consider the set of numbers whose simple continued fraction expansion terms are all 1 or 2. necessarily irrational.

inspired by the golden ratio being the most irrational number. this set is pretty irrational, no large terms.

smallest possible value is [1,2,1,2,1,2,...] = 1.366 = (1+sqrt(3))/2

[1,2,2,2...] = 1.414 = sqrt(2)

[1,1,1,...] = 1.618 = (1+sqrt(5))/2

[2,2,2...] = 2.414 = 1+sqrt(2)

[2,1,1,1,...] = 2.618 = 1 + (1+sqrt(5))/2

largest is [2,1,2,1,2,1,...] = 2.732 = 1+sqrt(3)

the set is (probably) a one-dimensional fractal like the Cantor set.

consider mapping binary bits to continued fraction terms: {0,1} -> {1,2}.

consider finite bitstrings and terminating continued fractions. tricky if the final continued fraction term is 1. perhaps repeating continued fractions.

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