Bicycling for Collatz

22Feb07

Q: How many people with ADD does it take to change a lightbulb?
A: HEY! Let’s ride bikes!


module Bikes where


Graphviz is fun. Graphviz is fair. You may already have one. You may already be there:


data Arr a = a :-> a


Graphviz plots graphs from .DOT files consisting of remarkably simple syntax:


instance (Show a) => Show (Arr a) where show (x:->y) = show x ++ " -> " ++ show y
class (Show a) => (Dot a) where
dot :: [a] -> String
dot = (++"}") . ("digraph untitled {"++) . unlines . map ((++";") . show)
instance (Show a) => Dot (Arr a)


One fun thing we can do with it is plotting graphs of integer-valued functions:


arr f = (\x-> x :-> f x)
rra f = (\x-> f x :-> x)


Let’s see what a simple invertible function looks like:


t2 = dot $ map (arr (*2)) [0..40]


Clearly there’s one and only one way to connect any given two nodes — which is what invertibility means. But let’s look at something fun now; say, the co-totient function. There’s a closed-form equation for this, but why not go full ape declarative?


totient x = x - (length $ [n | n<-[1..x], coprimes n x]) where coprimes a b = (gcd a b) == 1
t3 = dot $ map (arr totient) [1..100]


(Click to see full-size picture)

The first hierarchy line contains the prime numbers; their co-totient is always 1. These numbers are connected to numbers whose co-totient they are; from that graph you can find the co-totient of any number by looking at which number it’s connected to. At first glance it looks pretty tree-like, but it suffices to chase the series of powers of two to see it isn’t.

Now, being that the totient is the quantity of numbers “strange” to a number (in the sense their only common divisor is 1) and the co-totient is precisely a number minus its totient, higher co-totients seem to indicate a more “composite” structure. That’s not clearly represented by graph hierarchy, of course. So we try a different representation of compositeness in integers:


t4 = dot $ filter (\(x:->y) -> y /=1 && x/=y ) $ concat $ map (\y->(map (arr (gcd y)) [2..32])) [2..32]


This will connect all numbers that divide each other by as many arrows as there are ways to divide it:

Click here to see full image

That makes for a smashing bedroom poster.

Finally, as hopelessly amateur mathematicians we are, we just can’t resist poking the Collatz function!


t = dot $ map (rra (\x->if (odd x) then 3*x+1 else x `div` 2)) [1..384]


I’ve plotted Collatz graphs as far as 15000. But let’s keep it to manageable sizes; you can later experiment at home and see trust me that larger graphs keep the structure contained in this graph — there are relatively very few ways to get into the “main Collatz stream” down to 4-2-1 and most numbers are stuck in two-house citadels of their own, with a few managing to form their own substreams that sometimes even manage to join the main Collatz “stream”:

Abandon all hope ye who enter here

Yes, I know, I should be working in PFP and all, but when a boy’s mind is overcome with newfound love, he must clutch at any tiny window of intellectual focus he occasionally comes across by as his brain walks through the shadows of love.

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