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 Pi

2D plots based on the base b digits (1306 graphics)

Digits

Plots of the first 1000 digits of in various bases .

Cumulative digit counts

Plots of cumulative counts of occurrences of the digit among the first 1000 digits of in various bases .

Plots of the deviation of the cumulative counts of occurrences of the digit from the average among the first 1000 digits of in various bases .

Digit averages and moments

Cumulative sum of the deviations from the average of the digit sums of the first 10000 digits of for various bases .

Cumulative sum of the digit sums of the first 10000 digits of for various bases .

Logarithm of the cumulative sum of the moments of the deviations of the digits from the average of the first 1000 digits of for various bases .

Iterated averaged cumulative sums , of the digits of where in various bases .

Nested differences, sums, products, and quotients

Iterated pairwise sums (left) and differences (right) modulo 2 starting with the first 200 digits of in various bases .

Integer parts of the logarithm of the iterated pairwise products (left) and quotients (right) starting with the first 32 digits of in various bases .

Minima and maxima of the scaled iterated pairwise differences starting with the first 2000 scaled digits of in various bases . At each step , the paiwise differences are formed and the results are rescaled via

Cellular automata plots

Evolution of the elementary cellular automata with initial conditions where the are the digits of in base .

Eigenvalue plots

Eigenvalues in the complex plane of the square matrices formed from the first digits , of in various bases by partioning them into rows of length . The eigenvalues are colored from red () to blue ().

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

Eigenvalues in the complex plane of the square matrices formed from the first digits of in bases 2 to over the complex plane. The eigenvalues are colored from red () to blue ().

Eigenvalues in the complex plane of the square matrices of dimension with elements if and 0 otherwise. Here the are the digits of in various bases . The eigenvalues are colored cyclicly from red () to red ().

Root plots

Zeros of the polynomials of the digits of in various bases over the complex plane.

Fourier transforms

Plots of the discrete Fourier transform of the first 1024 digits of in various bases .

Plots of the absolute value and the argument of the averages of the discrete Fourier transform of the first 1024 digits of in various bases .

Plots of the fractional Fourier transform of the first 256 digits of in various bases .

Autocorrelation plots

Plots of the distances between nearest equal digits , of in various bases .

Plots of relative deviation of the autocorrelation from the average of the digits of for various bases .

Identical digit sequences

Cumulative counts of the number of occurrences of pairs of identical digits in the digits of in various bases .

Cumulative counts of the number of occurrences of triples of identical digits in the digits of .

Primes in digit sequences

Primes of the concatenated digits in the digits of in various bases .

Digit jump plots

Lines connecting integer lattice points of the form of the digits of for various bases . Red is base 2 and blue is base 200.

Lines connecting integer lattice points of the form of the digits of for various bases . Red is base 2 and blue is base 50.

Run products

Logarithm of the cumulative products of the runs of nonzero digits of in various bases .

Fourier series

Exponential Fourier series , over the complex plane. Here the are the digits of for various bases .

Exponential Fourier series , over the complex plane. Here the are the digits of for various bases .

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

Cumulative products

Logarithm of the cumulative products of the digits of for various bases .

Averages over bases

Deviation from the average of of the digits of .

Deviation from the average of of the digits of .

The numbers where are the digits of in base , .

The numbers where are the digits of in base , .

Digit permutations

Cumulative distribution of the spacings of all integers formed by concatenating the permutations of the first eight digits of in various bases .

Function values at the digits

Cumulative sums of the Möbius function values of the digits of for various bases .

Values of the Jacobi symbol at the digits of in various bases . If the value is , the square is blue; if the value is , the square is red.

Values of the binomial coefficients at the digits of in various bases modulo 2.

Values of the divisor sums at the digits of in various bases modulo 2.

q-products

Plot of the scaled exponents of the numerator expansion of a based on the digits of in various bases . The plot shows where . The color varies from red (base 2) to blue (base 48).

Integers in base Pi

Plots of the digits of first 200 integers in base for various functions .

Plots of the cumulative sums modulo 2 of the digits of first 200 integers in base for various functions .