middles” of certain chains from the Greene–Kleitman SCD for Bn. We also prove additional properties of the resulting SCD and show that this settles a related conjecture.
We show that symmetric Venn diagrams for n sets exist for every prime n, settling an open question. Until this time, n = 11 was the largest prime for which the existence of such diagrams had been proven, a result of Peter Hamburger. We show that the problem can be reduced to nding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice B n , and prove that such decompositions exist for all prime n. A consequence of the approach is a constructive proof that the quotient poset of B n , under the relation equivalence under rotation", has a symmetric chain decomposition whenever n is prime. We also show how symmetric chain decompositions can be used to construct, for all n, monotone Venn diagrams with the minimum number of vertices, giving a simpler existence proof.
In this paper we show that symmetric Venn diagrams for n sets exist for every prime n, settling an open question. Until this time, n = 11 was the largest prime for which the existence of such diagrams had been proven. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice n , and prove that such decompositions exist for all prime n. A consequence of the construction is that the quotient poset of n , under the relation "equivalence under rotation", has a symmetric chain decomposition whenever n is prime. We also show how SCDs can be used to construct, for all n, monotone Venn diagrams with the minimum number of vertices, giving a simpler existence proof. 1 Venn Diagrams For a survey of results on symmetric Venn diagrams, as well as the constuction of the first known symmetric Venn diagram for n = 11 sets see [Ham]. A Venn diagram for n sets (def. from Grunbaum? "Venn Diagrams and Independent Families of Sets", Mathematics Magazine, 48 (Jan-Feb 1975) 12-23.) is a collection of n simple closed curves C 1 ,C 2 ,...,C n in the plane with the property that for any S 2,...,n}, the region # cekillian@unity.ncsu.edu + savage@csc.ncsu.edu # Research supported by NSA grants MDA 904-01-0-0083 is nonempty and connected. A region of the Venn diagram is a maximal connected subset of R i=1 C i , where R denotes the set of all points in the plane. Thus, a Venn diagram partitions R C i into exactly 2 regions, one for each subset of 2,...,n}.Itis known that Venn diagrams for n-sets exist for all n 1. In fact the recursive construction of [?] yields simple Venn diagrams for all n.