On paper folding and Knaster continua - Auburn University
Transcript of 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
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Paper folding
In 2002 Britney Gallivan folded a large piece of toilet paper 12times
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Paper folding
In 2012 St-Marks highschool in Massachussets folded an evenlarger piece of toilet paper 13 times
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Paper folding
In 2012 St-Marks highschool in Massachussets folded an evenlarger piece of toilet paper 13 times
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Paper folding
In 2012 St-Marks highschool in Massachussets folded an evenlarger piece of toilet paper 13 times
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Paper folding curves
Heighway et al discovered that if you unfold toilet paper, then youget a curve in the plane:
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Paper folding curves
Heighway et al discovered that if you unfold toilet paper, then youget a curve in the plane:
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Paper folding curves
Heighway et al discovered that if you unfold toilet paper, then youget a curve in the plane:
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Paper folding curves
Heighway et al discovered that if you unfold toilet paper, then youget a curve in the plane:
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Heighway’s dragon curve
By unfolding St-Mark high school toilet paper this way, you get the13th iterate of Heighway’s dragon
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Heighway’s dragon curve
By unfolding St-Mark high school toilet paper this way, you get the13th iterate of Heighway’s dragon
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Coding the curve
In 1980 Alf van der Poorten and Michel Mendes-France proposedto code folded paper by +1 and −1.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Coding the curve
In 1980 Alf van der Poorten and Michel Mendes-France proposedto code folded paper by +1 and −1.
+1
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Coding the curve
In 1980 Alf van der Poorten and Michel Mendes-France proposedto code folded paper by +1 and −1.
+1 +1 −1
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Coding the curve
In 1980 Alf van der Poorten and Michel Mendes-France proposedto code folded paper by +1 and −1.
+1 +1 −1 +1 +1−1 −1
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Coding the curve
In 1980 Alf van der Poorten and Michel Mendes-France proposedto code folded paper by +1 and −1.
+1 +1 −1 +1 +1−1 −1
If you code Knaster’s curve then you get an infinite symbolicsequence:
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Coding the curve
In 1980 Alf van der Poorten and Michel Mendes-France proposedto code folded paper by +1 and −1.
+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 · · ·
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Coding the continuum
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Coding the continuum
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Coding the continuum
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Coding the continuum
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
or
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
or
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
or
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
or
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Substitutions
Now apply the substitution
a 7→ ab b 7→ cbc 7→ cd d 7→ ad
Iteration gives Heighway’s dragon curve
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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
to R2
Robbert 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
to R2
Robbert 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
to R2
Robbert 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
to R2
Robbert 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
to R2
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Topological description
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 .
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Topological description
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 .
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Topological description
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 .
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Topological description
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 .
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Topological description
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 .
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Topological description
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 .
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Topological description
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 .
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Topological description
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 .
Robbert Fokkink - TU Delft On paper folding and Knaster continua
Topological description
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 .
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
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.
Robbert Fokkink - TU Delft On paper folding and Knaster continua
End
Thank you for your attention
Robbert Fokkink - TU Delft On paper folding and Knaster continua