Enzymind Labs

"There's the Old Testament, the New Testament and the Handbook of Integer
Sequences", My Favorite Integer Sequences, Neil J. A. Sloane

"The relationships between number theory and dynamical systems are known to mathematics;
however, this knowledge has not yet escaped fairly narrow confines." Symmetries of Period-Doubling Maps, Linas Vepstas

The infinite sequences considered here, as well as others, have several direct relationships to number theory and combinatorics.

One of these relationships has to do with the sequence's expression as numbers. Fundamentally they are not necessarily numbers; they are infinite lists of abstract symbols. But if the symbols used are considered to be numerals (remember that numerals are ordered symbols, having meaning with respect to counting), then each infinite sequence can represent numbers -- strings of numerals with respect to a numeral system.

It is useful to consider these numbers as exiting in the range 0 to 1, by placing the decimal point at the head (left) of a sequence. Using the symbols [0 1] then gives a natural interpretation of the sequences as binary (base 2) numbers. With this interpretation, periodic (and constant) sequences represent rational numbers (and sometimes integers and natural numbers, which are subsets of rational numbers); rational numbers always have expansions that eventually repeat themselves. Non-periodic sequences represent irrational numbers (a subset of real numbers, those that aren't rational); irrational number expansions never repeat, i.e. their digit sequences are non-periodic.

[To Do: examples, 1000..., 1111..., Period Doubling number, Thue-Morse number, Euler's relation between fundamental irrational numbers]

It is striking to me that while the numeric equivalent of the Period Doubling and Thue-Morse sequences are irrational, they also have a definite connection to analytic number theory, which is concerned with prime integers and infinite sums of integers. I suppose it is not so striking to mathematicians -- irrational numbers are commonly expressed as infinite sums of terms formed from integers.

Examples of the relationship of integers to aperiodic patterns are given in:

Integral decimation of recurrent sequences

Also see:

Famous L-systems of mathematical history

"Cantor's manipulation of decimal expansions reflects a perception of digit expansions as a symbolic dynamical system. The symbolic states are simply the digits 0 through 9. A decimal expansion is a listing of the states over each generation; the number itself is the behavior of the dynamical system. This idea suggests all kinds of curious things to consider."

Patrick Morton
Connections between binary patterns and paperfolding
Journal de théorie des nombres de Bordeaux, 2 no. 1 (1990), p. 1-12 (PDF)

"... there is the beautiful fact discovered by Mendès France that [the 1-D Rudin-Shapiro sequence is exactly the direction sequence of the paperfolding sequence obtained by folding a rectangular piece of paper alternately under and over the left edge (which is held fixed). (See [2] or [4].)

[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. MR 684028 | Zbl 0493.10001

[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. MR 1004431 | Zbl 0694.10009

Distributions of Rationals on the Unit Interval (or, How to (mis)-Count Rationals), Linas Vepstas

Farey fractions, square lattices, and modular group fractals:

"The distribution of rationals on the unit interval is filled with surprises. As a child, one is told that the rationals are distributed ``uniformly'' on the unit interval. If one considers the entire set , then yes, in a certain narrow sense, this is true. But if one considers just subsets, such as the subset of rationals with ``small'' denominators, then the distribution is far from uniform and full of counter-intuitive surprises, some of which we explore below. This implies that using "intuition'' to understand the rationals and, more generally, the real numbers is a dangerous process. Once again, we see the footprints of the set-theoretic representation of the modular group at work.

This paper is part of a set of chapters that explore the relationship between the real numbers, the modular group, and fractals"