If you want to make several sequences of the Collatz function for turning it into a graph, you probably want to memorize, which parts you already calculated. What we try to do is to create a graph like this (image from xkcd):

When we would calculate the whole chain for each number until it (hopefully) reaches the end sequence 8,4,1 we do a lot of work over and over again. Therefore, we want an algorithm that when calculating 24 stops at 10 if this chain has already be calculated.

A moderately understandable solution is to use a `Module`

that contains a *function* which is used as `memory`

to store, whether a numbers was already seen. Additionally, we use a `Internal`Bag`

to store all the different chains. The following function takes a list of positive numbers and calculates the Collatz-sequence for each number. It stops each sequence, when it meets a number that has already be seen:

```
CollatzSequence[list_] := Module[{memory, tmp, chain, result = Internal`Bag[]},
memory[1] = False;
memory[n_] := (memory[n] = False; True);
Do[
chain = Internal`Bag[];
tmp = l;
While[memory[tmp],
Internal`StuffBag[chain, tmp];
tmp = If[EvenQ[tmp], tmp/2, 3 tmp + 1];
];
Internal`StuffBag[chain, tmp];
Internal`StuffBag[result, chain],
{l, list}];
Internal`BagPart[#, All] & /@ Internal`BagPart[result, All]
]
CollatzSequence[{10, 11, 12}]
(* {{10, 5, 16, 8, 4, 2, 1}, {11, 34, 17, 52, 26, 13, 40, 20,
10}, {12, 6, 3, 10}} *)
```

This can now easily be used to create a `Graph`

. It works even for a very large number of chains like say 50000. The only thing you have to do is to turn the list of numbers into list of edges:

```
Graph[
Flatten[(Rule @@@ Partition[#, 2, 1]) & /@
CollatzSequence[Range[50000]]],
PerformanceGoal -> "Speed",
GraphLayout -> {"PackingLayout" -> "ClosestPacking"},
VertexStyle -> Opacity[0.2, RGBColor[44/51, 10/51, 47/255]],
EdgeStyle -> RGBColor[38/255, 139/255, 14/17]]
```

Another very nice way to visualize Collatz-sequences is to draw them as path which makes left/right turns depending on the whether the number is odd or even. I got inspired by a reddit post and wrote my own version that uses this color scheme. The results look stunningly beautiful

Only for reference, let me give you my uncleaned code for a small `Manipulate`

that lets you change everything live.

```
SetAttributes[Collatz, {Listable}];
Collatz[n_, e_, a_, f_] := Module[{nn = n, bag = Internal`Bag[]},
While[nn =!= 1, Internal`StuffBag[bag, nn];
nn = If[EvenQ[nn], nn/2, 3 nn + 1]
];
Internal`StuffBag[bag, nn];
With[{seq = Reverse[Internal`BagPart[bag, All]]},
AnglePath[Transpose[{seq/(1 + seq^e), a*(f - 2 Mod[seq, 2])}]]]];
astroIntensity[l_, s_, r_, h_, g_] :=
With[{psi = 2 Pi (s/3 + r l), a = h l^g (1 - l^g)/2},
l^g + a*{{-0.14861, 1.78277}, {-0.29227, -0.90649}, {1.97294,
0.0}}.{Cos[psi], Sin[psi]}];
Manipulate[
DynamicModule[{seq},
seq = ControlActive[Collatz[Range[5000, 5020], e, a, f],
Collatz[RandomInteger[1000000, {n}], e, a, f]];
Graphics[{Opacity[o], Thickness[ControlActive[0.01, 0.003]],
Line[seq,
VertexColors -> (Table[
astroIntensity[l, s, r, h, g], {l, 0, 1,
1/(Length[#] - 1)}] & /@ seq)]}, ImageSize -> 500]
]
, "Colors", {{s, 2.49}, 0, 3}, {{r, 0.76}, 0, 5}, {{h, 1.815}, 0,
2}, {{g, 1.3}, 0.1, 2}, {{o, 0.5}, 0.1, 1},
Delimiter,
"Structure",
{{e, 1.3}, 0.9, 1.8},
{{a, 0.19}, 0.1, 0.3},
{{f, 0.7}, 0.1, 1.5},
{n, 300, 5000, 1}
]
```

Many more Collatz visualization strategies and analysis algorithms can be found in this blog post of user vzn.

1Your last line has too many brackets, and you give

`GraphPlot3D`

two arguments even though it only accepts one. Just to mention two problems. – C. E. – 2015-06-11T21:39:12.4771If this is supposed to be related to the collatz conjecture your implementation is way off. It doesn't do anything remotely like that. I also don't understand why you show us the Array function as it doesn't seem to be connected to the line above at all. – Sjoerd C. de Vries – 2015-06-11T22:04:12.460

Related: Collatz Tools

– dionys – 2015-06-12T07:54:10.590See also: http://mathematica.stackexchange.com/questions/69555/intelligent-vertex-placement-in-large-tree-graphs

– David G. Stork – 2015-06-12T10:22:07.607many more collatz visualization strategies & analysis algorithms here & hope to hear more from anyone in [chat]

– vzn – 2015-06-12T16:26:24.4371@vzn I have included a link to your blog in my answer since it contains many good links. – halirutan – 2015-06-12T17:09:50.730

3I'm sorry, this is really bothering me -- there's no such thing as "a collatz". Lothar Collatz is the name of the person who posed the Collatz conjecture. What you have there is a

function, and you are asking to visualize the adjacency graph of your function. (Which leads to another problem: functions don't have adjacency graphs.) – None – 2015-12-03T04:54:41.233