The Farey Room

• Continued Fractions and Gaps explores the nature and distribution of the fractally-distrubted cleavages in these images.
• Symmetries of Period-Doubling Maps identifies the Modular Group SL(2,Z) as the basic symmetry group of the cleavages.
• The Bernoulli Map, the Gauss-Kuzmin-Wirsing Operator and the Riemann Zeta shows a relationship between fractals and the Riemann Zeta.
• The Minkowski Question Mark and the Modular Group SL(2,Z) shows that the distribution of Farey Fractions transforms under the dyadic representation of the Modular Group.

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-Arnold'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. A cross-section of this picture (a horizontal line drawn through it) is a Cantor set.

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

Linas Vepstas February 1994

References

• Stern-Brocot Tree
• The Minkowski Question Mark provides another important bridge between Farey Trees and Fractals.

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