Re: fractal spirals

From: Gerry Myerson (gerry_at_maths.mq.edi.ai.i2u4email)
Date: 06/30/04 

Date: Wed, 30 Jun 2004 14:58:58 +1000

In article <200406300005.i5U05G723620@proapp.mathforum.org>,
 k.at-symbl.zxcvb.dot.org@mathforum.org (kevin doughty) wrote:

> Hello.
> I have posted here before and I am dismayed not to be able to find my
> old posts. I am an artist who has found an interesting and beautiful
> graph. I am trying to find out what it is called or where I can read
> about it.
>
> The formula is modified from the parametric description of a circle
> and is written as:
> x = sum (i = 1 to n) (r/i) sin (i*theta + i^2*pi*variable)
> y = sum (i = 1 to n) (r/i) cos (i*theta + i^2*pi*variable)

You may find this paper and the references useful.

MR1027862 (91d:11168)
 Moore, R. R.(5-MCQR); van der Poorten, A. J.(5-MCQR)
 On the thermodynamics of curves and other curlicues.
 Miniconference on Geometry and Physics (Canberra, 1989), 82--109,
 Proc. Centre Math. Anal. Austral. Nat. Univ., 22,
 Austral. Nat. Univ., Canberra, 1989.
11Z50 (11K06 11L03 58F99 82B99)

  The objective of the paper under review is to study "exponential
curves" defined as follows. Given a sequence of real numbers
$(\theta_n)$, the curve is a broken line whose $N$th vertex in the
complex plane has affix $S_N=\sum^{N-1}_{k=0}\exp 2i\pi \theta_k$. The
authors explore in detail the case $\theta_k= \tau k^2$, where
$\tau\in\bold R$ $(-\tfrac 12<\tau\leq \tfrac 12$ without loss of
generality). Let $S_N=S_N(\tau)$ be the associated exponential sum.
Poisson's formula implies $S_N(\tau)=|2\tau|^{-1/2}\exp 2i\pi\roman{sgn}
(\tau)S_{[2N\tau]}(-1/2\tau)\,+\,$error\break term.

 The graph $\Gamma_N(\tau)$ corresponding to $S_N(\tau)$ is thus
obtained from the simpler graph $\Gamma_{[2N\tau]} (-1/2\tau)$ which
consists of only $[2N\tau]\leq N$ edges. The error term in the above
formula takes into account the local behaviour of $\Gamma_N(\tau)$.
$\Gamma_{[2N\tau]}(-1/2\tau)$ then appears as "the skeleton" of
$\Gamma_N(\tau)$. Applying the same principle inductively enables one to
give a synthetic description of $\Gamma_\infty(\tau)$. The authors
profusely illustrate this idea and apply it for different values of
$\tau\:\tau=\pi-3=[0;7,15,1,292,1,\cdots]$,
$\tau=e-3=[-1;1,2,1,1,4,1,1,6,1,1,8,\cdots]$,
$\tau=\alpha-2=[-1;1,5,4,2,305,\cdots]$ where $\alpha$ is the real zero
of $\alpha^3-\alpha^2-\alpha-1$. The reader of this review will have
guessed that the map $\tau\to -1/2\tau$ renormalises
$\Gamma_\infty(\tau)$ and involves the continued fraction expansion of
$\tau$. The curve $\Gamma_\infty(\tau)$ thus depends heavily on the
arithmetic of $\tau$.

 All these ideas were, in essence, already described in previous
articles \ref[M. V. Berry and J. H. Goldberg, Nonlinearity 1 (1988),
no. 1, 1--26; MR0928946 (89b:58105); E. A. Coutsias and N. D.
Kazarinoff, Phys. D 26 (1987), no. 1-3, 295--310; MR0892449
(88h:11056); L. Callot and M. Diener, "Variations en spirales", Univ.
Oran, Oran, unpublished, 1984; per bibl.; J.-M. Deshouillers, in
Elementary and analytic theory of numbers (Warsaw, 1982), 75--82, PWN,
Warsaw, 1985; MR0840473 (88d:11069)].

 The novelty in the paper under review resides in the fine quality of
the graphs showing very explicitly the renormalisation principle. The
authors also apply their technique to higher order Gaussian sums and to
other sequences such as $\theta_n=(\log n)^d$ where $1\leq d\leq 6$ and
$\theta_n=n^{3/2}$.

 The study of exponential curves originates from joint papers written by
the reviewer with others in which he discussed the thermodynamics of
planar curves \ref[F. M. Dekking and the reviewer, J. Reine Angew. Math.
329 (1981), 143--153; MR0636449 (83b:10062); the reviewer, Phys. Rep.
103 (1984), no. 1-4, 161--172; MR0839681 (87i:58156)]. The article under
review actually recalls some of the reviewer's idea concerning
thermodynamics. Dimension, entropy and temperature of curves are
redefined. The authors misquote the reviewer on page 88 when they remark
that if the entropy is greater than $\tfrac 12$ then the curve must be
self-intersecting. This fortunately has no consequence, and the article
makes good reading.

 {For the entire collection see MR 90h:00015.}

Reviewed by M. Mendès France

© Copyright 2004, American Mathematical Society 

-- 
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)

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