few places, one has: 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 for integer n and m and as finite length of bits with 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 and stand for L and R. Function composition extends this to a monoid, in that one can write and generally, 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 as a general element of the monoid, there is a corresponding self-symmetry of the Cantor function: 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 curves. Other fractals possessing self-similarity are described with other kinds of monoids. The dyadic monoid is itself a sub-monoid of the modular group 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. Generalizations Let be the dyadic (binary) expansion of the real number 0 ≤ y ≤ 1 in terms of binary digits bk ∈ {0,1}. This expansion is discussed in greater detail in the article on the dyadic transformation. Then consider the function For z = 1/3, the inverse of the function x = 2 C1/3(y) is the Cantor function. That is, y = y(x) is the Cantor function. In general, for any z < 1/2, Cz(y) looks like the Cantor function turned on its side, with the width of the steps getting wider as z approaches zero. As mentioned above, the Cantor function is also the cumulative distribution function of a measure on the Cantor set. Different Cantor functions, or Devil's Staircases, can be obtained by considering different atom-less probability measures supported on the Cantor