Introduction
Minimum Area
Expansion
Omiting the Empty Set
Minimum Area Venn Diagrams
Bette Bultena
Department of Computer Science
University of Victoria
May 2009
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Outline
Basic Definitions
Extending a Venn Diagram
Using Polyominoes
Minimum Area
Minimum Area Venn Diagram Definition
Drawing the PolyVenn
Expansion
Half Set Systems
Sufficient Conditions
PolyVenns for all n
Omiting the Empty Set
Summary
What We Know
What We Don’t Know
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Basic Definitions
• A set of n closed curves.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Basic Definitions
• A set of n closed curves.
• Overlapping each other at single points.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Basic Definitions
• A set of n closed curves.
• Overlapping each other at single points.
• Dividing the plane into 2n regions.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Basic Definitions
• A set of n closed curves.
• Overlapping each other at single points.
• Dividing the plane into 2n regions.
• Each region is a unique subset of the interior of the curves.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Venn Diagram Examples
Venn diagrams as circles
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Venn Diagram Examples
Venn diagrams as ellipses
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Venn Diagram Examples
Venn diagrams as triangles
Thanks to Jeremy Carroll
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Introduction
Minimum Area
Expansion
Omiting the Empty Set
Venn Diagram Examples
Venn diagrams with minimum vertices.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Venn Diagrams for any Number of Curves
We can take any Venn diagram and extend it to another one by
finding a suitable curve.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Venn Diagrams for any Number of Curves
We can take any Venn diagram and extend it to another one by
finding a suitable curve.
• It must pass through each existing region exactly once.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Venn Diagrams for any Number of Curves
We can take any Venn diagram and extend it to another one by
finding a suitable curve.
• It must pass through each existing region exactly once.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Venn Diagrams for any Number of Curves
We can take any Venn diagram and extend it to another one by
finding a suitable curve.
• It must pass through each existing region exactly once.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Venn Diagrams for any Number of Curves
We can take any Venn diagram and extend it to another one by
finding a suitable curve.
• It must pass through each existing region exactly once.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Venn Diagrams for any Number of Curves
We can take any Venn diagram and extend it to another one by
finding a suitable curve.
• It must pass through each existing region exactly once.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Venn Diagrams for any Number of Curves
We can take any Venn diagram and extend it to another one by
finding a suitable curve.
• It must pass through each existing region exactly once.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Polyominoes as Curves
We consider using polyominoes whose perimeters represent
the curves.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Polyominoes as Curves
We consider using polyominoes whose perimeters represent
the curves.
Polyomino A simply connected set of unit squares on the
plane
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Polyominoes as Curves
We consider using polyominoes whose perimeters represent
the curves.
Polyomino A simply connected set of unit squares on the
plane
Domino Two unit squares.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Polyominoes as Curves
We consider using polyominoes whose perimeters represent
the curves.
Polyomino A simply connected set of unit squares on the
plane
Domino Two unit squares.
Triomino Three unit squares.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Polyominoes as Curves
We consider using polyominoes whose perimeters represent
the curves.
Polyomino A simply connected set of unit squares on the
plane
Domino Two unit squares.
Triomino Three unit squares.
Tetromino Four unit squares.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Minimum Area
What would a Minimum Area Venn diagram using polyominoes
look like?
• The perimeters of the polyominoes would intersect along
horizontal or vertical units.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Minimum Area
What would a Minimum Area Venn diagram using polyominoes
look like?
• The perimeters of the polyominoes would intersect along
horizontal or vertical units.
• Each unique region would take up one unit square.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Minimum Area
What would a Minimum Area Venn diagram using polyominoes
look like?
• The perimeters of the polyominoes would intersect along
horizontal or vertical units.
• Each unique region would take up one unit square.
Questions we answer:
• Can the polyomino Venn diagram (polyVenn) be confined
to a rectangular grid?
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Minimum Area
What would a Minimum Area Venn diagram using polyominoes
look like?
• The perimeters of the polyominoes would intersect along
horizontal or vertical units.
• Each unique region would take up one unit square.
Questions we answer:
• Can the polyomino Venn diagram (polyVenn) be confined
to a rectangular grid?
• If so, are there limits on the grid dimensions?
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Minimum Area
What would a Minimum Area Venn diagram using polyominoes
look like?
• The perimeters of the polyominoes would intersect along
horizontal or vertical units.
• Each unique region would take up one unit square.
Questions we answer:
• Can the polyomino Venn diagram (polyVenn) be confined
to a rectangular grid?
• If so, are there limits on the grid dimensions?
• Do polyVenns exist for all values of n?
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Minimum Area
What would a Minimum Area Venn diagram using polyominoes
look like?
• The perimeters of the polyominoes would intersect along
horizontal or vertical units.
• Each unique region would take up one unit square.
Questions we answer:
• Can the polyomino Venn diagram (polyVenn) be confined
to a rectangular grid?
• If so, are there limits on the grid dimensions?
• Do polyVenns exist for all values of n?
YES.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
PolyVenn Diagrams
AB
D
DE
D
C
EF D F
B
D F
AB
D F
B
DE F
BC
D F
C
C
BC
D
EF D F D
B
DE
AB
DE
AB
DE F
C
BC
BC
DE F DE
DE
AB
EF
AB C AB C AB C AB C
E F DE F DE
D
AB
F
B
EF
BC
C
F
C
A C A
D
D F
A
D
AB
AB
E
B
E
C
E
C
B C AB C A C AB C
DE F D F DE F DE F
E
E
AB C A C A
D F DE F
F
BC
B
EF
E
AB C A C A C A C A
F DE
F
EF
F
BC A
F DE
A
B
A
B
F
D
C A C
A
EF
E
A
E
A
B
C
D
E
F
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
PolyVenn Diagrams
AB
AB
D
DE
D
C
EF D F
D F
AB
D F
B
DE F
BC
D F
BC
C
C
EF D F D
D
B
DE
AB
DE
AB
DE F
BC
BC
C
DE F DE
DE
AB
EF
AB C AB C AB C AB C
E F DE F DE
D
F
B
B
EF
BC
C
F
C
A C A
D
D F
A
D
AB
AB
E
B
E
C
E
B C AB C A C AB C
C
E
E
DE F D F DE F DE F
AB C A C A
D F DE F
F
BC
B
EF
E
AB C A C A C A C A
F DE
EF
F
F
BC A
F DE
A
B
A
B
F
D
A
EF
A
E
C A C
E
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two ways to discover the polyVenn
1. The heuristic method:
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two ways to discover the polyVenn
1. The heuristic method:
• Fill each of the 2n subsets into the squares one by one.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two ways to discover the polyVenn
1. The heuristic method:
• Fill each of the 2n subsets into the squares one by one.
• Watch for disconnected elements and holes.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two ways to discover the polyVenn
1. The heuristic method:
• Fill each of the 2n subsets into the squares one by one.
• Watch for disconnected elements and holes.
2. The algorithmic method:
• Cover 2n−1 unit squares with a simply connected piece
while overlapping each previous piece on 2n−2 squares.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two ways to discover the polyVenn
1. The heuristic method:
• Fill each of the 2n subsets into the squares one by one.
• Watch for disconnected elements and holes.
2. The algorithmic method:
• Cover 2n−1 unit squares with a simply connected piece
while overlapping each previous piece on 2n−2 squares.
• Special thanks to Matt Klimesh who shared his algorithm
and was the inspiration for this type of construction.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Filling in the pieces
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
The Half Set System
Definition
An n-HSS is a set {S1 , S2 , . . . , Sn } of subsets of {1, 2, . . . , 2n }
with the property that for any nonempty subset A ⊆ {1, 2, . . . , n}
\ Si = 2n−|A| .
i∈A
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
The Half Set System
Theorem
Let {S1 , S2 , . . . , Sn } be a collection of subsets of {1, 2, . . . , 2n }.
For all A ⊆ {1, 2, . . . , n}
\ Si = 2n−|A|
i∈A
if and only if for all subsets B ⊆ {1, 2, . . . n}, there is a unique
element m ∈ {1, 2, . . . , 2n } such that

! 
\
\
{m} =
Si ∩  S i  .
i∈B
i6∈B
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
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
0
0
a 2r +r × 2c+c grid if the following is true:
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
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
0
0
a 2r +r × 2c+c grid if the following is true:
• There is a (n0 = r 0 + c 0 ) half set system for each of the
2r × 2c original unit squares.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
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
0
0
a 2r +r × 2c+c grid if the following is true:
• There is a (n0 = r 0 + c 0 ) half set system for each of the
2r × 2c original unit squares.
• The pieces of the half sets can be arranged so they are
simply connected on the larger grid.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
An example...
...of a disconnected piece that becomes connected in the larger
piece.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Two By Two Expansion
Theorem
If there is a polyVenn on a 2r × 2c grid, then there is a polyVenn
on a 2r +1 × 2c+1 grid.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Proof By Example
• Take an existing polyVenn.
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Introduction
Minimum Area
Expansion
Omiting the Empty Set
Proof By Example
• Take an existing polyVenn.
• Double both its rows and columns.
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Introduction
Minimum Area
Expansion
Omiting the Empty Set
Proof By Example
• We need two more pieces for the new polyVenn.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Proof By Example
• We need two more pieces for the new polyVenn.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Proof By Example
• We need two more pieces for the new polyVenn.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Proof By Example
• We need two more pieces for the new polyVenn.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Proof By Example
• We need two more pieces for the new polyVenn.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
A Vertical Expansion Example
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Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
Two By Four Expansion
Theorem
If there is a polyVenn on a 2r × 2c grid, then there is a polyVenn
on a 2r +1 × 2c+2 grid.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two By Four Expansion Proof
• Suppose we want to expand a 2r × 2c polyVenn.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two By Four Expansion Proof
• Suppose we want to expand a 2r × 2c polyVenn.
• And we had r + c Half Set Systems for a 2 × 4 grid.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two By Four Expansion Proof
• Suppose we want to expand a 2r × 2c polyVenn.
• And we had r + c Half Set Systems for a 2 × 4 grid.
• We use variations on the same one.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two By Four Expansion Proof
Expanding horizontally
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two By Four Expansion Proof
Expanding horizontally
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two By Four Expansion Proof
Expanding horizontally
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two By Four Expansion Proof
Expanding horizontally
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two By Four Expansion Proof
Expanding horizontally
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two By Four Expansion Proof
Expanding horizontally
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two By Four Expansion Proof
Expanding horizontally
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Introduction
Minimum Area
Expansion
Omiting the Empty Set
Two By Four Expansion Proof
Expanding vertically
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Grids With 2n − 1 Unit Squares
• What about h × w grids where 2n − 1 = hw?
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Grids With 2n − 1 Unit Squares
• What about h × w grids where 2n − 1 = hw?
• We know there is a polyVenn whenever h = 2n/2 − 1 and
w = 2n/2 + 1.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
The (2r − 1) × (2r + 1) polyVenn
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
What We Know So Far
1
1
2
4
8
16
32
64
128
256
512
T
2
4
8
16
32
64
128
256
512
T
T
X
X
X
X
X
X
X
T
T
B
B
U
U
U
U
U
K
K
B
B
B
U
U
U
K
K
B
B
B
B
U
K
K
B
B
B
B
K
K
B
B
B
K
K
B
B
K
K
B
K
K
K
KEY:
T trivial
K Klimesh discovery
B Bultena discovery
X not possible
U unknown
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
The Limits We Know
Lemma
A single row polyVenn does not exist for any n > 2.
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
The Limits We Don’t Know
Conjecture
For each r , there is a value cr for which a polyVenn on a 2r × 2c
grid exists and for which no polyVenn exists for more than 2cr
columns.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Summary
The Limits We Don’t Know
Conjecture
For each r , there is a value cr for which a polyVenn on a 2r × 2c
grid exists and for which no polyVenn exists for more than 2cr
columns.
Conjecture
The value of c1 is 4. In particular, there is no polyVenn on a
2 × 32 grid.
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Further Research
• Turn the LIMITS WE DON’T KNOW into the LIMITS WE
KNOW.
• Convexity of Polyominoes
• Properties of the Perimeters
Summary
Introduction
Minimum Area
Expansion
Omiting the Empty Set
Acknowledgements
A special thanks to:
• Frank Ruskey
• Matthew Klimesh
• Stirling Chow
• Khalegh Ahmadi
Summary
Introduction
Minimum Area
Expansion
Thank-you!
Omiting the Empty Set
Summary