Editing Cantor function (section)

==Self-similarity==
The Cantor function possesses several [[symmetry|symmetries]]. For <math>0\le x\le 1</math>, there is a reflection symmetry 
:<math>c(x)=1-c(1-x)</math>
and a pair of magnifications, one on the left and one on the right:
:<math>c\left(\frac{x}{3}\right) = \frac{c(x)}{2}</math>
and
:<math>c\left(\frac{x+2}{3}\right) = \frac{1+c(x)}{2}</math>
The magnifications can be cascaded; they generate the [[dyadic monoid]]. This is exhibited by defining several helper functions.  Define the reflection as
:<math>r(x)=1-x</math>
The first self-symmetry can be expressed as
:<math>r\circ c = c\circ r</math>
where the symbol <math>\circ</math> denotes function composition. That is, <math>(r\circ c)(x)=r(c(x))=1-c(x)</math> and likewise for the other cases. For the left and right magnifications, write the left-mappings
:<math>L_D(x)= \frac{x}{2}</math> and <math>L_C(x)= \frac{x}{3}</math>
Then the Cantor function obeys
:<math>L_D \circ c = c \circ L_C</math>
Similarly, define the right mappings as
:<math>R_D(x)= \frac{1+x}{2}</math> and <math>R_C(x)= \frac{2+x}{3}</math>
Then, likewise, 
:<math>R_D \circ c = c \circ R_C</math>
The two sides can be mirrored one onto the other, in that
:<math>L_D \circ r = r\circ R_D</math>
and likewise,
:<math>L_C \circ r = r\circ R_C</math>
These operations can be stacked arbitrarily. Consider, for example, the sequence of left-right moves <math>LRLLR.</math> Adding the subscripts C and D, and, for clarity, dropping the composition operator <math>\circ</math> in all but a few places, one has:
:<math>L_D R_D L_D L_D R_D \circ c = c \circ L_C R_C L_C L_C R_C</math>
Arbitrary finite-length strings in the letters L and R correspond to the [[dyadic rationals]], in that every dyadic rational can be written as both <math>y=n/2^m</math> for integer ''n'' and ''m'' and as finite length of bits <math>y=0.b_1b_2b_3\cdots b_m</math> with <math>b_k\in \{0,1\}.</math> Thus, every dyadic rational is in one-to-one correspondence with some self-symmetry of the Cantor function.

Some notational rearrangements can make the above slightly easier to express. Let <math>g_0</math> and <math>g_1</math> stand for L and R. Function composition extends this to a [[monoid]], in that one can write <math>g_{010}=g_0g_1g_0</math> and generally, <math>g_Ag_B=g_{AB}</math> for some binary strings of digits ''A'', ''B'', where ''AB'' is just the ordinary [[concatenation]] of such strings.  The dyadic monoid ''M'' is then the monoid of all such finite-length left-right moves. Writing <math>\gamma\in M</math> as a general element of the monoid, there is a corresponding self-symmetry of the Cantor function:
:<math>\gamma_D\circ c= c\circ \gamma_C</math>

The dyadic monoid itself has several interesting properties. It can be viewed as a finite number of left-right moves down an infinite [[binary tree]]; the infinitely distant "leaves" on the tree correspond to the points on the Cantor set, and so, the monoid also represents the self-symmetries of the Cantor set. In fact, a large class of commonly occurring fractals are described by the dyadic monoid; additional examples can be found in the article on [[de Rham curve]]s.  Other fractals possessing self-similarity are described with other kinds of monoids. The dyadic monoid is itself a sub-monoid of the [[modular group]] <math>SL(2,\mathbb{Z}).</math>

Note that the Cantor function bears more than a passing resemblance to [[Minkowski's question-mark function]]. In particular, it obeys the exact same symmetry relations, although in an altered form.