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→Self-similarity: fix bug and start wrapping it up. |
→Self-similarity: the concluding statement for this section |
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The first self-symmetry can be expressed as
:<math>r\circ c = c\circ r</math>
where the symbol <math>\circ</math>
:<math>L_D(x)= \frac{x}{2}</math> and <math>L_C(x)= \frac{x}{3}</math>
Then the Cantor function obeys
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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.
== Generalizations ==
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