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The Stern-Brocot Tree starts with “virtual vertices” (labeled 0/1 and 1/0 below) appended to the basic binary tree structure… Labels are constructed/assigned via a “mediant” process…add the numerators and denominators of the vertices on each side above the vertex… The fact that we can use information from only the previous row without reference to anyContinue reading “The Stern-Brocot Tree”

The idea of “discrete” can be modeled by a binary tree…by the parent/child relationship. The space between branches are the “holes” in the structure or the two parts separated by a boundary (“branches” in a binary tree: the two “children” of the one “parent”). The idea starts with a blank tree devoid of labels. WeContinue reading “The Binary Tree”

The title of this blog entry: in as much as it assumes there is such a “thing” as a beginning it assumes a certain relationship between the discrete and the continuous. Which comes “first”, discrete or continuous, how do they relate? Does the discrete “generate” the continuous as in parent/child?…or is there a “smooth” transition,Continue reading “The Beginning”

Read the previous blog post Towers of Hanoi, then answer the question,” how many cuts does it take to divide a strip of paper into 4 parts, 8 parts, 16 parts?” Likewise, how many jumps does it take to get from the first entry in our lists of code to the last entry? Answer: obviously,Continue reading “How we put Humpty back together again”

Check out this link to the On-Line Encyclopedia of Integer Sequences… https://oeis.org/A000975 There you will find all kinds of info related to the sequence 2,5,10,22,42,85,170,341,… The sequence is related to the discrete derivative/Gray’s coding that is the output of the DD tree. When examined closely the transformation of the order of the binary coding foundContinue reading “Towers of Hanoi”

The ultimate model is the world we live in….all of it, including us. There are no shortcuts…no map to crumble up and throw on the table so that we can cry,” aha….a fixed point!” …except in the mind of the one making the claim. Show me the map and I’ll show you the holes. TheContinue reading “The world we live in…”

So long as it is founded on assumption (read “axiom”) as opposed to model construction (see Skolem’s work), math is just religion….and religion is no more than a venue for working out who should be in charge, who should prosper economically, who should have access to resources, who should be allowed to publish, etc. Read…Continue reading “Math and Religion (or….math is religion)”

What appears below is copied from Section 2. at this link… https://plato.stanford.edu/entries/axiom-choice/ …closely related to the idea of an infinitesimal…less than any rational number, but not equal to 0. “…a countable collection of pairs of sets of real numbers fails to have a choice function”…….Let’s think about that… Cosets in R/Q+ determined by a pairContinue reading “A bit of Stanford Encyclopedia of Philosophy”

DD (discrete derivative) matters because the tree built to produce it row after row goes on to infinity. It would be impossible to talk about the whole set of binary sequences, both finite and infinite, if we tried to do it individually. No, we need to see the whole set at once. How else besidesContinue reading “Why does DD matter?”

Consider the modular form of the Stern-Brocot (S-B) tree. It is built starting with the 2×2 matrices a = [[00][10]] and b = [[01][00]]. (check out “Matrix Gateway to Geometric Algebra, Spacetime and Spinors by G. Sobczyk for an interesting connection) That pair of “vectors” when multiplied (it’s non-commutative) gives ab = [[10][00] and baContinue reading “Key idea”