Some space lling curves problems - Centre de Recherches ... · Robbert Fokkink (TU Delft) Some space lling curves problems. Final Question Out of any Eulerian circuit you can create

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





A word problem





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





A word problem





A word problem





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




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




Let’s go Eulerian




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


Carousels


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





Carousels





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






Final Question






Some space lling curves problems - Centre de Recherches ... · Robbert Fokkink (TU Delft) Some space lling curves problems. Final Question Out of any Eulerian circuit you can create

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