folding paper and thermodynamics

PHYSICS REPORTS (Review Section of Physics Letters) 103, Nos. 1-4 (1984) 161-172. North-Holland, Amsterdam

Folding Paper and Thermodynamics

Michel MENDES FRANCE Dept. de Math., Univ. Bordeaux I, 351 cours de la LiMration, 33405 Talence, France

I. The baker's transformation and paperfolding

The baker's transformation F is a mapping which takes the unit square onto itself and can be described as follows. First "squash" the square to obtain the rectangle 2 x ½, then "fold" the rectangle in order to reconstruct the unit square. One is usually interested in the iterates F, F 2, F 3, . . . . The sequence is known to be mixing (ergodic). See for example H6non and Pomeau [4] or any book on ergodic theory (H6non and Pomeau call this transformation the Poincar6 transformation).

$

I I I

Fig. 1.

Fig. 2.

We shall study this mapping from an unconventional point of view. Take a sheet of paper and fold it in two as previously shown. Then repeat n times and unfold. Reading from left to right, one meets a sequence of 2" - 1 ridges ^ and valleys v. For n = 3,

V V A V V A A .

An interesting fact, though trivial, is that if you now fold the sheet of paper (n + 1) times and then unfold, the new sequence of v's and ^'s is twice as long (actually 2 "+1 - 1) and the first half coincides with the previous 2" - 1 sequence. For example, taking n = 4

V V A V V A A V V V A A V A A

This allows us to define the infinite (v, ^)-sequence obtained by folding the sheet of paper infinitely often.

Instead of folding at each stage in the "positive" direction, one might fold in the negative direction

162 Common trends in particle and condensed matter physics

<---4 Fig. 3.

and choose at will at each step either F+ or F_. A more sophisticated argument shows that one can again define an infinite fold

i F ~ . = "" F~,F~2F~ (ej = -+1) n = l

and thus an infinite (v, ^)-sequence which we shall call a paperfolding sequence. They are discussed at length in a recent paper of Dekking, van der Poorten and myself [3]. In the following paragraphs we shall rapidly describe some of their properties and show to what extent the original baker transformation generates chaos.

2. A short incursion in N u m b e r Theory

Put v = 1 and A = 0. Then any paperfolding sequence can be viewed as an infinite sequence of O's and l's, and hence can be interpreted as the binary expansion of a real number.

Our first example, the positive infinite paperfolding sequence, generates the paperfolding number

• 1 1 0 1 1 0 0 1 1 1 0 0 1 0 0 . . .

T h e o r e m 1. All paperfolding numbers are transcendental (see Mendes France and van der Poorten [6] or Dekking, Mendes France and van der Poorten [3]).

This result is a first measure of the complexity of the baker's transformation.

3. Geometr ic propert ies and seifavoiding curves

We now map v onto the symbol L (left) and ^ onto R (right). Given a paperfolding sequence say

A = L L R L L R R L L L R R . . . ,

we construct on the lattice Z 2 the following infinite polygon. Starting at the origin O, follow the unit interval on the x-axis and then read the first instruction, namely "go left". At the next lattice point read

L ~ L

It. R O

Fig. 4.

M. Mendes France, Folding paper and thermodynamics 163

instruction number 2, "go left". The third instruction tells us to go r ight . . , and so on. We thus obtain an infinite polygon on the square lattice. In the same fashion, every paperfolding sequence generates an infinite "paperfolding" polygon.

Theorem 2. Paperfolding polygons are selfavoiding (see Davis and Knuth [1]).

In this context, selfavoiding means that no intervals are described twice. Fig. 5 represents the beginning of the (F+) = polygon and fig. 6 the beginning of (F_F+) =.

Fig. 6. Fig. 5.

Quite obviously, and in spite of nonavoidness, these curves seem quite erratic. We shall now try to see how to measure chaos in plane curves.

4. Thermodynamics on curves

Let F be a finite rectifiable curve. O(F) represents the set of straight lines D which intersect F. A straight line D depends on two parameters p and O. We endow g2(F) with the canonical probability measure



5. Infinite curves

In the previous paragraph, we discussed finite length curves. We now wish to extend our definitions to infinite length curves (locally rectifiable).

Let F be an infinite length curve and let Fs represent the beginning portion of F of length s. We have seen that the entropy of Fs is

S = S(s) = log + e t3' - 1

with obvious notations. At this point, it is convenient to normalize the entropy by dividing by the factor log 2s. By definition, the entropy of the curve F is

h(r): lim S(s) s ~ log 2s "

If the limit does not exist, we define the upper entropy/~(F) by taking the superior limit and the lower entropy h (F) with the inferior limit. For all curves F

0_< _h(r)_< #(r)_< 1.

Infinite straight lines, exponential spirals (p = a °, a > 1) or any curves which go "rapidly" to infinity have 0 entropy. On the contrary, infinite curves which stay in a bounded region of the plane, or which "hesitate" to go to infinity such as the logarithmic spiral p = log ,9, have maximal entropy. (The exponential spiral appears in biology linked with shells. That the entropy of such spirals is 0 is thus not surprising. Life is a fight for order. On the other hand, spirals can be related to turbulence. Think of whirls. Can this remark justify the high entropy of the logarithmic spiral?)

Theorem 4. Infinite paperfolding polygons have entropy equal to I.

The proof of the above statement is not obvious and can be found either in Mendes France and Tenenbaum [5] or in Dekking, Mendes France and van der Poorten [3].

Paperfolding polygons lie between order and chaos. It is easy to see that selfavoiding curves on a lattice have entropies bounded by ~ so that paperfolding polygons are nevertheless as chaotic as can be.

6. Exponential sums

We devote this last paragraph to some interesting curves associated to exponential sums [2]. Given an infinite sequence of real numbers

/,/ : (UO, U l , . . . )

define


21rl t~ = log 2 1 r l - 1SKI'

so that the temperature is

log 21r} Z = ( 2]FI_I~KI) -1 .

The partition function is then

2F 1 z - - I ~ K L ~ -

and the entropy

S = ~T (T log Z)

, 2IFI, 13 = l o g ~ e ~ _ l .

exp(1/T) - 1

(1)

The computation of S and T has been done assuming the curve is in equilibrium. We extend the definition to any curve of finite length.

Theorem 3. The only curves that "exist" at T = 0 are straight line segments. Then S = O. The proof is obvious.

The partition function depends only on T so it seems difficult to deduce the volume V and the pressure P from Z. Yet one is lead to postulate that the length [FI is to be equated with volume and [~K1-1 with pressure P. (The larger P the more F is "squeezed" in the convex hull K.) Equation (1) reads

T = (log 2fi-V-]-j2PV ~-1

o r

2 P V - 1 - exp(-1/T)"

This is the equation of state. At high temperature

P V ~ ½ T

which proves that at high temperature, curves behave like perfect gases.


5. Infinite curves

In the previous paragraph, we discussed finite length curves. We now wish to extend our definitions to infinite length curves (locally rectifiable).

Let F be an infinite length curve and let F, represent the beginning portion of F of length s. We have seen that the entropy of Fs is

2s 3, S = S(s) = log ~ - ~ + e"' "------1

with obvious notations. At this point, it is convenient to normalize the entropy by dividing by the factor log 2s. By definition, the entropy of the curve F is

h(F)= lim S(s) s-= log 2s "

If the limit does not exist, we define the upper entropy/~(F) by taking the superior limit and the lower entropy _h(F) with the inferior limit. For all curves F

0 -< _h (F) -</~(F) -< 1.

Infinite straight lines, exponential spirals (p = a s, a > 1) or any curves which go "rapidly" to infinity have 0 entropy. On the contrary, infinite curves which stay in a bounded region of the plane, or which "hesitate" to go to infinity such as the logarithmic spiral p = log 0, have maximal entropy. (The exponential spiral appears in biology linked with shells. That the entropy of such spirals is 0 is thus not surprising. Life is a fight for order. On the other hand, spirals can be related to turbulence. Think of whirls. Can this remark justify the high entropy of the logarithmic spiral?)

1 Theorem 4. Infinite paperfolding polygons have entropy equal to ~.

The proof of the above statement is not obvious and can be found either in Mendes France and Tenenbaum [5] or in Dekking, Mendes France and van der Poorten [31.

Paperfolding polygons lie between order and chaos. It is easy to see that selfavoiding curves on a lattice have entropies bounded by ½ so that paperfolding polygons are nevertheless as chaotic as can be.

6. Exponential sums

We devote this last paragraph to some interesting curves associated to exponential sums [2]. Given an infinite sequence of real numbers

U ~" (U0, U l , . . . )

define

N - 1

ZN = ~'~ exp2i~ruk. k=0


In the complex plane, plot the points Zo = 0, zl, z2, . . , and join adjacent vertices ZN, ZN+I. We thus obtain an infinite polygon F(u).

Theorem 5. Let

Iz~ = max{Izl[ . . . . . IzNI},

then the entropy of F(u) is

,. log N . Iz~ -1 Hm up logN

_h = lim inf log N . ]z?~ -1 N-~ log N

Even though the proof of this theorem is elementary, we will not give it here. We prefer to illustrate the theorem by choosing different sequences u and studying the related curve F(u).

Example 1. u, = na (a irrational). In this case, the exponential sum ZN is bounded hence h = 1. The pattern is nevertheless very regular. See fig. 10 where we have drawn 200 sides of the polygon generated by the sequence u, = n~/-i-7.

Fig. 10.

Example 2. u,, = n2a (a irrational). To different values of a correspond different kinds of curves. We have plotted the 4000 first sides of the polygon associated with three different values of a. See figs. 11, 12, and 13.

Fig. 11 corresponds to a = ~/2 whose continued fraction expansion is particularly simple: ~/2= (1, 2, 2, 2 . . . . ). The fact that the partial quotients are bounded implies that the entropy of F(u) is ~.

Fig. 12 corresponds to a -- e = (2, 1, 2, 1, 1, 4, 1 . . . . ,1, 2n, 1, . . . ) . Computation yields h-> ½. Fig. 13 corresponds to a = ~" whose continued fraction is unknown. Hence we do not know the value

of the entropy. The shape of I'(u) is here particularly remarkable. Notice the spiral effect which can be


Fig. 11. Fig. 12.

%

Fig, 13.

related to a Cornu spiral. More surprising is the local periodicity of length 113 which illustrates the approximate formula

~- ~ 355/113.

E x a m p l e 3. u, = n log n, h = ½ (fig. 14).

E x a m p l e 4. u, = n ~, 0 < a < 1.

The curve F is a spiral, the entropy of which is h = a (fig. 15).

E x a m p l e 5. u, nX /n , h _ 1

A remarkable regularity which was explained by J.M. Deshouillers (fig. 16).


/

Fig. 14.


Fig. 15.

E x a m p l e 6. u , = n 2 log n, _h -> ~. A rather inhomogeneous erratic behavior (fig. 17).

7. A final remark

I am aware of the limits of the thermodynamic approach to the study of plane curves. In particular, the partition function

Z = ~ e -t3n n = l

defined in paragraph 4 may seem artificial (not to mention "volume" and "pressure"). Yet in a paper under preparation, I believe I can entirely justify the choice of the partition function by defining first the entropy

S = - ~ Pk log Pk k = l

then deducing Z. A last comment. Usually, mathematicians give tools to the physicists in order to develop their

theories (Einstein and tensors), and physicists invent concepts which mathematicians then try to "legalize" (L. Schwartz and the theory of distributions). I feel I belong neither here nor there.


Fig. 16.


t

Fig. 17.

References

[l] Ch. Davis and D. Knuth, Number representation and dragon curves I, II, Jour. Recreational Math. 3 (1970) 61-81, 133-149. [2] F.M. Dekking and M. Mendes France, Uniform distribution modulo one: a geometrical viewpoint, Jour. fiir die Reine und Angewandte

Mathematik 329 (1981) 143-153. [3] F.M. Dekking, M. Mendes France and A. van der Poorten, Folds! Mathematical Intelligencer 4 (1982) 130-138, 173-181, 190-195. [4] M. Hrnon and Y. Pomeau, Two strange attractors with simple structure, in: Turbulence and Navier-Stokes equation, Springer Lecture Notes in

Mathematics 565 (1976) 29~68. [5] M. Mendes France and G. Tenenbaum, Dimension des courbes planes, papiers plies et suites de Rudin-Shapiro, Bull. Soc. Math. France 109

(1981) 207-215. [6] M. Mendes France and A. van Poorten, Arithmetic and analytic properties of paperfolding sequences, Bull. Austral. Math. Soc. 24 (1981)

123-131. [7] L.A. Santal6, Integral geometry and geometric probability, Encyclopedia of Mathematics (Addison Wesley, 1976). [8] H. Steinhaus, Length, Shape and Area, Colloquium Mathematicum 3 (1954) 1-13.

folding paper and thermodynamics

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