geodesic continued fractions - open university
TRANSCRIPT
continued fraction expansion of a rational?
How many shortest continued fractions are there?
Can we characterise shortest continued fractions?
Questions
continued fraction expansion of a rational?
How many shortest continued fractions are there?
Can we characterise shortest continued fractions?
Questions
continued fraction expansion of a rational?
How many shortest continued fractions are there?
Can we characterise shortest continued fractions?
Collaborators
Universal tessellations
D. Singerman
Die Lehre von den Kettenbrüchen
O. Perron
Ford circles
Ford circle
Ford circle
Ford circles
Continued fractions
b1 + 1
b2 + 1
b3 + 1
Continued fraction approximants
Continued fraction approximants
Continued fraction approximants
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
The Farey graph
Correspondence
Finite continued fractions ! nite paths from 1 in Farey graph.
Shortest continued fractions ! geodesics from 1 in Farey graph.
Correspondence
Finite continued fractions ! nite paths from 1 in Farey graph.
Shortest continued fractions ! geodesics from 1 in Farey graph.
The extended modular group
z 7!
az + b
Theorem.
There is a unique geodesic from 1 to x if and only if
jbi j > 3 for each i > 2.
Unique geodesics
Theorem. There is a unique geodesic from 1 to x if and only if
jbi j > 3 for each i > 2.
Unique geodesics
Theorem. There is a unique geodesic from 1 to x if and only if
jbi j > 3 for each i > 2.
The number of geodesics
Theorem.
Let d(1; x ) = n . Then there are no more than Fn
geodesics from 1 to x . For each positive integer n , the bound
Fn can be attained.
The number of geodesics
Then there are no more than Fn
geodesics from 1 to x . For each positive integer n , the bound
Fn can be attained.
The number of geodesics
Theorem. Let d(1; x ) = n . Then there are no more than Fn
geodesics from 1 to x .
For each positive integer n , the bound
Fn can be attained.
The number of geodesics
Theorem. Let d(1; x ) = n . Then there are no more than Fn
geodesics from 1 to x . For each positive integer n , the bound
Fn can be attained.
Non-geodesic paths
Description of geodesic continued fractions
Theorem. A continued fraction with coecients b1; b2; : : : ; bn
corresponds to a geodesic in the Farey graph if and only if
jbi j > 2 for each i > 2 and there is no substring of b2; b3; : : : ; bn of the form 2;3; 3;3; 3;3; : : : ; 3;2.
Description of geodesic continued fractions
Theorem. A continued fraction with coecients b1; b2; : : : ; bn corresponds to a geodesic in the Farey graph
if and only if
jbi j > 2 for each i > 2 and there is no substring of b2; b3; : : : ; bn of the form 2;3; 3;3; 3;3; : : : ; 3;2.
Description of geodesic continued fractions
Theorem. A continued fraction with coecients b1; b2; : : : ; bn corresponds to a geodesic in the Farey graph if and only if
jbi j > 2 for each i > 2
and there is no substring of b2; b3; : : : ; bn of the form 2;3; 3;3; 3;3; : : : ; 3;2.
Description of geodesic continued fractions
Theorem. A continued fraction with coecients b1; b2; : : : ; bn corresponds to a geodesic in the Farey graph if and only if
jbi j > 2 for each i > 2 and there is no substring of b2; b3; : : : ; bn of the form 2;3; 3;3; 3;3; : : : ; 3;2.
Cutting sequences and the modular surface
The modular group
z 7!
az + b
T (z ) = z + 1 X (z ) = 1=z
Fundamental region for the modular group
T (z ) = z + 1 X (z ) = 1=z
Fundamental region for the modular group
T (z ) = z + 1 X (z ) = 1=z
The modular surface
Farey triangles
Farey triangles
Dual graph
Dual graph
Generators of the modular group
=
z 7!
az + b
cz + d : a ; b; c; d 2 Z; ad bc = 1
= C2 C3
1+ 1=z 1 1
0 1
! 1 0
1 1
=
z 7!
az + b
cz + d : a ; b; c; d 2 Z; ad bc = 1
= C2 C3
1+ 1=z
=
z 7!
az + b
cz + d : a ; b; c; d 2 Z; ad bc = 1
= C2 C3
1+ 1=z 1 1
0 1
! 1 0
1 1
L b1R
1
L b1R
Words in L and R ! nite paths from 1
Can convert a path from 1 to x to a geodesic from 1 to x in
nitely many moves.
Correspondence
Words in L and R ! nite paths from 1
Can convert a path from 1 to x to a geodesic from 1 to x in
nitely many moves.
Correspondence
Words in L and R ! nite paths from 1
Can convert a path from 1 to x to a geodesic from 1 to x in
nitely many moves.
Correspondence
Words in L and R ! nite paths from 1
Can convert a path from 1 to x to a geodesic from 1 to x in
nitely many moves.
Hecke groups
Hecke groups
z
T (z ) =
z
T (z ) =
z
T (z ) =
1
Hecke groups
How many shortest continued fractions are there?
Can we characterise shortest continued fractions?
Questions
continued fraction expansion of a rational?
How many shortest continued fractions are there?
Can we characterise shortest continued fractions?
Questions
continued fraction expansion of a rational?
How many shortest continued fractions are there?
Can we characterise shortest continued fractions?
Collaborators
Universal tessellations
D. Singerman
Die Lehre von den Kettenbrüchen
O. Perron
Ford circles
Ford circle
Ford circle
Ford circles
Continued fractions
b1 + 1
b2 + 1
b3 + 1
Continued fraction approximants
Continued fraction approximants
Continued fraction approximants
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
Chain of Ford circles
The Farey graph
Correspondence
Finite continued fractions ! nite paths from 1 in Farey graph.
Shortest continued fractions ! geodesics from 1 in Farey graph.
Correspondence
Finite continued fractions ! nite paths from 1 in Farey graph.
Shortest continued fractions ! geodesics from 1 in Farey graph.
The extended modular group
z 7!
az + b
Theorem.
There is a unique geodesic from 1 to x if and only if
jbi j > 3 for each i > 2.
Unique geodesics
Theorem. There is a unique geodesic from 1 to x if and only if
jbi j > 3 for each i > 2.
Unique geodesics
Theorem. There is a unique geodesic from 1 to x if and only if
jbi j > 3 for each i > 2.
The number of geodesics
Theorem.
Let d(1; x ) = n . Then there are no more than Fn
geodesics from 1 to x . For each positive integer n , the bound
Fn can be attained.
The number of geodesics
Then there are no more than Fn
geodesics from 1 to x . For each positive integer n , the bound
Fn can be attained.
The number of geodesics
Theorem. Let d(1; x ) = n . Then there are no more than Fn
geodesics from 1 to x .
For each positive integer n , the bound
Fn can be attained.
The number of geodesics
Theorem. Let d(1; x ) = n . Then there are no more than Fn
geodesics from 1 to x . For each positive integer n , the bound
Fn can be attained.
Non-geodesic paths
Description of geodesic continued fractions
Theorem. A continued fraction with coecients b1; b2; : : : ; bn
corresponds to a geodesic in the Farey graph if and only if
jbi j > 2 for each i > 2 and there is no substring of b2; b3; : : : ; bn of the form 2;3; 3;3; 3;3; : : : ; 3;2.
Description of geodesic continued fractions
Theorem. A continued fraction with coecients b1; b2; : : : ; bn corresponds to a geodesic in the Farey graph
if and only if
jbi j > 2 for each i > 2 and there is no substring of b2; b3; : : : ; bn of the form 2;3; 3;3; 3;3; : : : ; 3;2.
Description of geodesic continued fractions
Theorem. A continued fraction with coecients b1; b2; : : : ; bn corresponds to a geodesic in the Farey graph if and only if
jbi j > 2 for each i > 2
and there is no substring of b2; b3; : : : ; bn of the form 2;3; 3;3; 3;3; : : : ; 3;2.
Description of geodesic continued fractions
Theorem. A continued fraction with coecients b1; b2; : : : ; bn corresponds to a geodesic in the Farey graph if and only if
jbi j > 2 for each i > 2 and there is no substring of b2; b3; : : : ; bn of the form 2;3; 3;3; 3;3; : : : ; 3;2.
Cutting sequences and the modular surface
The modular group
z 7!
az + b
T (z ) = z + 1 X (z ) = 1=z
Fundamental region for the modular group
T (z ) = z + 1 X (z ) = 1=z
Fundamental region for the modular group
T (z ) = z + 1 X (z ) = 1=z
The modular surface
Farey triangles
Farey triangles
Dual graph
Dual graph
Generators of the modular group
=
z 7!
az + b
cz + d : a ; b; c; d 2 Z; ad bc = 1
= C2 C3
1+ 1=z 1 1
0 1
! 1 0
1 1
=
z 7!
az + b
cz + d : a ; b; c; d 2 Z; ad bc = 1
= C2 C3
1+ 1=z
=
z 7!
az + b
cz + d : a ; b; c; d 2 Z; ad bc = 1
= C2 C3
1+ 1=z 1 1
0 1
! 1 0
1 1
L b1R
1
L b1R
Words in L and R ! nite paths from 1
Can convert a path from 1 to x to a geodesic from 1 to x in
nitely many moves.
Correspondence
Words in L and R ! nite paths from 1
Can convert a path from 1 to x to a geodesic from 1 to x in
nitely many moves.
Correspondence
Words in L and R ! nite paths from 1
Can convert a path from 1 to x to a geodesic from 1 to x in
nitely many moves.
Correspondence
Words in L and R ! nite paths from 1
Can convert a path from 1 to x to a geodesic from 1 to x in
nitely many moves.
Hecke groups
Hecke groups
z
T (z ) =
z
T (z ) =
z
T (z ) =
1
Hecke groups