Simple Symmetric Venn Diagrams with 11 and 13 Curves Khalegh - - PowerPoint PPT Presentation

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Simple Symmetric Venn Diagrams with 11 and 13 Curves Khalegh - - PowerPoint PPT Presentation

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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

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Simple Symmetric Venn Diagrams with 11 and 13 Curves

Khalegh Mamakani and Frank Ruskey

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Department of Computer Science, University of Victoria, Canada

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A closed curves divides the plane into two open subsets of the points.

 

WHAT IS A VENN DIAGRAM?

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WHAT IS A VENN DIAGRAM?

01 00 11 11 10 00

Regions are formed by the intersection of interior and exterior of the curves.

Rank of a region :

A binary number that indicates the curves containing the region.

Weight of a region :

The number of curves containing the region.

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100 010 001 101 110 011 111 000

Venn Diagram : There are exactly 2n regions where each region is in the interior

  • f a unique subset of the curves.

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No more than two curves intersect at any given point.

SIMPLE VENN DIAGRAMS

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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.

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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.

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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.

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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.

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ROTATIONAL SYMMETRY

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ROTATIONAL SYMMETRY

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Diagram remains invariant by a rotation of 2휋╱푛 radians about a centre point.

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ROTATIONAL SYMMETRY

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ROTATIONAL SYMMETRY

For any rotationally symmetric 푛-Venn diagram, 푛 must be prime (Henderson 1963).

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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).

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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

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REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS

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REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS

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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 2n − 2 columns.

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REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS

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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

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REPRESENTATION OF SIMPLE MONOTONE VENN DIAGRAMS

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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

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

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CROSSCUT SYMMETRY

Crosscut : A curve segment that sequentially crosses all other curves once.

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M4 (Ruskey) Hamilton (Edwards)

CROSSCUT SYMMETRY

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CROSSCUT SYMMETRY

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CROSSCUT SYMMETRY

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CROSSCUT SYMMETRY

Crosscut Symmetry : Reflective symmetry across the crosscut (ignoring the two regions at top and

bottom).

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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).

C5 intersections : [C4, C6, C3, C6, C4, C1, C4, C6, C3, C6, C4]

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CROSSCUT SYMMETRY THEOREM

Crossing sequence of the 7-V

  • ssing sequence of the 7-V
  • ssing sequence of the 7-Venn
  • ssing sequence of the 7-Venn

ρ α δ αr+ 1, 3, 2, 5, 4, 3, 2, 3, 4, 6, 5, 4, 3, 2, 5, 4, 3, 4

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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, αr+ where

  • ρ is 1, 3, 2, 5, 4, . . . , n − 2, n − 3 and δ is n − 1, n − 2, . . . , 3, 2.
  • |α| = |αr+| = (2n−1 − (n − 1)2)/n and α[i] ∈ {2, . . . , n − 3}.
  • αr+ is obtained by reversing α and incrementing each element by 1.

CROSSCUT SYMMETRY THEOREM

Crossing sequence of the 7-V

  • ssing sequence of the 7-V
  • ssing sequence of the 7-Venn
  • ssing sequence of the 7-Venn

ρ α δ αr+ 1, 3, 2, 5, 4, 3, 2, 3, 4, 6, 5, 4, 3, 2, 5, 4, 3, 4

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The First Simple Symmetric 11-Venn Diagram

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The First Simple Symmetric 11-Venn Diagram

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THE FIRST SIMPLE SYMMETRIC 11-VENN DIAGRAM

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.

α sequence :

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ITERATED CROSSCUT SYMMETRY

Simple Symmetric 7-Venn Diagram Hamilton (Edwards)

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ITERATED CROSSCUT SYMMETRY

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A simple symmetric 13-Venn diagram with iterated crosscut symmetry

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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

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α sequence of the simple symmetric 13-Venn :

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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

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α sequence of the simple symmetric 13-Venn :

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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

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α sequence of the simple symmetric 13-Venn :

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Thank you!

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