on paper folding and knaster continua - auburn university

On paper folding and Knaster continua

Robbert Fokkink - TU Delft

joint work with Wieb Bosma and Cor Kraaikamp

ECM Krakow 2012

Robbert Fokkink - TU Delft On paper folding and Knaster continua

Paper folding

In 2002 Britney Gallivan folded a large piece of toilet paper 12times

Paper folding

In 2002 Britney Gallivan folded a large piece of toilet paper 12times

Paper folding

In 2012 St-Marks highschool in Massachussets folded an evenlarger piece of toilet paper 13 times

Paper folding

Topology

In 1922 Bronislaw Knaster, then a PhD student at WarsawUniversity, constructed an indecomposable continuum by foldingan imaginary piece of toilet paper an infinite number of times

Topology

In 1922 Bronislaw Knaster, then a PhD student at WarsawUniversity, constructed an indecomposable continuum by foldingan imaginary piece of toilet paper an infinite number of times

Paper UNfolding

In 1967 NASA physicists John Heighway, Bruce Banks and WilliamHarter studied the creases of unfolded folded paper

Such creases can create beautiful patterns

Paper UNfolding

Paper folding curves

Heighway et al discovered that if you unfold toilet paper, then youget a curve in the plane:

Heighway’s dragon curve

By unfolding St-Mark high school toilet paper this way, you get the13th iterate of Heighway’s dragon

Heighway’s dragon curve

By unfolding St-Mark high school toilet paper this way, you get the13th iterate of Heighway’s dragon

Coding the curve

In 1980 Alf van der Poorten and Michel Mendes-France proposedto code folded paper by +1 and −1.

Coding the curve

+1 +1 −1

Coding the curve

+1 +1 −1 +1 +1−1 −1

Coding the curve

+1 +1 −1 +1 +1−1 −1

If you code Knaster’s curve then you get an infinite symbolicsequence:

Coding the curve

+1 +1 −1 +1 +1−1 −1

If you code Knaster’s curve then you get an infinite symbolicsequence:

+1 +1 −1 +1 +1 −1 −1 +1 +1 +1 −1 −1 +1 −1 −1 · · ·

Coding the curve

This coding process can be automated:

+1 ? −1 ? +1 ? −1 ? +1 ? −1 ? +1 ? −1 ? · · ·

Fill in the odd places by an alternate +1 and −1, starting with +1.Leave the even places open. Now fill in the spaces that are 1modulo 4, starting with +1.

+1 +1 −1 ? +1 −1 −1 ? +1 +1 −1 ? +1 −1 −1 ? · · ·

Next fill in the spaces that are 3 modulo 8, etc.

Mendes-France and Van der Poorten discovered that, starting withthe occasional -1, you can get a continuum of coded sequences.

Coding the curve

+1 ? −1 ? +1 ? −1 ? +1 ? −1 ? +1 ? −1 ? · · ·

+1 +1 −1 ? +1 −1 −1 ? +1 +1 −1 ? +1 −1 −1 ? · · ·

Coding the curve

+1 ? −1 ? +1 ? −1 ? +1 ? −1 ? +1 ? −1 ? · · ·

Fill in the odd places by an alternate +1 and −1, starting with +1.Leave the even places open.

Now fill in the spaces that are 1modulo 4, starting with +1.

+1 +1 −1 ? +1 −1 −1 ? +1 +1 −1 ? +1 −1 −1 ? · · ·

Coding the curve

+1 ? −1 ? +1 ? −1 ? +1 ? −1 ? +1 ? −1 ? · · ·

+1 +1 −1 ? +1 −1 −1 ? +1 +1 −1 ? +1 −1 −1 ? · · ·

Coding the curve

+1 ? −1 ? +1 ? −1 ? +1 ? −1 ? +1 ? −1 ? · · ·

+1 +1 −1 ? +1 −1 −1 ? +1 +1 −1 ? +1 −1 −1 ? · · ·

Coding the curve

+1 ? −1 ? +1 ? −1 ? +1 ? −1 ? +1 ? −1 ? · · ·

+1 +1 −1 ? +1 −1 −1 ? +1 +1 −1 ? +1 −1 −1 ? · · ·

Coding the curve

+1 ? −1 ? +1 ? −1 ? +1 ? −1 ? +1 ? −1 ? · · ·

+1 +1 −1 ? +1 −1 −1 ? +1 +1 −1 ? +1 −1 −1 ? · · ·

Coding the continuum

That coding can also be described by Knaster’s contimuum

Start out from a point on the x-axis and move upwards along ablue line. You move either left or right. You move alternately leftand right along blue lines. You move along the blue lines half thetime, the other movements remain question marks.

Start out from a point on the x-axis and move upwards along ablue line.

You move either left or right. You move alternately leftand right along blue lines. You move along the blue lines half thetime, the other movements remain question marks.

Start out from a point on the x-axis and move upwards along ablue line. You move either left or right. You move alternately leftand right along blue lines.

You move along the blue lines half thetime, the other movements remain question marks.

The general codings that start with the occasional -1 correspondto different embeddings of the continuum.

Readers of TopologyProceedings will recognize this.

You can also code other arcs in the continuum by starting on ?that are 3 mod 4 instead of 1 mod 4.

The general codings that start with the occasional -1 correspondto different embeddings of the continuum. Readers of TopologyProceedings will recognize this.

General paper folding

Heighway’s folding operation is a repetition of

In 2011 Michel Dekking studied more general folding operations,which he had introduced earlier in 1982. These folding operationsrepeat

Substitutions

Van der Poorten and Mendes-France discovered that you cangenerate the Heigway dragon by substitutions.

Paint the edges ofthe network by four colors, labeled a, b, c , d

Substitutions

Van der Poorten and Mendes-France discovered that you cangenerate the Heigway dragon by substitutions. Paint the edges ofthe network by four colors, labeled a, b, c , d

Substitutions

Van der Poorten and Mendes-France discovered that you cangenerate the Heigway dragon by substitutions. Paint the edges ofthe network by four colors, labeled a, b, c , d

Substitutions

Now apply the substitution

a 7→ ab b 7→ cbc 7→ cd d 7→ ad

Iteration gives Heighway’s dragon curve

Substitutions

Now apply the substitution

a 7→ ab b 7→ cbc 7→ cd d 7→ ad

Iteration gives Heighway’s dragon curveRobbert Fokkink - TU Delft On paper folding and Knaster continua

Group actions

Dekking’s generalized foldings can also be described by substitutions.

In2010 Donald Knuth discovered that such substitutions can be regarded asequivariant maps of G -spaces.The group G is generated by rotationsaround Z2 by right angles.

The edges of the network represent G . Knuth’s equivariant map extends

Group actions

Dekking’s generalized foldings can also be described by substitutions. In2010 Donald Knuth discovered that such substitutions can be regarded asequivariant maps of G -spaces.

The group G is generated by rotationsaround Z2 by right angles.

Group actions

Dekking’s generalized foldings can also be described by substitutions. In2010 Donald Knuth discovered that such substitutions can be regarded asequivariant maps of G -spaces.The group G is generated by rotationsaround Z2 by right angles.

Group actions

Topological description

There is a topological explanation for all this, which generalizes tounfoldings over arbitrary angles.

To make the +1 and -1 topologically visible, we need windingnumbers.

In other words, we need some toilet rolls

To unfold the toilet paper, we need to lift it to a (branched)universal covering space.

To make the +1 and -1 topologically visible, we need windingnumbers. In other words, we need some toilet rolls

So we have a graph with toilet paper that has been wrappedaround it

To unfold this toilet paper in Heighway style, we add relations tothe graph. One relation in each toilet roll

For right angles these relations are a4 and b4. We add anotherrelation combining a and b: (ab)2. This gives us

< a, b : a4, b4, (ab)2 >

but that is Knuth’s G .

< a, b : a4, b4, (ab)2 >

To unfold this toilet paper in Heighway style, we add relations tothe graph.

One relation in each toilet roll

< a, b : a4, b4, (ab)2 >

For right angles these relations are a4 and b4.

We add anotherrelation combining a and b: (ab)2. This gives us

< a, b : a4, b4, (ab)2 >

For right angles these relations are a4 and b4. We add anotherrelation combining a and b: (ab)2.

This gives us

< a, b : a4, b4, (ab)2 >

Universal cover

The folding operation lifts to a universal branched cover.

Knuth’sequivariance means that the lift commutes with the deck transformations.The substitutions requires 4 letters since the translation subgroup hasindex 4. This extends to unfolding over other angles, for instance if youuse 60 degree angles, then the universal cover is the hyperbolic plane

By extending unfolding to the hyperbolic plane, we can generalize all

previous work on folding toilet paper to arbitrary angles.

Universal cover

The folding operation lifts to a universal branched cover. Knuth’sequivariance means that the lift commutes with the deck transformations.

The substitutions requires 4 letters since the translation subgroup hasindex 4. This extends to unfolding over other angles, for instance if youuse 60 degree angles, then the universal cover is the hyperbolic plane

Universal cover

The folding operation lifts to a universal branched cover. Knuth’sequivariance means that the lift commutes with the deck transformations.The substitutions requires 4 letters since the translation subgroup hasindex 4.

This extends to unfolding over other angles, for instance if youuse 60 degree angles, then the universal cover is the hyperbolic plane

Universal cover

The folding operation lifts to a universal branched cover. Knuth’sequivariance means that the lift commutes with the deck transformations.The substitutions requires 4 letters since the translation subgroup hasindex 4. This extends to unfolding over other angles, for instance if youuse 60 degree angles, then the universal cover is the hyperbolic plane

Universal cover

Thank you for your attention

on paper folding and knaster continua - auburn university

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