Powers of 3 in binary

I was thumbing through A New Kind of Science [1] and one of the examples that jumped out at me was looking at the bits in the binary representation of the powers of 3. I wanted to reproduce the image myself and here’s the result.

Here a white square represents a 1 and a blue square a 0. There’s a white border on the right edge because all powers of 3 are odd, i.e. end in 1 when written in binary.

As with many of the images in Wolfram’s book, it’s somewhere between regular and chaotic.

I produced the image at the command line with the following Python code.

    from numpy import log2

    N = 60
    k = int(log2(3)*N)+1

    for i in range(N):
        s = format(3**i, str(k)+'b')
        s = s.replace('1', chr(0x2588))
        s = s.replace('0', ' ')
        print(s)

The code writes powers of three as binary strings of length k where k is calculated so the last number in the sequence will fit. It then replaces 1s with a solid block character and replaces 0s with a blank space. My terminal has a blue background and a white foreground, so 1s show up as white squares and 0s as blue.

The characters in my terminal are twice as high as they are wide, so I changed the aspect ratio after taking the screen shot so that the blocks come out more square.

Update: Other bases

What if we look at bases other than 3?

Here’s the revised code to let you specify any base b.

    from numpy import log2

    def out(s):
        s = s.replace('1', chr(0x2588))
        s = s.replace('0', ' ')
        print(s)    

    b = 5
    N = 50
    k = int(log2(b)*N)+1

    for i in range(N):
        s = format(b**i, str(k)+'b')
        out(s)

Here’s what powers of 5 look like:

And here’s what powers of 7 look like:

What if our base isn’t prime? It doesn’t make an obvious difference if the base is odd, but if it’s even we get a skewed image. Here’s the plot for p = 6.

This is just the plot for powers of 3 with the nth row shifted left by n positions.

Related posts

• Sierpinski strikes again
• Cellular automata with random initial conditions
• A complexity fallacy

[1] I’ve looked through the book before for fun, but this time I’m looking through it with a practical project in mind. I never thought that would happen.