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

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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.


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Cosine Transform 1 -> cos (z)


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Exponential Transform 1 -> exp (z)


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An attempt to create a symmetrized version


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


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Tall


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Farey Transforms

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Symmetric component of above image


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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.