The Unbereable Insolence of Prime Numbers or (Playing to be Ulam)

So rock me mama like a wagon wheel, rock me mama anyway you feel (Wagon Wheel, Old Crow Medicine Show)

This is the third iteration of Hilbert curve. I placed points in its corners. Since the curve has beginning and ending, I labeled each vertex with the order it occupies:Dark green vertex are those labeled with prime numbers and light ones with non-prime. This is the sixth iteration colored as I described before (I removed lines and labels):

Previous plot has 4.096 points. There are 564 primes lower than 4.096. What If I color 564 points randomly instead coloring primes? This is an example: Do you see any difference? I do. Let me place both images together (on the right, the one with primes colored):

The dark points are much more ordered in the first plot. The second one is more noisy. This is my particular tribute to Stanislaw Ulam and its spiral: one of the most amazing fruits of boredom in the history of mathematics.

This is the code:

library(reshape2)library(dplyr)library(ggplot2)library(pracma)opt=theme(legend.position="none",          panel.background = element_rect(fill="white"),          panel.grid=element_blank(),          axis.ticks=element_blank(),          axis.title=element_blank(),          axis.text=element_blank())hilbert = function(m,n,r) {  for (i in 1:n)  {    tmp=cbind(t(m), m+nrow(m)^2)    m=rbind(tmp, (2*nrow(m))^r-tmp[nrow(m):1,]+1)  }  melt(m) %>% plyr::rename(c("Var1" = "x", "Var2" = "y", "value"="order")) %>% arrange(order)}iter=3 #Number of iterationsdf=hilbert(m=matrix(1), n=iter, r=2)subprimes=primes(nrow(df))df %>%  mutate(prime=order %in% subprimes,               random=sample(x=c(TRUE, FALSE), size=nrow(df), prob=c(length(subprimes),(nrow(df)-length(subprimes))), replace = TRUE)) -> df#Labeled (primes colored)ggplot(df, aes(x, y, colour=prime)) +  geom_path(color="gray75", size=3)+  geom_point(size=28)+  scale_colour_manual(values = c("olivedrab1", "olivedrab"))+  scale_x_continuous(expand=c(0,0), limits=c(0,2^iter+1))+  scale_y_continuous(expand=c(0,0), limits=c(0,2^iter+1))+  geom_text(aes(label=order), size=8, color="white")+  opt#Non labeled (primes colored)ggplot(df, aes(x, y, colour=prime)) +  geom_point(size=5)+  scale_colour_manual(values = c("olivedrab1", "olivedrab"))+  scale_x_continuous(expand=c(0,0), limits=c(0,2^iter+1))+  scale_y_continuous(expand=c(0,0), limits=c(0,2^iter+1))+  opt#Non labeled (random colored)ggplot(df, aes(x, y, colour=random)) +  geom_point(size=5)+  scale_colour_manual(values = c("olivedrab1", "olivedrab"))+  scale_x_continuous(expand=c(0,0), limits=c(0,2^iter+1))+  scale_y_continuous(expand=c(0,0), limits=c(0,2^iter+1))+  opt