plotting Hamid Naderi Yeganehs parrot using ggplot

Question

I am a) new to stackoverflow and b) an advanced beginner to R ;-)

i saw some bird artworks of Yeganeh with the associated functions in the web Drawing Birds in Flight With Mathematics and wanted to reproduce them in R to experiment a bit with colouring and so on.

However, while this one yielded a quite good result:

k <- 1:9830

X <- function(k) {
  sin(pi * k / 20000) ^ 12 *
    (0.5 * cos(31 * pi * k / 10000) ^ 16 *
       sin(6 * pi * k / 10000) + (1 / 6 * sin(31 * pi * k / 10000)) ^ 20) +
    3 * k / 20000 + cos(31 * pi * k / 10000) ^ 6 *
    sin((pi / 2) * ((k - 10000) / 10000) ^ 7 - pi / 5)
}

Y <- function(k) {
  -9 / 4 * cos(31 * pi * k / 10000) ^ 6 *
    cos(pi / 2 * ((k - 10000) / 10000) ^ 7 - pi / 5) *
    (2 / 3 + (sin(pi * k / 20000) * sin(3 * pi * k / 20000)) ^ 6) +
    3 / 4 * cos(3 * pi * ((k - 10000) / 100000)) ^ 10 *
    cos(9 * pi * ((k - 10000) / 100000)) ^ 10 *
    cos(36 * pi * ((k - 10000) / 100000)) ^ 14 +
    7 / 10 * ((k - 10000) / 10000) ^ 2
}

R <- function(k) {
  sin(pi * k / 20000) ^ 10 *
    (1 / 4 * cos(31 * pi * k / 10000 + 25 * pi / 32) ^ 20 +
       1 / 20 * cos(31 * pi * k / 10000) ^ 2) +
    1 / 30 * (3 / 2 - cos(62 * pi * k / 10000) ^ 2)
}

bird <- data.frame(x = X(k), y = Y(k), r = R(k))

library(tidyverse)
library(ggforce)

q <- ggplot() +
  geom_circle(aes(x0 = x, y0 = y, r = r), 
              data = bird, 
              n = 30) + 
  coord_fixed() +
  theme_void()

the following code yielded some weird result which should basically be related to the difference in the function. (x-A(k))+(y-B(k))=(R(k)) for the parrot below, whlie the bird above "simply" consisted of the k-th circle (X(k), Y(k)) and the radius of the k-th circle R(k)

k <- -10000:10000

A <- function(k) {
  (3*k/20000)+(cos(37*pi*k/10000))*sin((k/10000)*(3*pi/5))+(9/7)*(cos(37*pi*k/10000))*(cos(pi*k/20000))*sin(pi*k/10000)
}

B <- function(k) {
  (-5/4)*(cos(37*pi*k/10000))*cos((k/10000)*(3*pi/5))*(1+3*(cos(pi*k/20000)*cos(3*pi*k/20000)))+(2/3)*(cos(3*pi*k/200000)*cos(9*pi*k/200000)*cos(9*pi*k/100000))
}

R <- function(k) {
  (1/32)+(1/15)*(sin(37*pi*k/10000))*((sin(pi*k/10000))+(3/2)*(cos(pi*k/20000)))
}

parrot <- data.frame(a = A(k), b = B(k), r = R(k))

q <- ggplot() +
  geom_circle(aes(x0 = a, y0 = b, r = r), 
              data = parrot,
             n=30) + 
  coord_fixed() +
  theme_void()
q

Any help would be very much appreciated. Cartesian coords already applied as [explained here] (https://www.wikiwand.com/en/Hamid_Naderi_Yeganeh). From the visual point of view, it seems like the function is plotted properly but the "view" on it needs to be changed...

Thanks in advance!