Collatz iterations
October 5th, 2009
Once again, I had a leisurely day at work. Decided to elaborate a simple alogirthm in Processing: the Collatz transformation. It’s really simple: if you can divide a number by 2, divide by 2. If not, multiply by 3 and add one. Repeat. Plot in a 1000 x 1000 grid in Processing. Result:
(click for full-sized graph)
I calculated the numer of iterations for each number from 1 to 1 million using a console application in VB.Net. Below is the (simple) recursive algorithm that uses a class variable to keep track of the number of iterations.
aantalIteraties = aantalIteraties + 1
If i Mod 2 = 0 Then
i = i / 2
Else
i = i * 3
i = i + 1
End If
If i = 1 Then Return
CalculateToOne(i)
End Sub
I saved numbers and iterations in a textfile, and created the above plot. Each point has a gray value depending on the number of iterations.
Even though I’m creating plots like this almost every week, and even though I know any patterns would have long been discovered, I’m still excited when I see visual patterns emerging. In this example there are patches that are darker, there’s the widening diagonal line from the top left, and horizontal streaks. If I were a briliant mathematician I’d probably flabbergast you with an explanation that’s harder to follow than the largest-but-one snowflake in a blizzard.
I’m not, though.
My name is Marco Hokke. My blog is about the things that interest me and things I might forget if I would not blog them.
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