The Farey Room contains pictures generated by means transformations of the Farey Number Mapping, and through transformations of the Continued Fraction Mapping. Click on the small icons to see larger, more detailed views.

Artistic apologies: This material was originally presented with graphic design sensibiliteis that were more sparse, neutral, mystical and suggestive, which helped highlight the intricately twisted patterns. I now think that more literate readers would rather enjoy a bit of the actual mathematical background that goes into these pictures. And so I've added a small dose of math to the descriptions. My aplogies to the artistic types who would prefer dark and mystery.

**Palms**
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_{z}(x) = 1/(a+z/(b+z/(c+z/(d+ ... )))) mod 1.
We refer to this f_{z}(x) as the "Farey Transform" of x.
This f_{z}(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 goes negative, it is technically undefined,
as the continued fraction rockets out of control. However, computationally,
x is always a rational, and so each continued fraction terminates.
Along the horizontal axis, real numbers. Along the
vertical axis, z, ranging from +1 to -1 (+1 at bottom, -1 at top, 0 along
the midline).
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 of x, and z != 1.

One can play more complex games. If x = 1/(a

**Cmap**
Cosine Transform f_{r}(x) with r_{n}=cos(nz). Parameter
z runs from zero at the bottom, to 2pi at the top.

**Emap**
Exponential Transform f_{r}(x) with r_{n}=exp(-nt).
Parameter t runs from one at the bottom to minus one at the top.

**Jmap**
Spherical Bessel (j_{0}) Transform
r_{n}=j_{0}(nz)

**Cn**
Sine Squared Transform r_{n}=(1+cos(nz)). The goal here is to
avoid the pathological divide-by-zero's that occur in
**Cmap** when cos(nz) = -1. As is clear here, the figure
is far better behaved than **Cmap**.

**Magic Third**
As above, except only the range 1/2 < x < 2/3 is shown (not to scale).

**The Brush**
A different 1/2 < x < 2/3 map.

**Road to the Horizon**
If x =
1/(a_{1}+1/(a_{2}+1/(a_{3}+1/(a_{4}+ ...))))
then consider
h_{g}(x) =
1/(g(a_{1})+1/(g(a_{2})+1/(g(a_{3})+1/(g(a_{4})+ ...))))
In the road to the horizon, we use g(z)=1/z

**Phat**
A 'trivial' reworking of the map: this shows x f_{z}(x). Note
since f_{1}(x) = x, that the bottom row of pixels is just
x^{2}.

**Crystaline**
An attempt to create a symmetrized version.

These show the basic map on an extended range ... either wider (x runs zero to two) or taller (z runs zero to 4), or both. Note that we accidentally flipped some of these "upside down".

The basic map, which is combined with itself to
show the symmetric and the anti-symmetric components.
The color at pixel (x,z) is simply the value of the Farey Transform
f_{z}(x), as defined above.
Again, black=0.0, red=1.0, and a spectrum from blue to green in between.
Note that z runs from +1 at the top, to -1 downwards. Thus, the top
edge of the first picture is simple the straight line
f_{1}(x)=x.

**Farey Transform**- To construct the Farey Transform of x, start with the
continued-fraction expansion of x when 0 < x < 1:
x = 1/(a+1/(b+1/(c+1/(d+ ...))))
where a,b,c,... are positive integers. Note that this
expansion is unique: for any rational or irrational x, there is
only one, unique sequence a,b,c...
Note that if x is a rational number, then the continued fraction
has only a finite number of terms.
Define f

_{z}(x) = 1/(a+z/(b+z/(c+z/(d+ ... )))) mod 1. We refer to this f_{z}(x) as the "Farey Transform" of x. Note that as long as 0 =< z =< 1, the continued fraction is convergent for all values of x, both rational and irrational. Note that f(x) is a continuous function of x only when z=1. Note that f(x) is always well-defined for all rational values of x, since whenever x is rational, then the continued fraction terminates after a finite number of terms, and therefore, any manipulation on is therefore finite. Note that for 1 < z, that f_{z}(x) for irrational x seems to be stable and convergent in all cases (Quickie proof: each term is bounded, and the bounds converge as well or better than the z=1 case). Note that for z < 0, that f_{z}(x) for irrational x is ill-defined, (although it may be possible to regularize it). **Hausdorf Measure**- When we say 'Hausdorf Measure' above, what we really mean is this: Take the real number line between 0 and 1, and divide it into N equal bins. We start with each bin being empty (zero). Next, generate a seqeunce of random numbers 'x' between 0 and 1. Apply a transform (f(x) mod 1) to each x. Then, if f(x) falls in the i'th bin, increment the value of the i'th bin by one. When properly normalized, the resulting 'density' on the real-number line converges to a stable limit as N is increased, and as the number of samples is increased. We loosely call this value of the density the 'Hausdorf measure', in that it describes a practical way of taking a well-defined limit.

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". The importance of Farey Trees to fractal phenomena was previously brought to my attention by Paul Cvitanovi\'c during some lectures in Paris in 1985.

Linas Vepstas February 1994

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- P. Cvitanovi\'c, B. Shraiman, and B. Söderberg, Scaling laws for mode locking in circle maps, Phys. Scripta 32, 263-270, 1985.

Copyright (c) 1994 Linas Vepstas All Rights Reserved.