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The modular group action¶
The library allows to build figures out of cycles which are obtained from each other by means of fraction-linear transformations (FLT). We are going to illustrate this by the action of the modular group $\mathrm{SL}_2(\mathbb{Z})$ on a single circle [1, Section 14.4]. We repeatedly apply FLT $T= \begin{pmatrix} 1&1\\0&1 \end{pmatrix}$ for translations and $S= \begin{pmatrix} 0&-1\\1&0 \end{pmatrix}$ for the inversion in the unit circle.
First we are loading the library and define the number of iterations.
from figure import *
%load_ext figure.figure_ipython
show_asy_off()
steps = 3
trans = 6
epsilon=0.00001 # square of radius for a circle to be ignored
We will use the metric associated to the figure, it can be extracted by [[get_point_metric]] method.
F = figure()
e = F.get_point_metric();
Firstly, we add to the figure an initial cycle and, then, add new generations of its shifts and reflections.
a = F.add_cycle(cycle2D([0, numeric(3, 2)], e, numeric(1, 4)), "a")
c = F.add_cycle(cycle2D([0, numeric(11, 6)], e, numeric(1, 36)), "c")
We want to shift all cycles in the previous generation. Their key are grasped by get_all_keys() method. Each cycle with the collected key is shifted horizontally by an
integer t in range [-trans, trans]. This done by moebius_transform() relations and it is our responsibility to produce proper Clifford-valued entries to the matrix, see \cite{Kisil05a}*{\S~2.1} for an advise.
for i in range(steps):
L = F.get_all_keys(2*i, 2*i)
for ck in L:
L1 = F.get_cycles(ck) # Each cycle node may have several cycles
for x in L1 :
for t in range(-trans, trans+1):
# We shift initial cycles by zero in order to have their copies in the this generation.
if ((t !=0 or i ==0) \
# To simplify the picture we are skipping circles whose radii
# would be smaller than the threshold.
and (x.det()-(pow(t, 2)-1)*epsilon).evalf() > 0):
b = F.add_cycle_rel(moebius_transform(ck, True, \
[dirac_ONE(), t*e.subs({e.op(1).op(0) : 0}), 0, dirac_ONE()]), "%s-%s-%s" % (str(x), i, t) )
# We want the colour of a cycle reflect its generation, smaller cycles
# also need to be drawn by a finer pen. This can be set for each cycle
# by [[set_asy_style]] method, see its description in~\ref{sec:two-dimens-graph} for details.
F.set_asy_style(b, "rgb(0,0,%.2f)+%.3f" % (1-1/(i+1.),1/(i+1.5)))
# Similarly, we collect all key from the previous generation cycles
# to make their reflection in the unit circle.
L = []
if i < steps-1:
L = F.get_all_keys(2*i+1,2*i+1)
for ck in L:
# This time we keep things simple and are using [[sl2_transform]]
# relation, all Clifford algebra adjustments are taken by the
# library. The drawing style is setup accordingly.
b = F.add_cycle_rel(sl2_transform(ck, True, [0, -1, 1, 0]), "%ss" % str(ck))
F.set_asy_style(b, "rgb(0, 0.7, %.2f)+%.3f" % (1-1/(i+1.), 1/(i+1.5)))
Finally, we draw the picture. This time we do not want cycles label to appear, thus the last parameter with_labels of asy_write() is False. We also want to reduce the size of \Asymptote\ file and will not print headers of cycles, thus specifying with_header=True. The remaining parameters are explicitly assigned their default values.
u = F.add_cycle(cycle2D([0, 0], e, numeric(1)), "u")
%asy_write F, 600, -2.17, 2.17, 0, 2, "modular-group", with_labels=False
