Approximation properties Scaled minimal distance of the nearest rational number with a given denominator to . The graphic shows . Convergence of some sum and limit representations Convergents of the representation for as a function of . Convergence of the representation as a function of and . Convergence of the representation as a function of . Here the are the Fibonacci numbers. Parametrized curves for for various over the complex plane. Here is the truncated sum of the generalized Bailey–Borwein–Plouffe algorithm. The black dot indicates the value of . Fractional parts of the Beatty sequences Fractional parts of the numbers of the Beatty sequences , , , that lie in the intervals . Nested binary operations Values over the complex plane of the nested binary operations addition, subtraction, multiplication, and division using the arguments , , , , , and . Pseudorandom structures Random binary trees based on the digits of in various bases . Starting with the root, recursively new branches are formed if is odd. The are used in their natural order, starting with . Points generated by iteratively reflecting a point on the th edge of an . The first point is the origin and the are the digits of in various base . Iterations of the map starting with homogeneously distributed points of the unit square. The points are colored from red to blue with increasing iteration number. Iterations of the map starting with homogeneously distributed points of the unit square. The points are colored from red to blue with increasing iteration number.
