The discrete derivative tree yields a reordering of labels on vertices in each row of the standard Binary Tree such that the elements of each row are strings that make a cycle of a single bit change between neighbors. Since the number of elements doubles from row n to row n+1 there is a ratio of 2:1 between periods. Progress through the tree is a “period doubling route to chaos”….the period of the limit string (the “leaves” of the tree) is infinite.
The doubling is a chain linked (timed) by the string of shortest period: …101010101010… (the form of the last input column in an “orderly” truth table and the sequence of branch labels across a row in the standard BinaryTree.) All other strings have a longer “wavelength” (period), lower “frequency”. The frequency of …101010101… is infinite. The frequency of the string of infinite length (the infinite cycle consists of a complementary pair that meets at infinity) is 1.
The complements in the limit string (there is only one…think of it as the “Hyper Webster” in one volume…see below) can be modeled as two loops that meet “tangentially” at infinity. We can think of it as two circles that rotate relative to each other like gears in a clock. The “crossing” over is indeterminate, as in “cannot be predicted”. The cycle in the circles is determinate, but the point at which the “complementary exchange” takes place is not since those finite strings (the rationals) can be of any length. That length approaches infinity as we move down the tree from row to row.
The exchange of information between cycles is bit by bit…governed by the fundamental frequency of each clock domain…each with its own period. “All the gears have the same pitch, but they have different diameters and hence number of teeth”…Brocot (of Stern-Brocot Tree fame) was a clock maker… it’s just a model.
We can also think of the limit row of the tree as consisting of a single circle where a “random” pair of opposite points on the circle are the points at which the complementary pair “begin and end”. (In projective space these two ends are identified giving us the connection to the model above.) There is exactly one “infinite cycle”, one master circle of two complements that contains every possible cycle as a partial string…the “endpoint/initial digit” is what varies…like a line sweeping around a circle determining “beginning and end”. One can count through an infinity of infinites just as Sharkovskii showed.
See the “proof” of the Banach-Tarski paradox… Here’s a good explanation using a four letter “dictionary”. Needless to say, that “proof” has a foundation in a model of ZFC which is not absolute. Take note that self-similarity plays a crucial role in the “proof”. The four letter “dictionary” part of the above explanation makes it clear. Here’s another link to an explanation via the HyperWebster….same idea using all 26 letters.
The first link is the best as it explains the connection to the DD tree….a single bit change can be up or down left or right…four possibilities for a relation between neighbors. Exactly what is described in the explanation where it shows the connection to the free group on two generators. Note this line…”So, by undoing a rotation, we seemingly create points out of thin air! This is the key trick that will let us duplicate a ball.” There is a connection here between what it takes to turn the shift-BITXOR process for producing the discrete derivative of a string into a 1-to-1 function via appending the initial bit (an element from the set {0,1}) to the output of shift-BITXOR.
The ordering of the “cycle” (the choice of bit where the “complements” meet) is the order determined via the DD tree. That tree determines both the period doubling and the ordering of the finite partial strings in the infinite product of the tree. In the infinite row there is a one bit change difference as the line sweeps around the circle….step by step, sequenced by the frequency of …10101010101… the “master string”. That “sequencing” is relative…the relation being that between periods (or frequency)….there is no “absolute” time, only an absolute shortest period. There is no meaning to be attached to “absolute time”.
The rows of finite strings are cycles that are related. Gauss proved that every binary code that is just a repeated finite string is the code of the continued fraction form of a quadratic number….a solution to a homogeneous quadratic equation. See the Minkowski ? function.
21st Century Paradox 