Pi-walk

Walking the pie

Heavily inspired by the talented Naideh Bremer.

The idea is:

Begin somewhere in the coordinate system. Take the first digit of pi, and take a step in a direction specified by that digit. Take the next digit of i. And take a new step i a direction specified by that digit.

Continue for 1000000 steps.

First - what direction should we take for a given digit?:

x <- cos(5/2*pi - digit/5*pi)
y <- sin(5/2*pi - digit/5*pi)

This should return the x,y step, based on a digit.

If we take two steps, the x-position at the end will be the sum of the two x-steps. Similar with the y-position.

Therefore we can calculate the position at step n by calculating the cumulative sum of x and y.

We will need to add (0,0) at the beginning - where we start our walk.

Taking pi as a string, we can do all that, and return a tibble:

library(tidyverse)
piPoints <- function(piString){
  # removing punctuation
  numbers <- paste0(substr(piString, 1, 1), substr(piString, 3, nchar(piString)))
  # Splitting numbers
  numbers <- as.integer(unlist(strsplit(numbers,"")))
  # calculating steps
  x <- cos(5/2*pi - numbers/5*pi)
  y <- sin(5/2*pi - numbers/5*pi)
  # calculating cumulative steps
  x <- cumsum(x)
  y <- cumsum(y)
  # adding start point
  x <- c(0,x)
  y <- c(0,y)
  # adding id for sequential coloring
  id <- 1:(length(y))
  # Assembling and returning dataframe
  tibble(x=x,y=y, id = id)
}

Next we’ll get pi with a million digits:

large_pi <- read_file("https://pi2e.ch/blog/wp-content/uploads/2017/03/pi_dec_1m.txt")

Pour that into the function:

df <- piPoints(large_pi)

Plot - and save

ggplot(df, aes(x,y,group="1")) +
  geom_path(aes(colour=id)) +
  scale_colour_distiller(type="seq", palette="Set1") +
  theme_bw() + 
  coord_fixed(ratio = 1) +
  theme(line = element_blank(),
        text = element_blank(),
        title = element_blank(),
        legend.position="none",
        panel.border = element_blank(),
        panel.background = element_blank())
ggsave("piwalk.png")  

There are several other mathematical constants. And there are also possibilities for variation in calculating the steps. For now I’m satisfied with the result.