the modular group. When discussing the modular group, the more common notation for g and r is T and S, but that notation conflicts with the symbols used here. The above three-dimensional representation is just one of many representations it can have; it shows that the blancmange curve is one possible realization of the action. That is, there are representations for any dimension, not just 3; some of these give the de Rham curves. Integrating the Blancmange curve Given that the integral of from 0 to 1 is 1/2, the identity allows the integral over any interval to be computed by the following relation. The computation is recursive with computing time on the order of log of the accuracy required. Defining one has that The definite integral is given by: A more general expression can be obtained by defining which, combined with the series representation, gives Note that This integral is also self-similar on the unit interval, under an action of the dyadic monoid described in the section Self similarity. Here, the representation is 4-dimensional, having the basis . Re-writing the above to make the action of g more clear: on the unit interval, one has . From this, one can then immediately read off the generators of the four-dimensional representation: and Repeated integrals transform under a 5,6,... dimensional representation. Relation to simplicial complexes Let Define the Kruskal–Katona function The Kruskal–Katona theorem states that this is the minimum number of (t − 1)-simplexes that are faces of a set of N t-simplexes. As t and N approach infinity, (suitably normalized) approaches the blancmange curve. See also Cantor function (also known as the Devil's staircase) Minkowski's question mark function Weierstrass function Dyadic transformation References Benoit Mandelbrot, "Fractal Landscapes without creases and with rivers", appearing in The Science of Fractal Images, ed. Heinz-Otto Peitgen, Dietmar Saupe; Springer-Verlag (1988) pp 243–260. Linas Vepstas, Symmetries of Period-Doubling Maps, (2004) Donald Knuth, The Art of Computer Programming, volume 4a. Combinatorial algorithms, part 1. . See pages 372–375. Further reading External links Takagi Explorer (Some properties of the Takagi function) De Rham curves Theory of continuous functions