Random Points on a Sphere (Generating Random Sequences III, Revisited)

While searching for old notes—that I haven’t found anyway—I returned to an old blog entry and I thought I was kind of unsatisfactory, with be best part being swept under the carpet with a bit a faery dust, and very handwavingly.

So let’s work-out how to uniformly distribute points on a sphere in a more satisfactory fashion.

Square roots (Part VI)

I’ve discussed algorithms for computing square roots a couple of times already, and then some. While sorting notes, I’ve came across something interesting: Archytas’ method for computing square roots.

Archytas’ method is basically the old Babylonian method, where you first set

,

,

and iterate

,

,

until desired precision is achieved (or the scribe is exhausted).

Unary numbers.

A positional number system needs a base that is either greater than one, or smaller than minus one—yes, we can have a negative base for a number system. The system, however, seems to break down if the base we chose is base 1.

If the base is 1, then there are no permissible digits since the digits , in a base system, must be . But we can still represent numbers using just 1s. That's the unary numeral system, and numbers are just represented as repeated 1s. 15? Fifteen ones: 111111111111111. Operations? Not very complicated, just… laborious.

Elementary Automata (Generating Random Sequences XI)

So 3-cells context elementary automata seem too “self-correcting” to be useful pseudo-random generators. What if we fixed that boundary problem and have the automaton run on a cylinder (with both end joined)? What if we augment the context from 3 to 5 cells?