Created
August 1, 2022 18:38
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Animation of frequency of digits in continued fractions of sqrt(n)
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| library(tidyverse) | |
| library(future) | |
| library(furrr) | |
| library(gganimate) | |
| plan(multisession, workers = 12) | |
| cf_init <- function(D) { | |
| r = as.integer(floor((sqrt(D)))) | |
| list( | |
| D = D, r = r, a = r, b = r, c = D - r ^ 2 | |
| ) | |
| } | |
| cf_next <- function(step) { | |
| a = as.integer(floor((step$r + step$b) / step$c)) | |
| b = a * step$c - step$b | |
| c = (step$D - b ^ 2) %/% step$c | |
| list( | |
| D = step$D, r = step$r, a = a, b = b, c = c | |
| ) | |
| } | |
| cf_all_steps <- function(D) { | |
| step <- cf_init(D) | |
| steps = step %>% as_tibble() | |
| if (step$c != 0) { | |
| while (step$a <= step$r) { | |
| step <- cf_next(step) | |
| steps <- bind_rows(steps, | |
| step %>% as_tibble()) | |
| } | |
| } | |
| steps[-1, ] | |
| } | |
| cf_get_cf <- function(D) { | |
| cf_all_steps(D) %>% select(D, a) %>% mutate(k = row_number()) | |
| } | |
| MAX_D <- 100000 | |
| STEPS <- 110 | |
| density <- 2:MAX_D %>% setNames(nm = .) %>% | |
| future_map(cf_get_cf) %>% | |
| future_map( ~ count(., a)) %>% | |
| future_map( ~ mutate(., density = n / sum(n))) %>% | |
| bind_rows(.id = "D") %>% | |
| mutate(D = as.integer(D)) | |
| allDs <- density$D %>% unique() | |
| Ds <- | |
| allDs[as.integer((length(allDs) ^ (1 / STEPS)) ^ (1:STEPS))] %>% | |
| unique() | |
| summarize_density <- function(density) { | |
| density %>% | |
| group_by(a) %>% | |
| summarize(sum_density = sum(density)) %>% | |
| mutate(density = sum_density / sum(sum_density)) %>% | |
| select(-sum_density) | |
| } | |
| density_summary <- Ds %>% setNames(nm = .) %>% | |
| map(~ filter(density, D <= .)) %>% | |
| map_df(summarize_density, .id = "D") %>% | |
| mutate(D = as.integer(D)) %>% | |
| mutate(parity = factor(a %% 2, labels = c("Even numbers", "Odd numbers"))) | |
| density_reference <- tibble(a = 1:max(density_summary$a)) %>% | |
| mutate(density = (2 * log(a + 1) - log(a) - log(a + 2)) / log(2)) | |
| equation_text <- | |
| "italic(y) == 2 * plain(log)[2](italic(x)+1) - plain(log)[2](italic(x)) - plain(log)[2](italic(x)+2)" | |
| title_text <- | |
| "Frequency of numbers appearing in continued fraction of fractional part of \U221A n" | |
| subtitle_text <- "For all \U221A n up to \U221A {closest_state}" | |
| animation <- density_summary %>% | |
| ggplot(aes(x = a, y = density)) + | |
| geom_point(aes(color = parity)) + | |
| geom_line(data = density_reference, mapping = aes(x = a, y = density)) + | |
| scale_y_log10( | |
| breaks = c(1, .1, .01, .001, .0001), | |
| labels = c("1", ".1", ".01", ".001", ".0001") | |
| ) + | |
| transition_states(D) + | |
| shadow_wake(.1) + | |
| labs( | |
| title = str_wrap(width = 70, title_text), | |
| subtitle = subtitle_text, | |
| x = element_blank(), | |
| y = "Frequency" | |
| ) + | |
| theme(legend.position = "bottom", | |
| legend.title = element_blank()) | |
| animation_log_scale <- animate( | |
| animation + | |
| scale_x_log10() + | |
| annotate( | |
| "text", | |
| label = equation_text, | |
| x = 40, | |
| y = .1, | |
| parse = TRUE | |
| ), | |
| nframes = 213, | |
| start_pause = 5, | |
| end_pause = 10, | |
| renderer = gifski_renderer() | |
| ) | |
| animation_low_values <- animate( | |
| animation + | |
| xlim(1, 30) + | |
| annotate( | |
| "text", | |
| label = equation_text, | |
| x = 20, | |
| y = .1, | |
| parse = TRUE | |
| ), | |
| nframes = 213, | |
| start_pause = 5, | |
| end_pause = 10, | |
| renderer = gifski_renderer() | |
| ) | |
| anim_save("cf_log_scale.gif", animation_log_scale) | |
| anim_save("cf_low_values.gif", animation_low_values) |
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