# The Farey Room

The Farey Room contains pictures generated by means transformations of the Farey Number Mapping, and through transformations of the Continued Fraction Mapping.

## Continued Fractions

The above shows the most basic transformation, where all occurances of "1" in the numerator of continued fraction expansions of the real number are replaced by "z". That is, if x = 1/(a+1/(b+1/(c+1/(d+ ...)))) then f(x) = z/(a+z/(b+z/(c+z/(d+ ... )))) mod 1. This f(x) is used to generate a Hausdorff measure for a given (x,z). The measure is shown as a color, with black=zero, blue=small, green=larger, yellow=large, red=larger still. Note that as z gets larger than one, it is technically undefined, as the continued fraction rockets out of control. However, computationally, x was always a rational (but barely ...), and so each continued fraction terminates. Along the horizontal axis, real numbers. Along the vertical axis, z, ranging from 0 to 2.

Note the psuedo-Sinai's Tongues which occur for all irrational values on the horizonal axis. This is essentially due to the fact that the mapping is discontinuous for all rational values.

Cosine Transform 1 -> cos (z)

Exponential Transform 1 -> exp (z)

An attempt to create a symmetrized version

Spherical Bessel (j0) Transform 1 -> j0(z)

Tall

Wide

## Farey Transforms

Symmetric component of above image

Anti-symmetric component of above image

Most of these images were generated during January and February of 1994, in Austin, Texas. The work was inspired by a Christmas reading of the "Contorted Fractions" chapter of John Conway's "On Numbers and Games".

Linas Vepstas February 1994