Enzymind Labs

Related papers, books, and on-line material:

Iterated origami examples

Modular Origami

papercraft from Haruki Nakamura, see the 3-D logarithmic spiral examle (bottom of page)

 

From Wikipedia's Penrose tiling entry:

The substitution scheme introduces φ as a scaling factor; its matrix is the square of the Fibonacci substitution matrix; implemented as a symbol sequence ( e.g. 1→101, 0→10) this substitution produces a series of words with lengths which are the Fibonacci numbers with odd index, F(2n+1) for n=1,2,3.., the limit being the infinite Fibonacci binary sequence.

On Canonical Substitution Tilings, Edmund O Harriss, 2004 PhD thesis, Department of Mathematics Imperial College

An excellent historical development and discussion of the relationship between symbol substitution systems and aperiodic tilings:

Abstract
This thesis is concerned with canonical substitution tilings. These are tilings
generated by the canonical projection method which admit substitution rules,
and include the famous Penrose tilings. We characterise all canonical substitution
tilings and consider the question what the set of all substitution rules is for a given
tiling. In many cases, including the Penrose tilings, we are able to characterise
all the substitution rules for the tiling. Our methods are constructive and give an
algorithm to construct the substitution rules and tilings.

The Mathematics of Long-Range Aperiodic Order (R. V. Moody, 1997)

In this book devoted entirely to the mathematics of long-range aperiodic order the reader will find survey and research articles on the major areas of mathematics and mathematical physics that are emerging in this new field, including tilings, discrete geometry, diffraction and harmonic analysis, self-similarity and symmetry, non-crystallographic root systems, the cut and project method, number theoretical considerations, aperiodic Ising models and Schrodinger operators.

Kauffman, Stuart quote:

"Pick up a pinecone and count the spiral rows of scales. You may find eight spirals winding up to the left and 13 spirals winding up to the right, or 13 left and 21 right spirals, or other pairs of numbers. The striking fact is that these pairs of numbers are adjacent numbers in the famous Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21... Here, each term is the sum of the previous two terms. The phenomenon is well known and called phyllotaxis. Many are the efforts of biologists to understand why pinecones, sunflowers, and many other plants exhibit this remarkable pattern. Organisms do the strangest things, but all these odd things need not reflect selection or historical accident. Some of the best efforts to understand phyllotaxis appeal to a form of self-organization. Paul Green, at Stanford, has argued persuasively that the Fibonacci series is just what one would expects as the simplest self-repeating pattern that can be generated by the particular growth processes in the growing tips of the tissues that form sunflowers, pinecones, and so forth. Like a snowflake and its sixfold symmetry, the pinecone and its phyllotaxis may be part of order for free."

Stuart Kauffman At Home in the Universe, Oxford University Press, 1995, p 151. (1) Available from Amazon.com

I am a Strange Loop (Wikipedia), I am a Strange Loop (Amazon), Douglas Hofstadter

From The Washington Post's Book World/washingtonpost.com
Reviewed by Peter D. Kramer

Okay, I think, therefore I am. But who gets to play that game? A newborn? A mosquito? A computer? If my thoughts are elsewhere, am I here or there? When I no longer think as I once did, am I the same person? What composes this "I," molecules or memories?

Questions about the boundaries, location, continuity and constituents of the self stand at the heart of philosophy, but a mathematician and physicist, René Descartes, set the terms of the discussion. Who better to bring us up to date than Douglas Hofstadter? Trained in math and physics, Hofstadter won a 1980 Pulitzer Prize for Gödel, Escher, Bach, a bravura performance linking logic, art and music. He returns now to apply a concept from that book, the strange loop, to the definition of self.

Like consciousness, the strange loop is elusive. When a brilliant author uses one slippery concept to clarify another, the result for the reader can be anxiety. Page after page, we may wonder whether we will reach the limit of our understanding and whether the journey will be worth the effort.

Fortunately, Hofstadter is a gifted raconteur and a master of metaphor. He conjures up a car with a 16-cylinder motor and what the salesman calls Racecar Power®. It's not as if you can get a model that has the engine without the trademark feature. Similarly, Hofstadter writes, "consciousness is not an [added] option" for beings evolved to engage in symbolic thought, recognize patterns, create categories, reason via analogies and wonder about the self. Consciousness is "the upper end of a continuous spectrum of self-perception levels that brains automatically possess as a result of their design."

Hofstadter's strange loop is the feedback loop. Point a video camera at a TV displaying the camera's output, and you will produce a receding corridor of screens. Pixels make up the picture, but our interest is in the image, the tunnel of rectangles. Identity resembles that phenomenon. Never mind the neurons that make up our brain. Our emotions, others' responses and our repeated looks outward to the world and inward to ourselves shape what we call our self. Nor is ours the only loop we contain. We know how our friends see things; our mind houses their perspectives -- it has the formulae for producing their thoughts.

However mechanistic, Hofstadter's account of the self emerges from deep emotion. In 1993, when she was 43, Hofstadter's wife and soul mate, Carol, died suddenly of a brain tumor. Three months into his mourning, Hofstadter initiated a heartfelt correspondence with the philosopher Daniel Dennett. What emerged was Hofstadter's understanding of self as distributed over many minds, a concept that explained how Carol's "personal sense of 'I' " lived on (in "low-resolution fashion") as a "loop" in Hofstadter's consciousness.

I have so far given a superficial account of Hofstadter's position. As his book title indicates, for Hofstadter the self is a strange loop. Strange loops are reflexive and paradoxical, like M.C. Escher's impossible image of right and left hands drawing each other into existence. Hofstadter's example of a real-world strange loop is a key construct in Kurt Gödel's incompleteness theorem, published in 1931, a proof that any seemingly comprehensive mathematical system will contain true statements that cannot be proven. The theorem is notoriously indigestible.

How elusive is this strange loop? I was a young math buff. Last year, when Discover magazine surveyed authors about science writing that had influenced them, atop my list was a popularization of Gödel's proof by Ernest Nagel and James Newman -- the same book that inspired Hofstadter in his teens. If I am, for that reason, an ideal audience for the strange loop theory, there's good news and bad. I found Hofstadter's explication of Gödel revelatory. There were implications of the proof that I had never appreciated. But then (here's evidence for discontinuity), I no longer quite understand the proof. Nor, given what I do grasp, am I convinced that the entities Gödel conceived are apt analogues of the self.

This difficulty does surprisingly little to diminish Hofstadter's achievement. Philosophers of mind are divided between those who see consciousness as a special quality (like Racecar Power®) and those who see it as irredeemably physical (like neural networks). Hofstadter points to another level at which self might exist, up among the symbols and patterns -- or rather, to various levels on which self exists simultaneously. His conclusions mesh well with those of psychotherapy. We are not selves first and social creatures later. It's through empathy that we develop a rich sense of self. Nor is the self neatly demarcated. We contain multitudes.

Godel, Escher, Bach, Douglas Hofstadter

Regarding scale-free networks and emergence, I've been re-reading Godel, Escher, Bach. It's extraodinarily rich on this topic. It makes a lot more sense to me now that I've been thinking a lot about formal systems and the number theory of sequences.

The trouble with five , a Plus Magazine article that explores five fold symmetry tiling, including Penrose aperiodic tilings, and substitution/deflation rules. Nice image of a five fold similarity tiling.

2006:

• • • • • The Ubiquitous Thue-Morse Sequence, Invited talk for the Research Experiences for Undergraduates program at University of North Carolina, Greensboro, July 14 2006.
      • New Directions in State Complexity, Invited talk for the DCFS 2006 conference, June 22 2006.
      • A series of 4 expository talks for the Summer School CANT (Combinatorics, Automata, and Number Theory) at the University of Liège, Belgium, May 8-12 2006.
        • Talk 1: Morphisms and morphic words (introduction)
        • Talk 2: Automatic sequences
        • Talk 3: Fixed points of morphisms
        • Talk 4: Periodicity, morphisms, and matrices

Schoenfeld, Alan (1992.) Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. [Chapter 15, pp. 334-370, of the Handbook for Research on Mathematics Teaching and Learning (D. Grouws, Ed.). New York: MacMillan, 1992.] Retrieved 4/27/06:

"...[math is] the science of patterns—systematic attempts, based on observation, study, and experimentation, to determine the nature or principles of regularities in systems … The tools of mathematics are abstraction, symbolic representation, and symbolic manipulation."

A Fractals Unit for Elementary and Middle School Students

Tess, tesselated drawing program:

"With Tess, you can quickly create attractive symmetric planar illustrations. While you draw, Tess will automatically maintain the symmetry group you have chosen; 24 rosette, all 7 frieze, and all 17 wallpaper groups are included."

Also Heesch tilings.

From Introduction to Dynamical Systems (see below):

"Axel Thue came up with it in 1912 [Thu12] in his study of formal languages (not the term he used then). Marston Morse rediscovered the same sequence in 1917 [Mor21] in studying the dynamics of 'geodesics' on surfaces."

Marston Morse. Recurrent geodesics on a surface of negative curvature. Trans. Amer. Math. Soc., 22:84-110, 1921.

Axel Thue. Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid.-Akad. Oslo Mat.-Natur. Kl. Skr. (N.S.), (1), 1912.

Introduction to Dynamical Systems, an excellent set of class notes:

• Introduction to Dynamical Systems
  • Symbolic dynamical systems
  • Discrete-time systems
  • Continuous-time systems
  • Fractal long-term behavior
• Computational aids
  • Netscape
  • MAPLE
  • FRACTINT
  • Supplementary program files
• L-systems
  • Basic definitions
  • Fibonacci L-system
  • Types of L-systems
  • Thue-Morse L-system
  • Paperfolding and the Dragon curve
  • Turtle graphics and L-systems
  • FRACTINT conventions
  • Branching and bracketed L-systems
  • Famous L-systems of mathematical history
  • Self-similarity and scaling
• Tilings as Dynamical Systems
  • Basic concepts
  • Euclidean similarities
  • Examples of self-similar tilings
  • Periodic and recurrent tilings
  • Quasicrystals
  • Self-similar tilings arising from projections
  • Complex numbers and similarity constants
• Dynamical Systems and Fractal Basics
  • What is a fractal?
  • Metric spaces
  • Dynamical systems, orbits and attractors
  • Rotations of a circle
  • Topological dimension
  • Fractal dimension
  • Dimension computed by growth and by self-similarity
  • Some sample computations
• Iterated Function Systems
  • Self-similarity and sets of affine maps
  • Contractions and Fixed Points
  • Linear Contractions
  • Contraction mappings and hyperbolic IFS's
  • Calculating the attractor by random iteration
  • Plotting IFS's with FRACTINT
  • Designing IFS's: the Collage Theorem
• Interval Self-mappings
  • Iteration of a function f(x)
  • Web diagrams and fixed points
  • Population dynamics and iteration of functions
  • Quadratic mappings of the unit interval
  • Down the waterfall
  • Falling all the way: Cascade diagrams
  • Sarkovskii's theorem and the islands of stability in the cascade diagram
• Complex Iteration
• Outside work
  • Extra Readings
  • Problems
• References
• About this document ...

The Morse-Thue Sequence, Ralph E. Griswold
Department of Computer Science
The University of Arizona, Tucson, Arizona

Short blurb summarizing interesting properties, with graphic representations.

Photonic band gaps analysis of Thue-Morse multilayers made of porous silicon

Luigi Moretti, Ilaria Rea, Lucia Rotiroti, Ivo Rendina, Giancarlo Abbate, Antigone Marino, and Luca De Stefano

Optics Express, Vol. 14, Issue 13, pp. 6264-6272

"Dielectric aperiodic Thue-Morse structures up to 128 layers have
been fabricated by using porous silicon technology. The photonic band gap
properties of Thue-Morse multilayers have been theoretically investigated
by means of the transfer matrix method and the integrated density of states.
The theoretical approach has been compared and discussed with the
reflectivity measurements at variable angles for both the transverse electric
and transverse magnetic polarizations of light. The photonic band gap
regions, wide 70 nm and 90 nm, included between 0 and 30°, have been
observed for the sixth and seventh orders, respectively."

Spectral properties of one dimensional quasi-crystals

Linas Vepstas' expositions about modular groups, Farey sequences and their relationship to period-doubling behavior in fractal dynamical systems

"An Introduction to the Theory of Numbers"

"This book was first published in 1938, and is still in print, with the latest edition being the 5th (1980). It is likely that almost every serious student and researcher into number theory has consulted this book, and probably has it on their bookshelf. It was not intended to be a textbook, and is rather an introduction to a wide range of differing areas of number theory which would now almost certainly be covered in separate volumes. The writing style has long been regarded as exemplary, and the approach gives insight into a variety of areas without requiring much more than a good grounding in algebra, calculus and complex numbers."

list of number theory and fractality references

From Symbolic Dynamics to a Digital Approach: Chaos and Transcendence

We review recent progress on Feigenbaum attractors and their inter-connection with Number Theory. We further enlight the relation between Chaos and Transcendence.

Remarks on the spectral properties of tight binding and Kronig-Penney models with substitution sequence, Anton Bovier, Jean-Michel Ghez

We comment on some recent investigations on the electronic properties of models associated to the Thue-Morse chain and point out that their conclusions are in contradiction with rigorously proven theorems and indicate some of the sources of these misinterpretations. We briefly review and explain the current status of mathematical results in this field and discuss some conjectures and open problems

[Fourier spectral properties of recursive substitution sequences]

"A New kind of Science", S. Wolfram (2002). Also see my comments on "A New Kind of Science".

Von Koch and Thue-Morse revisited

"Section 5 Automatic sequences and fractal objects
The purpose of this section is to emphasize that there is a general link between automatic
sequences and fractal objects, and not only between the Thue-Morse sequence and the von
Koch curve."

Turbulence 14.5.2 Feigenbaum Sequence, Kip Thorne

Relationship of the logistic equation and period doubling behavior to a route to chaos (onset of turbulence) in fluid mechanical systems.

APPLICATIONS OF CLASSICAL PHYSICS, Kip Thorne (the whole book, amazing)

Symbolic Dynamics, Entropy and Complexity of the
Feigenbaum Map at the Accumulation Point , WERNER EBELING and KATJA RATEITSCHAK

The paper relates the First Feigenbaum sequence (Period-doubling sequence) to the critical point ("accumulation point") of the logistic map dynamics. It considers this sequence a "...prototype of complex linear structures. Coming from parameters below the Feigenbaum point, the sequences show multiperiodicity and with decreasing distance to the accumulation point, more and more complicated structures arise. The structures at the accumulation point itself are hierachical and may be characterized by grammatical rules...". I am also considering this sequence, and the Thue-Morse sequence, a prototype of complex linear structure. It also considers an information theoretic characterization of the sequence, through an entropy metric. For example, given a subset of the sequence, how predictable are the surrounding values? It does not address how the "grammar" (production rules) can be gleaned from a sequence, except perhaps "...by the way conditional entropies decay."

Abstract: "This paper aims to make further contributions to the exploration of the symbolic dynamics generated by the logistic map at Feigenbaum accumulation point. In particular we are interested in the grammar of these sequences; completing earlier studies we study here arbitrary partitions also. Our main aim is the investigation of the special grammars which characterize the long-range correlations between letters. Considering these sequences as standard examples of a complex system, we introduce and discuss a complexity function derived from the conditional entropies. Further we discuss local predictabilities."

"A new approach in the investigation of the Feigenbaum sequence is based on a symbol sequence generator [6,7]. A symbol sequence generator is a set of deterministic and stochastic grammatical rules to construct symbol sequences. If one applies the replacement rules
1 -> 10
0 -> 11

to an infinitely long Feigenbaum sequence then one gets the same symbol sequence." [Note: It doesn't see to me that this is anything new. Of course a sequence can be investigated via the rules that generate the sequence.]

"Prime Obsession" by John Derbyshire (2002), a very good popular and technical presentation of many ideas in analytic number theory. Its core theme is the Reimann Hypothesis. Half biographical sketches and history of math/science, and half the real meat of the math behind the hypothesis. Best if the reader has a background up to or beyond basic calculus, but readable for any critical thinker.

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104254130

http://etd.uwaterloo.ca/etd/m2wang2004.pdf

Counting the occurrences of generalized patterns in words generated by a morphism

A Strange Recursive Sequence ( { 0; 0->01, 1->011 }, includes a non-recursive formula for the nth digit)

Non-commutative geometry, by Alain Connes.

The Ubiquitous Prouhet-Thue-Morse Sequence. Allouche, J.-P.; Shallit, J. O. Many applications and some history

MathWorld: Thue-Morse Sequence. Some other applications

Thue-Morse Sequence (sequence A010060 in OEIS)

Thue-Morse Sequence over (1,2) (sequence A001285 in OEIS)

When Thue-Morse meets Koch A paper showing an "astonishing" similarity between the Thue-Morse Sequence and the Koch snowflake (but see Von Koch and Thue-Morse revisited below, which points out that this is not astonishing, and that TM is related to a broad class of fractals)

Reducing the influence of DC offset drift in analog IPs using the Thue-Morse Sequence A technical application of the Thue-Morse Sequence

http://www.mth.msu.edu/~sagan/Papers/Old/ccm.pdf

http://arxiv.org/PS_cache/math/pdf/0407/0407326v1.pdf

Sequences:

{ 0; 0->1, 1->01 } The number of symbols in each generation of the construction of these words follows the Fibonacci sequence. The resulting two (infinite length) words that result from the recursion are periodic points of the recursion with period 2.