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==Self-similarity==
The Cantor function posses several [[symmetry|symmetries]]. For <math>0\le x\le 1</math>, there is a reflection symmetry
:<math>c(x)=1-c(1-x)</math>
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:<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; additonal 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>
== Generalizations ==
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