Some space filling curves problems
Robbert Fokkink (TU Delft)
Montreal 2017
Robbert Fokkink (TU Delft)
Some space filling curves problems
Can you fill the shape with a coil?
Robbert Fokkink (TU Delft)
Some space filling curves problems
Can you fill the shape with a coil?
Robbert Fokkink (TU Delft)
Some space filling curves problems
Can you fill the shape with a coil?
Robbert Fokkink (TU Delft)
Some space filling curves problems
Can you fill the shape with a coil?
Given a patch of a grid
Does the graph have a coiled Hamiltonian path? Or a coiled
Hamiltonian circuit?
Robbert Fokkink (TU Delft)
Some space filling curves problems
Can you fill the shape with a coil?
Given a patch of a grid
Does the graph have a coiled Hamiltonian path? Or a coiled
Hamiltonian circuit?
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Exercise Suppose f (x) = x 2 is a group homomorphism. Then the
group is abelian.
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Exercise Suppose f (x) = x 2 is a group homomorphism. Then the
group is abelian.
Solution We have f (xy ) = f (x)f (y ).
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Exercise Suppose f (x) = x 2 is a group homomorphism. Then the
group is abelian.
Solution We have f (xy ) = f (x)f (y ). In other words,
xyxy = xxyy , simplify the equation yx = xy .
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Exercise Suppose f (x) = x 2 is a group homomorphism. Then the
group is abelian.
Solution We have f (xy ) = f (x)f (y ). In other words,
xyxy = xxyy , simplify the equation yx = xy .
That was too easy, let’s look at something more difficult.
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Exercise Suppose f (x) = x 3 is an injective group homomorphism.
Then the group is abelian.
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Exercise Suppose f (x) = x 3 is an injective group homomorphism.
Then the group is abelian.
Solution We now have xyxyxy = xxxyyy , simplify the equation
yxyx = xxyy .
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Exercise Suppose f (x) = x 3 is an injective group homomorphism.
Then the group is abelian.
Solution We now have xyxyxy = xxxyyy , simplify the equation
yxyx = xxyy . Combine into a single word y −2 x −2 yxyx.
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Exercise Suppose f (x) = x 3 is an injective group homomorphism.
Then the group is abelian.
Solution We now have xyxyxy = xxxyyy , simplify the equation
yxyx = xxyy . Combine into a single word y −2 x −2 yxyx.
Convert the word into a tile.
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Exercise Suppose f (x) = x 3 is an injective group homomorphism.
Then the group is abelian.
Solution We now have xyxyxy = xxxyyy , simplify the equation
yxyx = xxyy . Combine into a single word y −2 x −2 yxyx.
Convert the word into a tile.
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Our word problem is: do x and y commute?
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Our word problem is: do x and y commute? Or, by injectivity, do
x 3 and y 3 commute?
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Our word problem is: do x and y commute? Or, by injectivity, do
x 3 and y 3 commute?
Now convert the word problem to a tile problem.
Robbert Fokkink (TU Delft)
Some space filling curves problems
A word problem
Our word problem is: do x and y commute? Or, by injectivity, do
x 3 and y 3 commute?
Now convert the word problem to a tile problem.
Robbert Fokkink (TU Delft)
Some space filling curves problems
Equivalence
Word Problem
Let F2 be the free group on two elements. Let N be a normal
divisor which contains y −2 x −2 yxyx and x −2 y −2 xyxy . Does N
contain y −9 x −9 y 9 x 9
⇓ Van Kampen Diagram
⇑ Conway Tile
Tile Problem
Can we tile a 9 × 9 square by chair tiles?
Robbert Fokkink (TU Delft)
Some space filling curves problems
The quality of a problem
A problem is as good as its solution.
Robbert Fokkink (TU Delft)
Some space filling curves problems
The quality of a problem
A problem is as good as its solution.
Can you tile a triangle by chairs?
Robbert Fokkink (TU Delft)
Some space filling curves problems
The quality of a problem
A problem is as good as its solution.
Can you tile a triangle by chairs?
This innocent problem motivated Conway to define his tiles
Robbert Fokkink (TU Delft)
Some space filling curves problems
Let’s go Eulerian
Let’s look at the patch again:
Robbert Fokkink (TU Delft)
Some space filling curves problems
Let’s go Eulerian
Let’s look at the patch again:
Robbert Fokkink (TU Delft)
Some space filling curves problems
Let’s go Eulerian
Let’s look at the patch again:
Does the graph have a coiled Eulerian circuit? Of course it does.
Robbert Fokkink (TU Delft)
Some space filling curves problems
Let’s go Eulerian
Let’s look at the patch again:
Does the graph have a coiled Eulerian circuit? Of course it does.
Robbert Fokkink (TU Delft)
Some space filling curves problems
Let’s go Eulerian
Let’s look at the patch again:
Does the graph have a coiled Eulerian circuit? Of course it does.
Can we generate the circuit by a substitution?
Robbert Fokkink (TU Delft)
Some space filling curves problems
Let’s go Eulerian
Let’s look at the patch again:
Does the graph have a coiled Eulerian circuit? Of course it does.
Can we generate the circuit by a substitution? That depends on
what you mean.
Robbert Fokkink (TU Delft)
Some space filling curves problems
Geometric substitution
The substitution a → abad can be converted to geometry
Robbert Fokkink (TU Delft)
Some space filling curves problems
Geometric substitution
The substitution a → abad can be converted to geometry
Extend to a box by imposing directions
Robbert Fokkink (TU Delft)
Some space filling curves problems
Geometric substitution
The substitution a → abad can be converted to geometry
Extend to a box by imposing directions
Robbert Fokkink (TU Delft)
Some space filling curves problems
Geometric substitution
The substitution a → abad can be converted to geometry
Extend to a box by imposing directions
Robbert Fokkink (TU Delft)
Some space filling curves problems
Geometric substitution
Fill up the plane by boxes
Robbert Fokkink (TU Delft)
Some space filling curves problems
Geometric substitution
Fill up the plane by boxes
Robbert Fokkink (TU Delft)
Some space filling curves problems
Geometric substitution
Fill up the plane by boxes
Extend the substitution to a map on the plane
Robbert Fokkink (TU Delft)
Some space filling curves problems
Geometric substitution
The result is a plane filling substitution!
Robbert Fokkink (TU Delft)
Some space filling curves problems
Carousels
Start with a cross
Robbert Fokkink (TU Delft)
Some space filling curves problems
Carousels
Start with a cross
Robbert Fokkink (TU Delft)
Some space filling curves problems
Carousels
Start with a cross
Apply the substitution.
Robbert Fokkink (TU Delft)
Some space filling curves problems
Carousels
Robbert Fokkink (TU Delft)
Some space filling curves problems
Carousels
Robbert Fokkink (TU Delft)
Some space filling curves problems
Carousels
Dekking’s Carousel Theorem (2010)
Apply two substitutions to the cross, then you know if the carousel
fills the plane.
Robbert Fokkink (TU Delft)
Some space filling curves problems
Carousels
Dekking’s Carousel Theorem (2010)
Apply two substitutions to the cross, then you know if the carousel
fills the plane.
This theorem applies to rotated geometric substitutions. Perhaps
that rotation is not needed at all.
Robbert Fokkink (TU Delft)
Some space filling curves problems
Carousels
Dekking’s Carousel Theorem (2010)
Apply two substitutions to the cross, then you know if the carousel
fills the plane.
This theorem applies to rotated geometric substitutions. Perhaps
that rotation is not needed at all.
Robbert Fokkink (TU Delft)
Some space filling curves problems
Carousels
Dekking’s Carousel Theorem (2010)
Apply two substitutions to the cross, then you know if the carousel
fills the plane.
This theorem applies to rotated geometric substitutions. Perhaps
that rotation is not needed at all.
Robbert Fokkink (TU Delft)
Some space filling curves problems
Final Question
Out of any Eulerian circuit you can create a planefilling
substitution, by dividing the circuit into four paths.
Robbert Fokkink (TU Delft)
Some space filling curves problems
Final Question
Out of any Eulerian circuit you can create a planefilling
substitution, by dividing the circuit into four paths.
A Carousel problem for you
If you apply two such substitutions to the cross, do you still know
if the carousel fills the plane?
Robbert Fokkink (TU Delft)
Some space filling curves problems
Final Question
Out of any Eulerian circuit you can create a planefilling
substitution, by dividing the circuit into four paths.
A Carousel problem for you
If you apply two such substitutions to the cross, do you still know
if the carousel fills the plane?
Literature
Conway, Lagarias, Tiling with polyominoes, 1990
Dekking, Paperfolding morphisms, Arxiv 2013.
Robbert Fokkink (TU Delft)
Some space filling curves problems