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Introduction Minimum Area Expansion Omitting the Empty Set Summary Minimum Area Venn Diagrams Bette Bultena, Matthew Klimesh, Frank Ruskey University of Victoria & California Institute of Technology June 12, 2013 CanaDAM, St. Johns

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Introduction Minimum Area Expansion Omitting the Empty Set Summary

Minimum Area Venn Diagrams

Bette Bultena, Matthew Klimesh, Frank Ruskey

University of Victoria & California Institute of Technology

June 12, 2013 CanaDAM, St. John’s

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Introduction Minimum Area Expansion Omitting the Empty Set Summary

Outline

Introduction Venn diagrams Minimum Area Definition Small examples Expansion Sufficient conditions One by eight expansion Omitting the Empty Set Summary What we know What we don’t know

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Types of curves

Venn diagrams as circles

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Types of curves

Venn diagrams as ovals

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Types of curves

Venn diagrams as triangles

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Types of curves

$n = 4$ $n=2$ $n = 3$ $n = 5$ $n = 6$

Venn diagrams with minimum intersections

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Types of curves

A B C A B A A B B C C C

A B C A A A B B B C C C

polyomino Venn diagrams

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Venn diagram definition

  • A Venn diagram
  • is a collection of simple closed curves C = C1, C2, . . . , Cn

drawn on the plane

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Venn diagram definition

  • A Venn diagram
  • is a collection of simple closed curves C = C1, C2, . . . , Cn

drawn on the plane

  • such that each of the 2n sets X1 ∩ X2 ∩ . . . ∩ Xn is a

nonempty and connected region where Xi is either the bounded interior or unbounded exterior of Ci.

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Venn diagram definition

  • A Venn diagram
  • is a collection of simple closed curves C = C1, C2, . . . , Cn

drawn on the plane

  • such that each of the 2n sets X1 ∩ X2 ∩ . . . ∩ Xn is a

nonempty and connected region where Xi is either the bounded interior or unbounded exterior of Ci.

We call the second requirement of the statement the region property.

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Minimum area

What is a minimum area polyomino Venn diagram?

  • A diagram
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Minimum area

What is a minimum area polyomino Venn diagram?

  • A diagram
  • with polyominoes,
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Introduction Minimum Area Expansion Omitting the Empty Set Summary

Minimum area

What is a minimum area polyomino Venn diagram?

  • A diagram
  • with polyominoes,
  • where each set
  • i∈I

interior(Pi) ∩

  • i /

∈I

exterior(Pi) together with a base region of unit squares, is a single unit square,

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Minimum area

What is a minimum area polyomino Venn diagram?

  • A diagram
  • with polyominoes,
  • where each set
  • i∈I

interior(Pi) ∩

  • i /

∈I

exterior(Pi) together with a base region of unit squares, is a single unit square,

  • for all I ⊆ [n].
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Base regions that are rectangles

  • A (r, c)-polyVenn is a minimum area polyomino Venn

diagram confined to a 2r × 2c base rectangle.

A B A A A A B B B

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Base regions that are rectangles

  • A (r, c)-polyVenn is a minimum area polyomino Venn

diagram confined to a 2r × 2c base rectangle.

  • The number of polyominoes is n = r + c.

A B A A A A B B B

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Base regions that are rectangles

  • A (r, c)-polyVenn is a minimum area polyomino Venn

diagram confined to a 2r × 2c base rectangle.

  • The number of polyominoes is n = r + c.
  • The number of unit squares on the grid is 2n.

A B A A A A B B B

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Base regions that are rectangles

  • A (r, c)-polyVenn is a minimum area polyomino Venn

diagram confined to a 2r × 2c base rectangle.

  • The number of polyominoes is n = r + c.
  • The number of unit squares on the grid is 2n.

Example

little polyVenns:

A B A A A A B B B

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Are there any more single row polyVenns?

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Are there any more single row polyVenns?

Place the leftmost polyomino

A A A A

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Are there any more single row polyVenns?

Only one choice for the second polyomino

B A A B B A A B

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Are there any more single row polyVenns?

No place for the third

B A A B B A A B

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The known polyVenns

20 21 22 23 24 25 26 27 28 · · · 20 F F F × × × × × × · · · 21 F F ? ? ? ? ? ? · · · 22 ? ? ? ? ? ? ? · · · 23 ? ? ? ? ? ? · · · 24 ? ? ? ? ? · · · 25 ? ? ? ? · · · 26 ? ? ? · · · 27 ? ? · · · 28 ? · · · . . . ...

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A (1, 3)-polyVenn (symmetric)

D C B A D D D D D D D D C C C C C C C C B B B B B B B B A A A A A A A A

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A (1, 4)-polyVenn

A B C D E A A A BC A A A A A A A A A A A A A B B B B B B B B B B B B B B B C C C C C C C C C C C C C C C D D D D D D D D D D D D D D D D E E E E E E E E E E E E E E E E

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The known polyVenns

20 21 22 23 24 25 26 27 28 · · · 20 F F F × × × × × × · · · 21 F F F F ? ? ? ? · · · 22 ? ? ? ? ? ? ? · · · 23 ? ? ? ? ? ? · · · 24 ? ? ? ? ? · · · 25 ? ? ? ? · · · 26 ? ? ? · · · 27 ? ? · · · 28 ? · · · . . . ...

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We can expand when...

Suppose we have a 2r × 2c grid that holds an existing

  • polyVenn. Then we can expand this polyVenn to create one on

a 2r+r ′ × 2c+c′ grid if the following is true:

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We can expand when...

Suppose we have a 2r × 2c grid that holds an existing

  • polyVenn. Then we can expand this polyVenn to create one on

a 2r+r ′ × 2c+c′ grid if the following is true:

  • A non-empty set of mini-systems exist that meet the

required region property on small bounding rectangles of dimensions 2r ′ × 2c′.

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We can expand when...

Suppose we have a 2r × 2c grid that holds an existing

  • polyVenn. Then we can expand this polyVenn to create one on

a 2r+r ′ × 2c+c′ grid if the following is true:

  • A non-empty set of mini-systems exist that meet the

required region property on small bounding rectangles of dimensions 2r ′ × 2c′.

  • There is a way to connect elements of these mini-systems
  • n the large grid to create connected polyominoes.
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How it’s done

The first polyomino is created by expanding the existing polyVenn.

In this case each unit square is expanded into a 2 × 4 mini-grid.

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How it’s done

Then find a system that meets the region requirement on the mini-grid.

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Then lay these out

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Another known polyVenn

20 21 22 23 24 25 26 27 28 · · · 20 F F F × × × × × × · · · 21 F F F F ? ? ? ? · · · 22 E ? ? ? ? ? ? · · · 23 ? ? ? ? ? ? · · · 24 ? ? ? ? ? · · · 25 ? ? ? ? · · · 26 ? ? ? · · · 27 ? ? · · · 28 ? · · · . . . ...

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One by eight expansion

Theorem

If there is a (2, c)-polyVenn, then there is a (2, c + 3)-polyVenn.

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Find the systems

( ❑❑❏❏ , ▼▲▼▲ , ❑❏❏❑ ), ( ❏❏❑❑ , ▼▲▼▲ , ❑❏❏❑ ), ( ❏▼❑▲ , ❑❏❑❏ , ▼▲▼▲ ), ( ▼❑▲❏ , ❏❑❏❑ , ▼▲▼▲ ), ( ❏❑❏❑ , ▼❑▲❏ , ▼▲▼▲ ), ( ❑❏❑❏ , ❏▼❑▲ , ▼▲▼▲ ), ( ▼▲▼▲ , ❏❏❑❑ , ❏❑❑❏ ), ( ▼▲▼▲ , ❑❑❏❏ , ❏❑❑❏ ).

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Connect the mini-grids

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An expansion example

X(P3) X(P1) X(P4) X(P2) E1 E2 E3 P1 P2 P3 P4

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Some more polyVenns

20 21 22 23 24 25 26 27 28 · · · 20 F F F × × × × × × · · · 21 F F F F ? ? ? ? · · · 22 E ? ? A ? ? A · · · 23 ? ? ? ? ? ? · · · 24 ? ? ? ? ? · · · 25 ? ? ? ? · · · 26 ? ? ? · · · 27 ? ? · · · 28 ? · · · . . . ...

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Two by two expansion

Theorem

If there is a (r, c)-polyVenn, then there is a (r + 1, c + 1)-polyVenn.

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Find the systems

Dominoes

{ ❊ , ● } { ● , ❉ } { ❋ , ❊ } { ❉ , ❋ }

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Connect the minigrids

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Connect the minigrids

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Connect the minigrids

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Connect the minigrids

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Expand the original

  • Our existing polyVenn.
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Expand the original

  • Expanded.
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From the beginning

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From the beginning

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From the beginning

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From the beginning

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Another layout

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The known polyVenns

20 21 22 23 24 25 26 27 28 · · · 20 F F F × × × × × × · · · 21 F F F F ? ? ? ? · · · 22 E B B A A A A · · · 23 B B B B B B · · · 24 B B B B B · · · 25 B B B B · · · 26 B B B · · · 27 B B · · · 28 B · · · . . . ...

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Grids With 2n−1 unit squares

Theorem

PolyVenn diagrams where the empty set does not exist within the bounding rectangle, exist when within a dimension 2n/2−1 × 2n/2+1.

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Grids With 2n−1 unit squares

Theorem

PolyVenn diagrams where the empty set does not exist within the bounding rectangle, exist when within a dimension 2n/2−1 × 2n/2+1.

  • 1. Take the square version of the 2 × 2 expansion.
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Grids With 2n−1 unit squares

Theorem

PolyVenn diagrams where the empty set does not exist within the bounding rectangle, exist when within a dimension 2n/2−1 × 2n/2+1.

  • 1. Take the square version of the 2 × 2 expansion.
  • 2. Strip the first row of grids.
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Grids With 2n−1 unit squares

Theorem

PolyVenn diagrams where the empty set does not exist within the bounding rectangle, exist when within a dimension 2n/2−1 × 2n/2+1.

  • 1. Take the square version of the 2 × 2 expansion.
  • 2. Strip the first row of grids.
  • 3. Rotate it 90 degrees.
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Grids With 2n−1 unit squares

Theorem

PolyVenn diagrams where the empty set does not exist within the bounding rectangle, exist when within a dimension 2n/2−1 × 2n/2+1.

  • 1. Take the square version of the 2 × 2 expansion.
  • 2. Strip the first row of grids.
  • 3. Rotate it 90 degrees.
  • 4. Glue it to the end of the last column.
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Grids With 2n−1 unit squares

Theorem

PolyVenn diagrams where the empty set does not exist within the bounding rectangle, exist when within a dimension 2n/2−1 × 2n/2+1.

  • 1. Take the square version of the 2 × 2 expansion.
  • 2. Strip the first row of grids.
  • 3. Rotate it 90 degrees.
  • 4. Glue it to the end of the last column.
  • 5. Remove the top unit square, which is the square

representing the empty set.

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An example

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What we know

Lemma

A (0, c)-polyVenn does not exist for any c > 2.

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What we know

Lemma

A (0, c)-polyVenn does not exist for any c > 2.

Theorem

For every n > 0 and for every r, c > 2 where r + c = n, there exists a minimum area polyomino Venn diagram bound in a 2r × 2c bounding rectangle.

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Open problems

Conjecture

A (1, c)-polyVenn does not exist for c > 4.

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Open problems

Conjecture

A (1, c)-polyVenn does not exist for c > 4. Is there a construction for another infinite set of minimum area polyVenns where the empty set is omitted?

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Thank-you