Connections between binary patterns and paperfolding

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

P

ATRICK

M

ORTON

Connections between binary patterns and paperfolding

Journal de Théorie des Nombres de Bordeaux, tome 2, n

o

_{1 (1990), p. 1-12}

<http://www.numdam.org/item?id=JTNB_1990__2_1_1_0>

© Université Bordeaux 1, 1990, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Connections

between

_binary

_patterns

and

_{paperfolding.}

par PATRICK MORTON

1. Introduction. This talk will be an overview of some recent work on

binary

patterns

and their

_relationship

to

_paperfolding

sequences. Much of

this work was motivated

by the Rudin-Shapiro

_sequence

(aii (n))

defined

by

where

_ell(n~

is the number of occurrences of the

_pattern

11 in the

binary

representation

of n. On the one

hand,

this definition is _easy to

_generalize.

One can consider the

_analogous

_sequence

where P is _any

_pattern

of 0’s and l’s and

_ep(n)

counts occurrences of P in

n. ,

On the other

_hand,

there is the beautiful fact discovered

_by

Mend6s France that ail is

exactly

the direction sequence of the

paperfolding

se-quence obtained

by

folding

a

rectangular

piece

of _paper

alternately

under

and over the left

edge

(which

is held

fixed). (See

[2]

or

_[4].)

This raises the

_question:

is this result about all isolated or does it

_point

to some

deeper

connection between

binary

_patterns

and

paperfolding?

2.

_Properties

of The results of this section

_represent

_joint

work with David

_Boyd

and Janice Cook.

_{(See [1].)}

Let me

_begin

by recalling

some of the

_properties

of the _sequence

aP(n)

defined

_by

_(1),

where P _{is any}

_pattern

of 0’s and 1’s. First define

Then the

_following

theorem holds

_concerning

the behavior of

_sp(x).

THEOREM 1.

_(See

The

_partial

sum function

sp(x)

has the

repre-sentation

, , _.. _ "’" ,

where

_t(x)

is continuous and

_{so(x) is}

bounded.

_Further,

where

_{the ~k}

are the roots outside the unit circle of the

_polynomial

and

(By

definition k is a

period

of P = if and

only if p; =

_{p;+ k} for

0~~d-l--~)

Finally,

the continuous functions of x &#x3E; 0

_satisfying

As a

corollary

we have

where T =

logf/log2

and f

is

the

maximum modulus of the _roots of

P(x).

For all but 14

_patterns

_(in

_particular,

if d &#x3E;

_{5) ~ _}

_~1

is a root of

_P(x).

From this follows"the formula

so in these cases

A(x)

=

A, (x)

(for

1 x

₂₎

_represents

the

_limiting

behavior of

_sp(x)lx’

on the intervals

[2~2~},

as k - oo. This limit

function _possesses two rather nice

_properties:

1)

is nondifferei tiable almost

_everywhere,

2) x

E

_{Q #}

_xTa(x)

E

_Q(~).

Property

1)

also

_implies

that

_sP(x)

= _so that xT is the _correct

The

_polynomials

_P(x)

in theorem 1 also

_satisfy

a remarkable set of

recur-sions. In order to state these

_recursions,

let P and P’ be

_patterns

of

_length

d and

d ~- 1

_{recpectively,}

and let II and IT denote their sets of

_periods,

_e.g.

k E II if and

_only

if k is a

period

of P.

If

_P(x)

and

_P’(x)

denote the

_{corresponding}

_polynomials,

as in

(3),

then

P(x)

and

_P’(x)

_{depend only}

on the

_period

sets

_{II, IT,}

and for _every

_period

set IT there is a

_unique

_period

set 11 for which

We called these formulae the red and blue

_rules,

_{respectively.}

_They

show that the

_polynomials

_P(x)

form a

_tree,

with red or blue branches

depending

on the rule which connects

_P(x)

to

_P~(x).

In

_[1]

we show that this tree

has a fractal-like structure. It has a _sequence of

periodic

subtrees which

are

_isomorphic

to

_larger

and

_larger

initial

_pieces

of the whole

_tree;

thus the

tree

_reproduces

itself in its subtrees.

The _sequences ap are _very fundamental.

They

can be used to

_study

arbitrary

sequences of fl’s. If a = is any sequence of

fl’s,

then

there is a

_unique

_sequence

_{Pk}

of

_binary

_patterns

with

_increasing

values

(the

value of a

_pattern

is the

_integer

it

represents)

and no

leading

_zeros, for

which

This infinite

_product

is defined

_using

the

_topology

in which two _sequences

are close if a

large

number of their initial terms _agree. Let us call the set

the

_binary

_spectrum

of the _sequence a.

Now consider the

_question:

is it

_possible

to characterize the _{sequences a} which have a finite

_spectrum?

3. The arithmetic f’ractal groups

Fk(G).

,

The answer to the last

_question

is in fact _yes, and

_depends

on the

follow-ing

definition.

_(See

Morton and Mourant

_[4]

for the results of this

_section.)

DEFINITION. Let G be an abelian group, a = _an infinite

the associated _sequence of

_{kq -segments}

of a. These vectors

_partition

the

sequence a into vectors of

_length

kq. We define

is

_periodic

with

_period

ll~ for all

_{q &#x3E;}

_0}

Here is defined

_using

scalar

_{multiplication}

_(or

scalar addition if the

_operation

in G is

_addition).

The least

_period

M of a _sequence a in

rk(G)

is called its conductor.

The set

_rk(G)

is an abelian _group under

_{componentwise}

_{multiplication.}

As an

_example,

let P be a

_pattern

of

_digits

base k and consider the

sequence

ep(n)

which counts the number of occurrences of P in the base-k

representation

of n. Then _eP E

_rk(Z)

with conductor

_dividing

_kd-1,

where d is the number of

_digits

of P.

_(This

holds as

_long

as P is not a

_pattern

of

all

_0’s).

This fact follows from the

_equation

If k =

_2,

_reducing

el modulo 2

gives

the Thue-Morse sequence

ei,

which

satisfies

--- __,. -L. J ’- I - -

’-hence

_e1

E

_{rk(Z2 ).}

This last formula is a restatement of the familiar fact that

_ei

is invariant under the substitution

The

_{corresponding}

_equation

for the sequence of

integers

e1 also holds in

characteristic 0:

.I’ ,

The related _sequences ap =

(eP, where (

is _{a root} of

_unity,

generate

a

special subgroup

of

_{rk( ( &#x3E; ),}

_namely

the

_subgroup

with conductor

_dividing

a _power of

k},

which

_provides

an answer to the

_question

we raised in section 2.

THEOREM 2.

_{(See ~4J.)}

Tlie _sequences in

_{A2 (+ 1)}

are

_exactly

the

se-quences which have a finite

binary

_spectrum.

Of course an

analogous

result characterizes _sequences in

_&#x3E;)

The _{sequences in}

_rk(G)

can be

thought

of as arithmetic

_analogues

of

fractals. As an

_example

consider the

_{Rudin-Shapiro}

_{sequence a11,} whose first four terms are

We have

Since ali E

ra(~1)

with M =

_2,

the definition

(4)

implies

that the

relations

₍₅₎

_persist

for

_2q-segments

at all levels:

This can be used to

_generate

all

by considering

the first four terms as the

elements of

_XOI

and

_{Xi .}

_Using

₍₆₎

_{with q}

= 1

_gives

the _next four _terms of

the _sequence:

- - -

-For q

= 2 we then

_get

8 more terms :

etc. The same rule con:aects 29

_-egments

at ever

_{increasing levels,}

in

analogy

to the familiar constructions used to

_generate

certain

types

of ge-ometric fractals. The same remarks hold for _{any sequence in}

rk(G)

for all

k and G.

4.

_Properties

of

_rk(G).

_Among

the

_properties

_{of these groups I} want to

_particularly

mention three. For

_proofs

of 1. and 2. see

[4].

1. In the definition of

_rk(G)

we

only

need to

_require

that

_a-1(n)Xn

be

periodic

of

_period

M. The fact that has

_period

M for all

_{q &#x3E;}

0 follows.

2. If G = U is the _group of

complex

_roots of

unity,

contains _an

isomorphic

copy of every finite abelian group.

Every

finite abelian group is

a

homomorphic image

of

_rk(Z),

for _a,ny k.

3. If G is

_finite,

the _sequences in

_rk(G)

are all k-automatic.

I will _prove this last

_property

_using

the

_following

_maps

_Ti

on _sequences:

From

_property

1. above we have the

following

characterization of sequences

LEMMA. The _sequence a lies in

_rk(G)

if and

_only

if a-1Tia is

periodic

Proof. The _sequence is

_periodic

if and

_only

if all its compo-nent _sequences are

periodic,

and its i-th

_component

is

just

a-ltia.

If

_{Til’ ..., Ti,.}

are _any r of these _maps

_(with

possible

_{repetitions),}

then

obviously

Hence we can use these _maps to characterize _sequences in

_{rkq (G) :}

these are _sequences a for which

a-lta

is

periodic

for all monomials T

in

_{To, · · · ,}

_Tk-l

of

_degree

_q.

As is

_well-known,

a _sequence a is k-automatic if and

_only

if the set of

sequences +

_i)}

is a finite set.

_Using

the _maps T we can state

a is k-automatic if and

only

if there are

only

finitely

_many

images Ta,

as

T runs over all monomials in

_{To , ~ · ~ ,}

_{Tk-i .}

Using

this we _prove

THEOREM 3. If G is a finite abelian _group, the _sequences in are

a~l k-automatic.

Proof.

_By

_definition,

a E has the

_property

that is

_periodic

Tia

= _api,

where _pi is a

_periodic

_sequence, of

_{period M,}

_say. If S is _any monomial in

To; .. ,

and ,S’ _

_{S’Ti, then}

where

_{p’ =}

_S(pi)

is also

_periodic

with

_period

M.

_Peeling

off one term

_Ti

at

a time

_gives

_finally

that

where

_p

is

_periodic

with

_period

M. But there are

only

finitely

_many

_periodic

sequences of a

_given

period

with terms that are taken from the

_given

finite

set G. Hence there are

only finitely

_many distinct _sequences

_{sa, implying}

A different

_proof

of this theorem _appears in

_[5],

where we show that

the _sequences in

_rk(G)

are

essentially

fixed

_points

of kr- substitutions for

suitable r. Shallit

_{(private communication)}

has found

_yet

a third

_proof.

If

_Ak(G)

is the set of all k-automatic _sequences taken from the

_alphabet

G,

then it is a _consequence of theorem 3 that

and so

It is not hard to exhibit automatic _sequences which are not in

rkq

for _any

q. We do this

for q

= 2 in the

following

Example.

Let G =

Z2

and let u be the Baum-Sweet _sequence, defined

by

These formulae ma.y be written as follows:

It is _easy to check that

To show

_{that u v r29(Z2)}

for _{any q,} we show that -u + Su = _{u +} Su is _not

periodic,

for a suitable monomial S of

degree

_q. It suffices to

take q

&#x3E; 2

and S = Then

which is not

_periodic,

since the

_{series ~~ o}

_unxn

satisfies an irreducible

These considerations also raise the

_following

_{question :}

Question.

Since

_rk(G)

and

_Ak(G)

are abelian groups,

_they

have

asso-ciated

_rings

of

_{endomorphisms.}

What are

It is obvious that

_Ti

E

E2

and not

hard

to check that

Ti

E

El

for all

i =

0,1, ~ ~ ~ , J~ -

1. Since the

Ti

do _{not commute} this shows that

Ei

and

E2

are non-commutative

_rings.

5.

_Paperfolding

_sequences. I turn now to the connection between

binary

patterns

and

_paperfolding

_sequences. First a little notation. I want to consider

_paperfolding

_sequences

_{corresponding}

to an infinite _sequence of

folding

instructions

where

_{§n =}

o or u and _{o, u} denote the

_operations

of

_folding

a

rectangular

piece

of _paper

_respectively

over or under the left

edge, (see

[2],[3]).

If we

is the finite initial word of w of

length

_m, then there are two _sequences of

±1’s associated to wm .

The

_first,

_fw.,

encodes the _sequence of up or down folds obtained

by

unfolding

the _paper after

_applying

_according

to the rules

The

_second,

denoted encodes horizontal and vertical directions in the

plane

curve obtained

by

making

all the folds

_equal

to

_{right angles}

and

looking

at the paper

edge-on.

In

dwm’

horizontal and vertical directions

are coded

_by

For

_example,

The limit

_points

of the

_{1~, ~d~,,~ , rra &#x3E;}

_1}

in the set of all

sequences of are

_respectively

called

_paperfolding

_sequences and direc-tion _sequences

_{corresponding}

to the word z,v. The

_topology

with

_respect

to

which these limits are to be taken is the

_topology

in which two sequences

are close if a

_large

number of their initial terms _agree. It is not hard to show that a

_subsequence

or _converges in this

_topology

if and

only

if the

_subsequence

of

_{corresponding}

words wm is

reverse-convergent,

i.e. if and

_only

if an

_increasing

number of the final instructions of the

agree.

Using

this fact it is _easy to see that

paperfolding

_sequences and

direction _sequences of w are in 1-1

_{correspondence}

with the reverse infinite

words

for which are subwords of iv. This _sequence is called a _sequence

of

_unfolding

_{instructions,}

_{(see [2]}

or

[3]).

Let me denote the

paperfolding

and direction _sequences

_{corresponding}

to w

by /~

and

de..;.

If one defines

then one has

and the

_following

result holds:

THEOREM 4..

_(See

theorem

_13.)

Let the reverse infinite _sequence ú)

be

_periodic,

with

_period

A. Then

In fact sw and

dw

have conductor

2,

so

they

lie in the

respective

subgroups

and

_A2~(::i:l).

It follows

_by

theorem 2 and

_analogous

results that sw

and

_dfJ)

are

respectively

a. sum and

product

of

_pattern

_counting

_sequences.

For

_example,

where

and the

_digits

_{i, j}

in this formula are taken base 8. This leads to the

representation

where ap =

Theorem 4 raises the

_question:

for which

_unfolding

_{sequen ces w} is

dw

E

We

_phrase

the answer to this

_question

in terms of the _map r defined as

follows. Let

Dw

denote the set of direction _sequences which arise from a

given

infinite

_{word w,}

and set

Then T :

Dw

2013~

Dw

and in

[4]

we show that

for two fixed vectors

_{X, Y}

of

_length

_two; hence the _sequence is a

"sign sequence"

for

_d,

with

_respect

to

_segments

of

_length

two.

_By

_iterating

the _map T we _prove that is a

_sign

_sequence for

dw

with

_respect

to

segments

of

_length

_2k,

and this fact leads to the

_following

_result,

_{(see [4],}

theorems

_17-18):

THEOREM 5.

I~

The _sequence

_dw

lies in

_{r 2’B (:i: 1)}

if and

_only

a

periodic

sequence.

2)

If dw

E

_F2’(±l),

then

_dW

E

_{A2,B, (:1:1),}

for a suitable

multiple

A’ of A.

3)

dw

E

_{A2.B, (:i:1)}

if and

only

if

for some n &#x3E; o.

Hence direction _sequences in

_correspond

to

_pre-periodic

_points

of the _{map T, and the groups}

_r2~(:l:l)

arise

_naturally

in connection with the

discrete

_dynamical

_system

_{(Dw, T).}

THEOREME 6.

_(See

_~4),

theorem

_19.)

The direction _sequence

_dtN

is a finite

product

of

_pattern

_sequences base

2A

_(and

so has a finite

_spectrum

base

2À)

if and only if

or

where

_{W2 is}

obtained from W2

by switching

o’s and u’s.

As

_examples

with A = 1 let me note

where the

_patterns

in these formulae are all

_binary

_patterns.

From these

relations it is

_possible

to

_compute

the

_spectra

of all direction _sequences which lie in

_A2(~1).

For

_example,

where the

_product

is over all

binary

_patterns

beginning

with 1 and

having

length

k ~-1.

This shows that _every

_binary

_pattern

_(with

a

_{leading 1)}

occurs

in the

_binary

_spectrum

of some

deN.

I will finish

_by

_recalling

a recent theorem of Mend6s France and Shallit

_[3]

(see

their theorem 4.2 for the

_special

case of

paperfolding;

note that their

sequence of turns is here what we have called a

paperfolding

sequence):

THEOREM. The

_paperfolding

sequence is 2-automatic if and

only

if the

sequence w of

unfolding

instructions is

_{ultimately periodic.}

Together

with theorem

_6,

this

_gives

THEOREM 7. A

_paperfolding

sequence

feN

is 2-automatic if and

only

if its

associated direction _sequence

_dw

has a finite

_spectrum

base

2..B ,

for some A.

ACKNOWLEDGEMENTS : The author

_gratefully

_acknowledges

the

_support

of the Alexander von Humboldt

Foundation,

the

C.N.R.S.,

the D.R.E.T.

and the Mathematics

_Departements

at the

_University

of

_Bordeaux,

the

University

of Paris-Sud and the

_University

of

_Heidelberg

_during

the

_period

this article was written.

REFERENCES

[1]

D.W. BOYD, J. COOK and P. MORTON, On sequences of ± 1’s defined by binary

patterns. Dissertationes Math. _283, to _appear.

[2]

M. _DEKKING, M. MENDÈS FRANCE and A. van der POORTEN, Folds I,II III, Math. _{Intelligencer} 4

_(1982),

130-138, 173-181, 190-195.

[3]

M. MENDÈS FRANCE and J. O.

_SHALLIT,

Wire bending, J. Comb. _Theory 50

(1989),

1-23.

[4]

P. MORTON and W.J. _{MOURANT, Paper folding, digit patterns} and groups of

arithmetic fractals, Proc. London Math. Soc. _{59, 2,}

_(1989),

253-293.

[5]

P. MORTON and W.J. _{MOURANT, Digit} patterns and transcendental numbers. submitted.

Mots c2e&#x26;: _{Paperfolding,} occurrences of words, fractals.

1980 Mathematics _{subject classi,fications: 10A30, 10L10,} 68D22.

Mathematisches _Institut, Universitat _Heidelberg

Heidelberg

(GERMANY)

and

Dept. of _{Mathematics, Wellesley College,}