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Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch Talk, Thursday, September 20, 2018 Slides available at https://egunawan.github.io/talks/smith18 1/ 18

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Page 1: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Friezes, triangulations,continued fractions, and binary numbers

Emily GunawanUniversity of Connecticut

Smith College Lunch Talk,Thursday, September 20, 2018

Slides available at

https://egunawan.github.io/talks/smith18

1/ 18

Page 2: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Frieze patternsA frieze pattern is an image that repeats itself along one direction.The name comes from architecture, where a frieze is a decorationrunning horizontally below a ceiling or roof.

Figure: M. Ascher, Ethnomathematics, p162.

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Page 3: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Conway-Coxeter frieze patterns

Definition

A (Conway-Coxeter) frieze pattern is an array such that:

1. the top row is a row of 1s2. every diamond

ba d

c

satisfies the rule ad − bc = 1.

Example (an integer frieze)

1 1 1 1 1 1 1 · · ·Row 2 · · · 3 1 2 2 1 3 1

2 2 1 3 1 2 2 · · ·· · · 1 1 1 1 1 1 1

Note: every frieze pattern is completely determined by the 2nd row.

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Page 4: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Glide symmetryA glide symmetry is a combination of a translation and a reflection.

1 1 1 1 1 1 1 · · ·Row 2 · · · 3 1 2 2 1 3 1

2 2 1 3 1 2 2 · · ·· · · 1 1 1 1 1 1 1

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Page 5: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Practice

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Page 6: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Practice: Answer Key

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Page 7: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

What do the numbers around each polygon count?

3

1

22

1

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Page 8: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Conway and Coxeter (1970s)

Theorem

Finite frieze patterns with positive integer entries ←→triangulations of polygons

1 1 1 1 1 1 1 · · ·Row 2 · · · 3 1 2 2 1 3 1

2 2 1 3 1 2 2 · · ·· · · 1 1 1 1 1 1 1

v1

v2

v3v4

v5

3

1

22

1

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Page 9: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Conway and Coxeter (1970s)

Theorem

Finite frieze patterns with positive integer entries ←→triangulations of polygons

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Page 10: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Primary school algorithm

E

q

i

E

q

i

E

q

i10/ 18

Page 11: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Broline, Crowe, and Isaacs (BCI, 1970s)

Theorem

Entries of a finite frieze pattern ←→ edges between two vertices.

1 1 1 1 1 1 · · ·Row 2 · · · 3 1 2 2 1 3

2 2 1 3 1 2 · · ·· · · 1 1 1 1 1 1

· · ·

· · ·

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Page 12: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Glide symmetry (again)A glide symmetry is a combination of a translation and areflection. If we forget the arrows’ orientation, the diagonals haveglide symmetry.

1 1 1 1 1 1 1 · · ·Row 2 · · · 3 1 2 2 1 3 1

2 2 1 3 1 2 2 · · ·· · · 1 1 1 1 1 1 1

· · ·

· · ·

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Page 13: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Binary numbers

A binary number is a number expressed in the base-2 numeralsystem, with digits consisting of 1s and 0s. For example,

I 1 = 1 ∗ 20 (in decimal) is written as 1 (in binary).

I 2 = 1 ∗ 21 (in decimal) is written as 10 (in binary).

I 4 = 1 ∗ 22 (in decimal) is written as 100 (in binary).

I 5 = 1 ∗ 22 + 1 ∗ 20 (in decimal) is written as 101 (in binary).

I 29 = 16 + 8 + 4 + 1 = 1 ∗ 24 + 1 ∗ 23 + 1 ∗ 22 + 1 ∗ 20 iswritten as 11101 (in binary).

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Page 14: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Subwords of binary numbers (G.)

We can think of “11101” as a word in thealphabets {0, 1}. All subwords of “11101” which start with “1”:

I “11101” (itself)

I “11101”

I “11101”

I “11101”

I “11101”

I “11101”

I “11101”

I “11101”

I “11101”

I “11101”

I “11101”

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Page 15: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Continued fractions (Canakcı, Schiffler)

s tγ

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Page 16: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Broline, Crowe, and Isaacs (BCI, 1970s)

v1

v2

v3v4

v5

γ

T and γv1

v2

v3v4

v5A B C

(t1, t2) = (A,C )v1

v2

v3v4

v5A B C

(t1, t2) = (B,C )

Definition (BCI tuple)

Let R1, R2, . . . , Rr be the boundary vertices to the right of γ. ABCI tuple for γ is an r -tuple (t1, . . . , tr ) such that:

(B1) the i-th entry ti is a triangle of T having Ri as a vertex.

(B2) the entries are pairwise distinct.

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Page 17: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

How many BCI tuples are there?

R1

L1 L2 L3

R2

L4

s tγ

4(R1) 4(R2)

R1 R2

s t

s t s t

s t s t

s t s t

s t s t

s t s t

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Page 18: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Cluster algebras (Fomin and Zelevinsky, 2000)Definition A cluster algebra is a commutative ring with adistinguished set of generators, called cluster variables.Theorem (Caldero-Chapoton (2006): The cluster variables of acluster algebra from a triangulated polygon (type A) form a friezepattern.· · · 1 1 1 1 1 1 · · ·

1+a+ba b a 1+b

a1+ab b 1+a+b

a b

· · · 1+ab b 1+a+b

a b a 1+ba

1+ab · · ·

1 1 1 1 1 1 1

(Example: type A2)

v1

v2

v3v4

v5ab

3

1

22

1

I Remark: If the variables are specialized to 1, we recover theinteger frieze pattern.

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Page 19: Friezes, triangulations, continued fractions, and binary …Friezes, triangulations, continued fractions, and binary numbers Emily Gunawan University of Connecticut Smith College Lunch

Thank you

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