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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?

Given a patch of a grid

Does the graph have a coiled Hamiltonian path? Or a coiledHamiltonian circuit?

Given a patch of a grid

Does the graph have a coiled Hamiltonian path? Or a coiledHamiltonian circuit?

A word problem

Exercise Suppose f (x) = x2 is a group homomorphism. Then thegroup 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.

A word problem

Solution We have f (xy) = f (x)f (y).

In other words,xyxy = xxyy , simplify the equation yx = xy .

A word problem

Exercise Suppose f (x) = x3 is an injective group homomorphism.Then the group is abelian.

Solution We now have xyxyxy = xxxyyy , simplify the equationyxyx = xxyy . Combine into a single word y−2x−2yxyx .

Convert the word into a tile.

A word problem

Solution We now have xyxyxy = xxxyyy , simplify the equationyxyx = xxyy .

Combine into a single word y−2x−2yxyx .

A word problem

Our word problem is: do x and y commute?

Or, by injectivity, dox3 and y3 commute?

Now convert the word problem to a tile problem.

A word problem

Our word problem is: do x and y commute? Or, by injectivity, dox3 and y3 commute?

A word problem

Equivalence

Word Problem

Let F2 be the free group on two elements. Let N be a normaldivisor which contains y−2x−2yxyx and x−2y−2xyxy . Does Ncontain y−9x−9y9x9

⇓ Van Kampen Diagram ⇑ Conway Tile

Tile Problem

Can we tile a 9× 9 square by chair tiles?

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

Let’s go Eulerian

Let’s look at the patch again:

Does the graph have a coiled Eulerian circuit? Of course it does.

Let’s go Eulerian

Can we generate the circuit by a substitution? That depends onwhat you mean.

Let’s go Eulerian

Does the graph have a coiled Eulerian circuit? Of course it does.Can we generate the circuit by a substitution?

That depends onwhat you mean.

Let’s go Eulerian

Does the graph have a coiled Eulerian circuit? Of course it does.Can we generate the circuit by a substitution? That depends onwhat you mean.

Geometric substitution

The substitution a→ abad can be converted to geometry

Extend to a box by imposing directions

Fill up the plane by boxes

Extend the substitution to a map on the plane

The result is a plane filling substitution!

Carousels

Start with a cross

Apply the substitution.

Carousels

Start with a cross

Carousels

Start with a cross

Carousels

Dekking’s Carousel Theorem (2010)

Apply two substitutions to the cross, then you know if the carouselfills the plane.

This theorem applies to rotated geometric substitutions. Perhapsthat rotation is not needed at all.

Carousels

Final Question

Out of any Eulerian circuit you can create a planefillingsubstitution, by dividing the circuit into four paths.

A Carousel problem for you

If you apply two such substitutions to the cross, do you still knowif the carousel fills the plane?

LiteratureConway, Lagarias, Tiling with polyominoes, 1990Dekking, Paperfolding morphisms, Arxiv 2013.

Final Question

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