Well, my timing for telling people about my things is still off… clearly.

But look! I self-published a coloring book! A FRACTAL coloring book!

It’s got 3 different varieties of image. A large portion of the pages are line-drawings of fractal patterns for the Color-er to fill in. There are also grey-scale fractals that produce interesting gradient effects when colored in. And finally, there are a select few 3D fractal patterns ready for coloring.

Hey remember MONTHS ago when I posted those fractal images and spouted ideas of calendars and stuff? I FINALLY got my act together and made a Squarespace page to sell them!

Really excited, it took me a few days of hard work to get it all put together but I can’t wait to make more cool fractal things!

The “stick figure” is an image which is both easily recognizable and easy to draw for just about anyone. Both in physical and digital format, the stick figure has been utilized in various ways, from artistic expression to class-room style explanatory videos. In prehistoric times, we drew them on the walls of caves, and in modern times we use Adobe Flash.

Despite the simplicity of the drawing, it is still possible to add context and character to a stick figure. Here are a few of my personal favorite Stick-figure Influences:

CGP Grey – A YouTube personality best know for his “Grey Explains” videos, in which Grey breaks down many topics ranging from politics to the sciences using illustrations, animations and info-graphic-filled videos. In addition, Grey has recently started a popular podcast called “Hello Internet” with Brady Haran, another popular YouTube personality. Grey himself is never pictured, and instead uses a stick-figure representation of himself in his videos (and when he guest stars in other Channels’ videos).

xkcd – Though creator Randall Munroe has no set list (that I could find), he does a wonderful job of bringing forth specific attitudes and behaviors in the reoccurring characters of his web-comic:

Animator vs Animation – Using Adobe Flash, animator Alan Becker produced this trilogy of short films starring a digitally drawn Stick Man about the conflicts that arise between the Animator and the Animation. This Frankenstein-gone-Adobe animation also features appearances of the AIM-man (another Iconic cultural Stick-Figure), the “FireFox” and a host of other computer icons and programs. Found on Alan Becker’s website is a sneak peek at #4 which has recently been funded by Kickstarter and expected to be completed late July 2014.

Stickdeath.com – This overly crude website utilized the stick man to illustrate graphic violence and other grotesque and offensive scenes. Produced by Rob Lewis, these Flash videos (and games) feature non-specific Stick Men in a world of hate and violence.

ScreenCap from StickDeath

A note on style:

One thing I found interesting was the different shapes and proportions people constructed their stick figures with:

All 4 side by side, all images captured from their respective sites

There are subtle differences between them such as head size and shape, the presence of a neck and the addition of facial features and hair.

This is going to be a bit different than my other posts.

A few weeks ago, I stumbled across a video about the Collatz Conjecture. It was from one of the YouTube channels I follow, TippingPointMath. The video very clearly explains the Conjecture and gives a little background on the history of the problem.

Out of pure curiosity I began to play with the (as I now knew it) “3N+1 Conjecture” myself. I did not look much into other people’s methods of solving (though I did a quick search to ensure it was still an “unproven” conjecture), but attacked it myself from two specific angles.

The first attack was to evaluate what, in my mind, was a similar proposal, but one that is much easier to prove and observe. I looked at the “N+1 Conjecture”. That is, I formulated an algorithm similar to Collatz’s that goes:

{ N is a positive, whole integer

If N is even, N’=N/2; if N is odd, N’=N+1.}

I did not write out a formal proof, but I think it is easy to see that this algorithm always collapses to N=1, and then perpetuates a cycle of {1,2}. If you start out with an even N, N’ is obviously less than N; If N is odd, N’ will be greater than N by a value of 1, but this guarantees that N’ will be even, so then N”=N’/2=(N+1)/2. It is easy to see that N” will be less than N for all integers N>2.

A few examples of number chains this algorithm would produce:

Starting with N=64: {64,32,16,8,4,2,1,2,1…}

Starting with N=99: {99,100,50,25,26,13,14,7,8,4,2,1,2,1…}

Even starting with obscure primes and odd numbers, it is easy to see the decline of the chains to the {1,2} cycle.

Another way to look at both the N+1 and the 3N+1 algorithms, is in reverse. In reverse the N+1 algorithm becomes:

{If N’ is odd, N=2*N’

If N’ is even, N=2*N’ AND N’-1}

By looking at them this way, a tree like structure emerges, where N’ can branch off into more than one Ns. For example, if N’ is 4, it could be that N is 8 and was divided by 2, or that N is 3 and 1 was added to N. Both situations will result in N’=4. This is best shown in plot format, where it clears displays a branching tree structure:

18 generations of the N+1 tree

15 generations of the N+1 tree

Though these plots show the tree structure, it does look very chaotic and as if there is no pattern to the way it all reduces to 1. After noticing a pattern in the “evenness” of the numbers in the chains I thought of an idea. I plotted the tree again, but on different axes. I plotted the points such that the value p of each number in the tree can be found by the equation:

p=x*2^{y}

In order to still maintain the visualization of generations, I made it into a gif as well, with each step showing a separate generation.

each point can be given by p=x*2^y

Then I made a few 20-generation plots to show how this structure looks on a grand scale:

20 generations of the N+1 tree

20 generations of the N+1 tree, zoomed out to show all data

20 generations zoomed in to show structure

20 generations zoomed in to show structure

Now I began to attack the 3N+1 problem using similar techniques. First I figured out an algorithm that generates the reverse tree:

{If N’-1 is thirdable* and not even

N=^{1}⁄_{3}(N’-1) AND 2*N’

Otherwise,

N=2*N’ }

(*Sidenote: I use the word “thirdable” to describe the quality of a number to be divided by 3 evenly, with no remainder. Another way of saying this is that if a number is “thirdable” it has 3 as one of its factors)

From this I got a tree that looks like:

Generation tree for the 3N+1 tree

20 generations of the 3N+1 tree

Again, you can see the branching tree-like organization, but it looks like chaos.

It did not do me any good to plot it on the same axes as I did for the N+1 algorithm, but after examining the ways the numbers branch off, I thought of a similar way to illustrate an analogous structure. I plotted the tree in a 3-Dimensional plot, such that the value of each point p can be given by the equation:

p=x*2^{y}*3^{z}

Again, I made a gif of how this tree “grows” such that each step shows the next generation of branches. This goes through 30 generations:

30 generations of the 3N+1 tree

It was taking my computer a very long time to compute the higher generations so I wrote a different function which took N’ to be all integers less than a set value A. Here is a gif which inputs A to be various high values, the largest being N=10000.

Reverse Collatz “tree” in web form, largest N=10000

And here is a zoomed out perspective with the largest N being 500:

Though the appearance of the branches growing has dissolved, it is my opinion that this is an illustration of structure. If this structure can be defined, it could lead to a proof of the Collatz Conjecture.

What do you all think, is there structure here? Am I on to something? I used Octave to create all of these plots, and then Photoshop to make them into gifs. If anyone is interested in seeing the functions I wrote to create them, please contact me.