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 Pi

Plots based on the continued fraction digits (280 graphics)

Continued fraction digits

Plots of the simple continued fraction digits of .

Plots of of the simple continued fraction digits of . For almost all numbers, this sequence converges to the Khinchin constant .

Plots of of the simple continued fraction digits of in various bases. For almost all numbers, this sequence converges to the Khinchin harmonic mean .

Plots of where and are the simple continued fraction digits of .

Ratio between the actual number of occurrences of the digit in the simple continued fraction expansion of to the expected number according to the Gauss–Kuzmin distribution law.

Continued fraction convergents

Difference of the scaled convergents of the simple continued fraction expansion of .

The th root of the denominator of the convergents of the simple continued fraction expansion of . For almost all numbers, this sequence converges to the Khinchin–Levy constant .

The number of correct decimal digits minus of the th convergents of the simple continued fraction expansion of . By Lochs' theorem for almost all numbers, the th convergent gives correct decimal digits.

Nested differences, sums, products, and quotients

Iterated pairwise sums (left) and differences (right) modulo 2 starting with the first 200 simple continued fraction digits of .

Cellular automata plots

Evolution of the elementary cellular automata with initial conditions where the ere the simple continued fraction digits of .

Continued fraction values of base b digits

Values obtained by interpreting the base digits of as the simple continued fraction digits of a number.

Values obtained by interpreting the base digits of as the continued fraction digits of a number.

Values obtained by interpreting the base digits of as the continued fraction digits of a number for noninteger bases.

Eigenvalue plots

Eigenvalues in the complex plane of the square matrices formed from the first digits of the simple continued fraction expansion of , . The eigenvalues are colored from red () to blue ().

Eigenvalues in the complex plane of the square matrices formed from the first digits of the simple continued fraction expansion of . The matrix formed is . The eigenvalues are colored from red () to blue ().

Root plots

Zeros of the polynomials of the digits of the simple continued fraction expansion of over the complex plane.

Fourier series

Fejer averages of the exponential Fourier series , over the complex plane. Here the are the digits of the simple continued fraction of . The coloring of the curves varies from red () to violet ().

Pascal triangle modular plots

Density plot of the digits of the simple continued fraction of . The first digits are extracted and arranged in a triangle and colored according to their value modulo , two and three.

Power series plots

3D plots of the power series . Here, is the th digit of the simple continued fraction of . The left graphic shows the real part and the coloring is according to the imaginary part. The right graphic shows a contour plot of the argument over the complex plane and the coloring is cyclic.

Density plots of coefficients of of the power series . Here, is the th digit of the simple continued fraction of . The left graphic shows the coefficients modulo 2 and the right graphic shows the coefficients modulo .