# 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

Copyright (c) 1994 Linas Vepstas All Rights Reserved.