Part One: Totemic π, φ and √2 contrasted with their algorithmic output streams
Standing behind the static icon π, there is a whole family of algorithms, each of which generates the same dynamic output stream, the one that begins with 3.1415... Standing behind the icon φ, there is a family of algorithms whose output is the torrent of digits that begins with 1.6180... Similarly, behind iconic √2 and e, there exist algorithm families that generate 1.4142... and 2.7182... respectively. Forever. My aim here is to focus on algorithms in preference to their respective icons, which are a kind of fool’s gold whose glitter and word-magic are dangerous distractions.
To conserve space, I present an analysis of √2 only, but my method of inquiry applies equally to π and φ. (Totemic e follows as a separate case whose mystique can be deprogrammed by a shorter kind of analysis. In Berlinski page 280, e is hailed as “the black jewel of the calculus.” Now I acknowledge that e is invaluable as a tool, and notable for being ubiquitous, but the fact is that y=ex is a slope which must occur somewhere between the curves y=2x and y=3x; no mystery at all. See Boyce 2013 Appendix E, which is devoted to thus demystifying e.)
I choose √2 as representative of the whole group because √2 marks a turning point in history, not literally as ‘the square root of two’ or ‘the surd √2’ (neither of which was present in the vocabulary of the ancients) but as the first of the Hellenic incommensurables (Jourdain 13, 60; Dedekind 9). Taking a closer look, we find that the crisis was not about ‘the unit square diagonal’ (as often stated) but a problem intimately related to that: Suppose each of the two Pythagorean ‘legs’ is the unit square; clearly this means the ‘head’ is a square with area 2. But what is the edge of the large square? It would seem to measure just a shade greater 7/5 (aka 7:5 by Egyptian reckoning or 1.4 in contemporary notation). But how much greater exactly than 7/5? That is the question that embarrassed the Pythagoreans and started the ball rolling toward the iconic (but fallacious) √2 of today. Please refer to Figures 1a and 1b.