Ulam Spirals in R and ggplot

November 29, 2011
By

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Having seen a twitter post speed by about Ulam Spirals I started to read up.  As the story goes in 1963 Stanislaw Ulam was bored at conference and started scribbling numbers in a spiral. What he discovered was a strange diaginal pattern of Prime Numbers. I was a little bored as well, so i decided to write up some R to plot these patterns.

Here is the main function. I am sure there are more elegant ways of calculating a Ulam spiral but here is my hack. First we calculate the diagonals and then fill in the gaps.

 
Ulam.Spiral<-function(N){
  if (even(N)){
     cat(sprintf("Error: function only accepts odd integers because it a poorly written and fragile piece of code.\n"))
  }else{
m <- matrix(NA, nrow=(N), ncol=N)
    top.left<-c(1,1)
    bottom.right<-c(N,N)
    top.right<-c(1,N)
    bottom.left<-c(N,1)
    n<-N
    a<-N
    m[median(1:N),median(1:N)]<-1
 
while(a>=3){
# This is an adaptation of a Euler Problem 28 solution. It calculates the diaginals.
    m[bottom.right[1],bottom.right[2]]<- a^2 #bottom right
    m[top.left[1],top.left[2]]<- a^2 - 2*a + 2 # top left
    m[top.right[1],top.right[2]]<- a^2 - 3*a + 3 # top right
    m[bottom.left[1],bottom.left[2]]<- a^2 - a + 1 #bottom left
 
#Both Horizontal Rows
    m[top.left[1],(top.left[2]+1):(top.right[2]-1)]<-seq( m[top.left[1],top.left[2]]-1,m[top.left[1],top.right[2]]+1) #Fill in top row
    m[bottom.left[1],(bottom.left[2]+1):(bottom.right[2]-1)]<-seq(m[bottom.left[1],bottom.left[2]]+1, m[bottom.left[1],bottom.right[2]]-1) #Fill in bottom row
#Left Vertical Rows
    m[(top.left[1]+1):(bottom.left[1]-1), top.left[1]] <- seq(m[top.left[1],top.left[1]]+1, m[bottom.left[1],top.left[1]]-1)#Left Hand Row
#Right Verical Row
   m[(top.right[1]+1):(bottom.right[2]-1),top.right[2]] <-seq(m[top.right[1],top.right[2]]-1 , m[top.right[1],top.right[2]] - (a-2) )
#drop down on square and repeat
    a<-a-2
    top.left<-c(top.left[1]+1,top.left[2]+1)
    bottom.right<-c(bottom.right[1]-1,bottom.right[2]-1)
    top.right<-c(top.right[1]+1,top.right[2]-1)
    bottom.left<-c(bottom.left[1]-1,bottom.left[2]+1)
}
return(m)
}
}

Here is the output for a 5X5 spiral

     [,1] [,2] [,3] [,4] [,5]
[1,]   17   16   15   14   13
[2,]   18    5    4    3   12
[3,]   19    6    1    2   11
[4,]   20    7    8    9   10
[5,]   21   22   23   24   25

The code is on git hub. https://github.com/jofusa/ulam-spirals-R

Many Thanks to http://librestats.wordpress.com/2011/08/20/prime-testing-function-in-r/ for a useful IsPrime() function and saving me some time.

Here is an example Plot- Prime Number are in red:

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