Padovan Sequence and sierpinski trianagles

padovan.R

library(ggplot2)
library(magrittr)
library(parallel)

# Padovan Sequence
# 1, 1, 1, 2, 2, 3, 4, ...

padovan <- function(n) {
  n <- floor(n)
  if (n <= 3) stop("n needs to be an integer greater than 3")

  x <- integer(n)
  x[1:3] <- c(1, 1, 1)

  for( i in seq(4, n, by = 1) ) {
    x[i] <- x[i - 2] + x[i - 3]
  }
  x
}

padovan(25)


sierpinski <- function(x, depth = 0) {
  if (depth < 0 | depth > 10) stop()

  if (depth == 0) 
  {
    x[["tri"]] <- runif(1)
    return(x) 
  } else {

    va <- x[1, c("x", "y")]
    vb <- x[2, c("x", "y")]
    vc <- x[3, c("x", "y")]

    mab <- data.frame(x = mean(c(va$x, vb$x)), y = mean(c(va$y, vb$y)))
    mac <- data.frame(x = mean(c(va$x, vc$x)), y = mean(c(va$y, vc$y)))
    mbc <- data.frame(x = mean(c(vb$x, vc$x)), y = mean(c(vb$y, vc$y)))

    do.call(rbind,
    list(
         sierpinski(data.frame(x = c(va$x, mab$x, mac$x), y = c(va$y, mab$y, mac$y)), depth = depth - 1),
         sierpinski(data.frame(x = c(mab$x, vb$x, mbc$x), y = c(mab$y, vb$y, mbc$y)), depth = depth - 1),
         sierpinski(data.frame(x = c(mac$x, mbc$x, vc$x), y = c(mac$y, mbc$y, vc$y)), depth = depth - 1)
    )
    )

  } 
}




# starting at the origin
vertices <-
  list(
       v1 = data.frame(x =  0.0, y = 0.0),
       v2 = data.frame(x =  0.5, y = sin(pi / 3)),
       v3 = data.frame(x =  1.0, y = 0.0),
       v4 = data.frame(x =  0.5, y = -sin(pi / 3)),
       v5 = data.frame(x = -0.5, y = -sin(pi / 3))
  )

vertices$v6$x <- vertices$v5$x - 1
vertices$v6$y <- vertices$v2$y

vertices$v7$x <- vertices$v5$x
vertices$v7$y <- vertices$v2$y + 2 * sin(pi / 3)

vertices$v8$x <- vertices$v7$x + 3
vertices$v8$y <- vertices$v7$y

vertices$v9$x <- vertices$v4$x + 4
vertices$v9$y <- vertices$v4$y

vertices$v10$x <- mean(c(vertices$v5$x, vertices$v9$x))
vertices$v10$y <- vertices$v9$y - 5 * sin(pi / 3) 

vertices$v11$x <- vertices$v10$x - 7 
vertices$v11$y <- vertices$v10$y

padovan(10)
vertices$v12$x <- vertices$v7$x - 9
vertices$v12$y <- vertices$v7$y

padovan(11)
vertices$v13$x <- mean(c(vertices$v12$x, vertices$v8$x))
vertices$v13$y <- vertices$v7$y + 12 * sin(pi / 3)

padovan(12)
vertices$v14$x <- vertices$v13$x + 16
vertices$v14$y <- vertices$v13$y

vertices <- dplyr::bind_rows(vertices, .id = "vertex")

triangles <-
  list(
       t001 = dplyr::mutate(dplyr::filter(vertices, .data$vertex %in% c("v1", "v2", "v3")), padovan = 1),
       t002 = dplyr::mutate(dplyr::filter(vertices, .data$vertex %in% c("v1", "v3", "v4")), padovan = 1),
       t003 = dplyr::mutate(dplyr::filter(vertices, .data$vertex %in% c("v1", "v4", "v5")), padovan = 1),
       t004 = dplyr::mutate(dplyr::filter(vertices, .data$vertex %in% c("v5", "v6", "v2")), padovan = tail(padovan(4), 1)),
       t005 = dplyr::mutate(dplyr::filter(vertices, .data$vertex %in% c("v6", "v2", "v7")), padovan = tail(padovan(5), 1)),
       t006 = dplyr::mutate(dplyr::filter(vertices, .data$vertex %in% c("v7", "v3", "v8")), padovan = tail(padovan(6), 1)),
       t007 = dplyr::mutate(dplyr::filter(vertices, .data$vertex %in% c("v8", "v9", "v4")), padovan = tail(padovan(7), 1)),
       t008 = dplyr::mutate(dplyr::filter(vertices, .data$vertex %in% c("v5", "v9", "v10")), padovan = tail(padovan(8), 1)),
       t009 = dplyr::mutate(dplyr::filter(vertices, .data$vertex %in% c("v6", "v10", "v11")), padovan = tail(padovan(9), 1)),
       t010 = dplyr::mutate(dplyr::filter(vertices, .data$vertex %in% c("v7", "v11", "v12")), padovan = tail(padovan(10), 1)),
       t011 = dplyr::mutate(dplyr::filter(vertices, .data$vertex %in% c("v12", "v13", "v8")), padovan = tail(padovan(11), 1)),
       t012 = dplyr::mutate(dplyr::filter(vertices, .data$vertex %in% c("v13", "v9", "v14")), padovan = tail(padovan(12), 1))
  )

triangles <-
  triangles %>%
  unname %>%
  dplyr::bind_rows(., .id = "n") %>%
  dplyr::mutate(n = as.numeric(n))


ggplot() +
  theme_bw() +
  # theme_dark() + 
  # theme_minimal() + 
  aes(x = x, y = y) +
  geom_polygon(data = triangles, mapping = aes(group = n), color = "black") +
  # geom_text(data = vertices, mapping = aes(label = vertex)) +
  geom_text(data = dplyr::summarize(dplyr::group_by(triangles, n, padovan), x = mean(x), y = mean(y)),
            mapping = aes(x = x, y = y, group = n, label = padovan), color = 'white') +
  coord_equal() +
  guides(fill = FALSE) +
  theme(
        axis.title = element_blank(),
        axis.text = element_blank(),
        axis.ticks = element_blank(),
        axis.line = element_blank(), 
        panel.border = element_blank(),
        panel.grid = element_blank()
        ) +
  ggtitle("Padovan Sequence")


mclapply(unique(triangles$n), function(nn) { sierpinski(dplyr::filter(triangles, n == nn), depth = 7) },
         mc.cores = 10L) %>%
dplyr::bind_rows(., .id = "pad_seq") %>% 
ggplot(.) +
theme_bw() +
aes(x = x, y = y, group = tri, fill = pad_seq) +
geom_polygon() +
coord_equal()+
guides(fill = FALSE) +
theme(
      axis.title = element_blank(),
      axis.text = element_blank(),
      axis.ticks = element_blank(),
      axis.line = element_blank(), 
      panel.border = element_blank(),
      panel.grid = element_blank()
      )