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.
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 fz(x) = 1/(a+z/(b+z/(c+z/(d+ ... )))) mod 1.
We refer to this fz(x) as the "Farey Transform" of x.
This fz(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
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.
Cmap Cosine Transform fr(x) with rn=cos(nz). Parameter z runs from zero at the bottom, to 2pi at the top.
Emap Exponential Transform fr(x) with rn=exp(-nt). Parameter t runs from one at the bottom to minus one at the top.
Jmap Spherical Bessel (j0) Transform rn=j0(nz)
Cn Sine Squared Transform rn=(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/(a1+1/(a2+1/(a3+1/(a4+ ...)))) then consider hg(x) = 1/(g(a1)+1/(g(a2)+1/(g(a3)+1/(g(a4)+ ...)))) In the road to the horizon, we use g(z)=1/z
Phat A 'trivial' reworking of the map: this shows x fz(x). Note since f1(x) = x, that the bottom row of pixels is just x2.
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 fz(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 f1(x)=x.
Define fz(x) = 1/(a+z/(b+z/(c+z/(d+ ... )))) mod 1. We refer to this fz(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 fz(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 fz(x) for irrational x is ill-defined, (although it may be possible to regularize it).
Linas Vepstas February 1994