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Simple Symmetric Venn Diagrams with 11 and 13 Curves Khalegh Mamakani and Frank Ruskey Department of Computer Science, University of Victoria, Canada 1 Tuesday, 9 July, 13 WHAT IS A VENN DIAGRAM? A closed curves divides the plane into two

  1. Simple Symmetric Venn Diagrams with 11 and 13 Curves Khalegh Mamakani and Frank Ruskey Department of Computer Science, University of Victoria, Canada 1 Tuesday, 9 July, 13

  2. WHAT IS A VENN DIAGRAM? A closed curves divides the plane into two open subsets of the points.   2 Tuesday, 9 July, 13

  3. WHAT IS A VENN DIAGRAM? Regions are formed by the intersection of interior and exterior of the curves. 00 Rank of a region : 01 A binary number that indicates the curves containing the region. 11 00 11 Weight of a region : The number of curves containing the region. 10 3 Tuesday, 9 July, 13

  4. 000 100 101 110 111 011 001 010 Venn Diagram : There are exactly 2 n regions where each region is in the interior of a unique subset of the curves. 4 Tuesday, 9 July, 13

  5. SIMPLE VENN DIAGRAMS No more than two curves intersect at any given point. 5 Tuesday, 9 July, 13

  6. MONOTONE VENN DIAGRAMS Every region of weight k is adjacent to at least one region of weight k+1 and is also adjacent to at least one region of weight k-1. 6 Tuesday, 9 July, 13

  7. MONOTONE VENN DIAGRAMS Every region of weight k is adjacent to at least one region of weight k+1 and is also adjacent to at least one region of weight k-1. 6 Tuesday, 9 July, 13

  8. MONOTONE VENN DIAGRAMS Every region of weight k is adjacent to at least one region of weight k+1 and is also adjacent to at least one region of weight k-1. 6 Tuesday, 9 July, 13

  9. MONOTONE VENN DIAGRAMS Every region of weight k is adjacent to at least one region of weight k+1 and is also adjacent to at least one region of weight k-1. 6 Tuesday, 9 July, 13

  10. ROTATIONAL SYMMETRY 7 Tuesday, 9 July, 13

  11. ROTATIONAL SYMMETRY Diagram remains invariant by a rotation of 2 휋 ╱ 푛 radians about a centre point. 7 Tuesday, 9 July, 13

  12. ROTATIONAL SYMMETRY 8 Tuesday, 9 July, 13

  13. ROTATIONAL SYMMETRY For any rotationally symmetric 푛 -Venn diagram, 푛 must be prime (Henderson 1963). 8 Tuesday, 9 July, 13

  14. ROTATIONAL SYMMETRY For any rotationally symmetric 푛 -Venn diagram, 푛 must be prime (Henderson 1963). The existence of non-simple symmetric Venn diagrams has been proved for any prime number of curves (Griggs, Killian and Savage 2004). 8 Tuesday, 9 July, 13

  15. ROTATIONAL SYMMETRY For any rotationally symmetric 푛 -Venn diagram, 푛 must be prime (Henderson 1963). The existence of non-simple symmetric Venn diagrams has been proved for any prime number of curves (Griggs, Killian and Savage 2004). The largest prime number for which a simple symmetric Venn diagram was known : 7 8 Tuesday, 9 July, 13

  16. REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS 9 Tuesday, 9 July, 13

  17. REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS 9 Tuesday, 9 July, 13

  18. REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 A binary matrix of n − 1 rows and 2 n − 2 columns. 9 Tuesday, 9 July, 13

  19. REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 10 Tuesday, 9 July, 13

  20. REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 Crossing Sequence (Wendy Myrvold) : 1, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 2 10 Tuesday, 9 July, 13

  21. CROSSCUT SYMMETRY Crosscut : A curve segment that sequentially crosses all other curves once. 11 Tuesday, 9 July, 13

  22. CROSSCUT SYMMETRY M4 (Ruskey) Hamilton (Edwards) 12 Tuesday, 9 July, 13

  23. CROSSCUT SYMMETRY 13 Tuesday, 9 July, 13

  24. CROSSCUT SYMMETRY 13 Tuesday, 9 July, 13

  25. CROSSCUT SYMMETRY Crosscut Symmetry : Reflective symmetry across the crosscut (ignoring the two regions at top and bottom). 13 Tuesday, 9 July, 13

  26. CROSSCUT SYMMETRY Crosscut Symmetry : Reflective symmetry across the crosscut (ignoring the two regions at top and bottom). Curve intersections are palindromic (except for the crosscut). C 5 intersections : [ C 4 , C 6 , C 3 , C 6 , C 4 , C 1 , C 4 , C 6 , C 3 , C 6 , C 4 ] 13 Tuesday, 9 July, 13

  27. CROSSCUT SYMMETRY THEOREM Crossing sequence of the 7-V ossing sequence of the 7-V ossing sequence of the 7-Venn ossing sequence of the 7-Venn ρ α δ α r+ 1, 3, 2, 5, 4, 3, 2, 3, 4, 6, 5, 4, 3, 2, 5, 4, 3, 4 14 Tuesday, 9 July, 13

  28. CROSSCUT SYMMETRY THEOREM Crossing sequence of the 7-V ossing sequence of the 7-Venn ossing sequence of the 7-Venn ossing sequence of the 7-V ρ α δ α r+ 1, 3, 2, 5, 4, 3, 2, 3, 4, 6, 5, 4, 3, 2, 5, 4, 3, 4 A simple monotone rotationally symmetric n -Venn diagram is crosscut-symmetric if and only if it can be represented by a crossing sequence of the form ρ , α , δ , α r + where • ρ is 1 , 3 , 2 , 5 , 4 , . . . , n − 2 , n − 3 and δ is n − 1 , n − 2 , . . . , 3 , 2 . • | α | = | α r + | = (2 n − 1 − ( n − 1) 2 ) /n and α [ i ] ∈ { 2 , . . . , n − 3 } . • α r + is obtained by reversing α and incrementing each element by 1. 14 Tuesday, 9 July, 13

  29. The First Simple Symmetric 11-Venn Diagram 15 Tuesday, 9 July, 13

  30. The First Simple Symmetric 11-Venn Diagram 15 Tuesday, 9 July, 13

  31. THE FIRST SIMPLE SYMMETRIC 11-VENN DIAGRAM α sequence : 3, 2, 3, 4, 3, 4, 5, 4, 3, 2, 3, 4, 3, 4, 5, 4, 3, 4, 5, 4, 5, 6, 5, 4, 5, 6, 5, 6, 7, 6, 5, 4, 3, 2, 5, 4, 3, 4, 6, 5, 4, 5, 6, 7, 6, 7, 8, 7, 6, 5, 6, 5, 4, 3, 4, 5, 7, 6, 5, 4, 6, 5, 8, 7, 6, 5, 4, 5, 7, 6, 5, 6, 8, 7, 6, 5, 4, 6, 5, 7, 6, 5, 6, 7. 16 Tuesday, 9 July, 13

  32. ITERATED CROSSCUT SYMMETRY Simple Symmetric 7-Venn Diagram Hamilton (Edwards) 17 Tuesday, 9 July, 13

  33. ITERATED CROSSCUT SYMMETRY 17 Tuesday, 9 July, 13

  34. A simple symmetric 13-Venn diagram with iterated crosscut symmetry 18 Tuesday, 9 July, 13

  35. 18 Tuesday, 9 July, 13

  36. α sequence of the simple symmetric 13-Venn : 3 2 4 3 5 4 3 2 4 3 5 4 6 5 4 3 5 4 6 5 7 6 5 4 3 2 3 4 3 4 5 4 5 6 5 4 3 6 5 6 5 4 5 4 7 6 5 4 6 5 7 6 8 7 6 5 4 3 7 8 6 7 5 6 7 8 5 6 5 6 7 6 7 4 5 6 7 6 5 6 5 4 5 4 9 8 7 6 5 4 3 2 3 4 3 4 5 4 5 6 5 4 3 5 4 6 5 4 5 6 7 6 5 4 5 6 5 6 7 6 5 6 7 6 7 8 7 6 5 4 3 5 4 6 5 7 6 5 4 6 5 7 6 8 7 8 7 6 5 4 5 6 7 6 5 4 7 6 8 7 6 5 7 6 5 8 7 6 9 8 7 6 5 4 8 7 8 7 6 7 6 5 9 8 7 6 8 7 6 5 9 8 7 6 10 9 8 7 6 5 4 3 7 8 9 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 8 6 7 8 9 7 8 5 6 7 8 7 6 5 6 7 8 9 8 9 7 8 6 7 5 6 7 8 6 7 5 6 4 5 6 7 8 9 8 7 8 7 6 7 8 7 6 7 6 5 6 7 8 7 6 5 6 7 5 6 4 5 6 7 6 5 6 5 4 5 4 19 Tuesday, 9 July, 13

  37. α sequence of the simple symmetric 13-Venn : 3 2 4 3 5 4 3 2 4 3 5 4 6 5 4 3 5 4 6 5 7 6 5 4 3 2 3 4 3 4 5 4 5 6 5 4 3 6 5 6 5 4 5 4 7 6 5 4 6 5 7 6 8 7 6 5 4 3 7 8 6 7 5 6 7 8 5 6 5 6 7 6 7 4 5 6 7 6 5 6 5 4 5 4 9 8 7 6 5 4 3 2 3 4 3 4 5 4 5 6 5 4 3 5 4 6 5 4 5 6 7 6 5 4 5 6 5 6 7 6 5 6 7 6 7 8 7 6 5 4 3 5 4 6 5 7 6 5 4 6 5 7 6 8 7 8 7 6 5 4 5 6 7 6 5 4 7 6 8 7 6 5 7 6 5 8 7 6 9 8 7 6 5 4 8 7 8 7 6 7 6 5 9 8 7 6 8 7 6 5 9 8 7 6 10 9 8 7 6 5 4 3 7 8 9 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 8 6 7 8 9 7 8 5 6 7 8 7 6 5 6 7 8 9 8 9 7 8 6 7 5 6 7 8 6 7 5 6 4 5 6 7 8 9 8 7 8 7 6 7 8 7 6 7 6 5 6 7 8 7 6 5 6 7 5 6 4 5 6 7 6 5 6 5 4 5 4 19 Tuesday, 9 July, 13

  38. α sequence of the simple symmetric 13-Venn : 3 2 4 3 5 4 3 2 4 3 5 4 6 5 4 3 5 4 6 5 7 6 5 4 3 2 3 4 3 4 5 4 5 6 5 4 3 6 5 6 5 4 5 4 7 6 5 4 6 5 7 6 8 7 6 5 4 3 7 8 6 7 5 6 7 8 5 6 5 6 7 6 7 4 5 6 7 6 5 6 5 4 5 4 9 8 7 6 5 4 3 2 3 4 3 4 5 4 5 6 5 4 3 5 4 6 5 4 5 6 7 6 5 4 5 6 5 6 7 6 5 6 7 6 7 8 7 6 5 4 3 5 4 6 5 7 6 5 4 6 5 7 6 8 7 8 7 6 5 4 5 6 7 6 5 4 7 6 8 7 6 5 7 6 5 8 7 6 9 8 7 6 5 4 8 7 8 7 6 7 6 5 9 8 7 6 8 7 6 5 9 8 7 6 10 9 8 7 6 5 4 3 7 8 9 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 8 6 7 8 9 7 8 5 6 7 8 7 6 5 6 7 8 9 8 9 7 8 6 7 5 6 7 8 6 7 5 6 4 5 6 7 8 9 8 7 8 7 6 7 8 7 6 7 6 5 6 7 8 7 6 5 6 7 5 6 4 5 6 7 6 5 6 5 4 5 4 19 Tuesday, 9 July, 13

  39. Thank you! 20 Tuesday, 9 July, 13

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