Merge pull request #45 from cryptic-io/happy-tree

added happy-tree post

Author: Brian Picciano

_data/companies.yml

@@ -12,3 +12,6 @@
 
 - name: Google
   site: "https://www.google.com/intl/en/about/"
+
+- name: Admiral
+  site: "http://getadmiral.com"

_data/members.yml

@@ -12,7 +12,7 @@
   github: mediocregopher
   author-img: "https://avatars.githubusercontent.com/u/933154"
   bio: "Coding, climbing, longboarding, movie watcher, ginger. I love go, clojure, redis, and sometimes erlang."
-  companies: [Grooveshark]
+  companies: [Grooveshark, Admiral]
   google-plus: https://plus.google.com/u/0/+BrianPiccianoWillNeverBeAbleToGiveOutThisURL/posts
   website: http://mediocregopher.com
   email: mediocregopher@gmail.com

_posts/2015-11-21-happy-trees.md

@@ -0,0 +1,238 @@
+---
+layout: post
+title: Happy Trees
+tags: golang math color
+username: mediocregopher
+source: "https://github.com/mediocregopher/happy-tree"
+---
+
+This project was inspired by [this video](https://www.youtube.com/watch?v=_DpzAvb3Vk4),
+which you should watch first in order to really understand what's going on.
+
+My inspiration came from his noting that happification could be done on numbers
+in bases other than 10. I immediately thought of hexadecimal, base-16, since I'm
+a programmer and that's what I think of. I also was trying to think of how one
+would graphically represent a large happification tree, when I realized that
+hexadecimal numbers are colors, and colors graphically represent things nicely!
+
+## Colors
+
+Colors to computers are represented using 3-bytes, encompassing red, green, and
+blue. Each byte is represented by two hexadecimal digits, and they are appended
+together. For example `FF0000` represents maximum red (`FF`) added to no green
+and no blue. `FF5500` represents maximum red (`FF`), some green (`55`) and no
+blue (`00`), which when added together results in kind of an orange color.
+
+## Happifying colors
+
+In base 10, happifying a number is done by splitting its digits, squaring each
+one individually, and adding the resulting numbers. The principal works the same
+for hexadecimal numbers:
+
+```
+A4F
+A*A + 4*4 + F*F
+64 + 10 + E1
+155 // 341 in decimal
+```
+
+So if all colors are 6-digit hexadecimal numbers, they can be happified easily!
+
+```
+FF5500
+F*F + F*F + 5*5 + 5*5 + 0*0 + 0*0
+E1 + E1 + 19 + 19 + 0 + 0
+0001F4
+```
+
+So `FF5500` (and orangish color) happifies to `0001F4` (A darker blue). Since
+order of digits doesn't matter, `5F50F0` also happifies to `0001F4`. From this
+fact, we can make a tree (hence the happification tree). I can do this process
+on every color from `000000` (black) to `FFFFFF` (white), so I will!
+
+## Representing the tree
+
+So I know I can represent the tree using color, but there's more to decide on
+than that. The easy way to represent a tree would be to simply draw a literal
+tree graph, with a circle for each color and lines pointing to its parent and
+children. But this is boring, and also if I want to represent *all* colors the
+resulting image would be enormous and/or unreadable.
+
+I decided on using a hollow, multi-level pie-chart. Using the example
+of `000002`, it would look something like this:
+
+![An example of a partial multi-level pie chart](/imgs/happy-tree/partial.png)
+
+The inner arc represents the color `000002`. The second arc represents the 15
+different colors which happify into `000002`, each of them may also have their
+own outer arc of numbers which happify to them, and so on.
+
+This representation is nice because a) It looks cool and b) it allows the
+melancoils of the hexadecimals to be placed around the happification tree
+(numbers which happify into `000001`), which is convenient. It's also somewhat
+easier to code than a circle/branch based tree diagram.
+
+An important feature I had to implement was proportional slice sizes. If I were
+to give each child of a color an equal size on that arc's edge the image would
+simply not work.  Some branches of the tree are extremely deep, while others are
+very shallow. If all were given the same space, those deep branches wouldn't
+even be representable by a single pixel's width, and would simply fail to show
+up. So I implemented proportional slice sizes, where the size of every slice is
+determined to be proportional to how many total (recursively) children it has.
+You can see this in the above example, where the second level arc is largely
+comprised of one giant slice, with many smaller slices taking up the end.
+
+## First attempt
+
+My first attempt resulted in this image (click for 5000x5000 version):
+
+[![Result of first attempt](/imgs/happy-tree/happy-tree-atmp1-small.png)](/imgs/happy-tree/happy-tree-atmp1.png)
+
+The first thing you'll notice is that it looks pretty neat.
+
+The second thing you'll notice is that there's actually only one melancoil in
+the 6-digit hexadecimal number set. The innermost black circle is `000000` which
+only happifies to itself, and nothing else will happify to it (sad `000000`).
+The second circle represents `000001`, and all of its runty children. And
+finally the melancoil, comprised of:
+
+```
+00000D -> 0000A9 -> 0000B5 -> 000092 -> 000055 -> 00003 -> ...
+```
+
+The final thing you'll notice (or maybe it was the first, since it's really
+obvious) is that it's very blue. Non-blue colors are really only represented as
+leaves on their trees and don't ever really have any children of their own, so
+the blue and black sections take up vastly more space.
+
+This makes sense. The number which should generate the largest happification
+result, `FFFFFF`, only results in `000546`, which is primarily blue. So in effect
+all colors happify to some shade of blue.
+
+This might have been it, technically this is the happification tree and the
+melancoil of 6 digit hexadecimal numbers represented as colors. But it's also
+boring, and I wanted to do better.
+
+## Second attempt
+
+The root of the problem is that the definition of "happification" I used
+resulted in not diverse enough results. I wanted something which would give me
+numbers where any of the digits could be anything. Something more random.
+
+I considered using a hash instead, like md5, but that has its own problems.
+There's no gaurantee that any number would actually reach `000001`, which isn't
+required but it's a nice feature that I wanted. It also would be unlikely that
+there would be any melancoils that weren't absolutely gigantic.
+
+I ended up redefining what it meant to happify a hexadecimal number. Instead of
+adding all the digits up, I first split up the red, green, and blue digits into
+their own numbers, happified those numbers, and finally reassembled the results
+back into a single number. For example:
+
+```
+FF5500
+FF, 55, 00
+F*F + F*F, 5*5 + 5*5, 0*0 + 0*0
+1C2, 32, 00
+C23200
+```
+
+I drop that 1 on the `1C2`, because it has no place in this system. Sorry 1.
+
+Simply replacing that function resulted in this image (click for 5000x5000) version:
+
+[![Result of second attempt](/imgs/happy-tree/happy-tree-atmp2-small.png)](/imgs/happy-tree/happy-tree-atmp2.png)
+
+The first thing you notice is that it's so colorful! So that goal was achieved.
+
+The second thing you notice is that there's *significantly* more melancoils.
+Hundreds, even. Here's a couple of the melancoils (each on its own line):
+
+```
+00000D -> 0000A9 -> 0000B5 -> 000092 -> 000055 -> 000032 -> ...
+000D0D -> 00A9A9 -> 00B5B5 -> 009292 -> 005555 -> 003232 -> ...
+0D0D0D -> A9A9A9 -> B5B5B5 -> 929292 -> 555555 -> 323232 -> ...
+0D0D32 -> A9A90D -> B5B5A9 -> 9292B5 -> 555592 -> 323255 -> ...
+...
+```
+
+And so on. You'll notice the first melancoil listed is the same as the one from
+the first attempt. You'll also notice that the same numbers from the that
+melancoil are "re-used" in the rest of them as well. The second coil listed is
+the same as the first, just with the numbers repeated in the 3rd and 4th digits.
+The third coil has those numbers repeated once more in the 1st and 2nd digits.
+The final coil is the same numbers, but with the 5th and 6th digits offset one
+place in the rotation.
+
+The rest of the melancoils in this attempt work out to just be every conceivable
+iteration of the above. This is simply a property of the algorithm chosen, and
+there's not a whole lot we can do about it.
+
+## Third attempt
+
+After talking with [Mr. Marco](/members/#marcopolo) about the previous attempts
+I got an idea that would lead me towards more attempts. The main issue I was
+having in coming up with new happification algorithms was figuring out what to
+do about getting a number greater than `FFFFFF`. Dropping the leading digits
+just seemed.... lame.
+
+One solution I came up with was to simply happify again. And again, and again.
+Until I got a number less than or equal to `FFFFFF`.
+
+With this new plan, I could increase the power by which I'm raising each
+individual digit, and drop the strategy from the second attempt of splitting the
+number into three parts. In the first attempt I was doing happification to the
+power of 2, but what if I wanted to happify to the power of 6? It would look
+something like this (starting with the number `34BEEF`):
+
+```
+34BEEF
+3^6 + 4^6 + B^6 + E^6 + E^6 + E^6 + F^6
+2D9 + 1000 + 1B0829 + 72E440 + 72E440 + ADCEA1
+1AEB223
+
+1AEB223 is greater than FFFFFF, so we happify again
+
+1^6 + A^6 + E^6 + B^6 + 2^6 + 2^6 + 3^6
+1 + F4240 + 72E440 + 1B0829 + 40 + 40 + 2D9
+9D3203
+```
+
+So `34BEEF` happifies to `9D3203`, when happifying to the power of 6.
+
+As mentioned before the first attempt in this blog was the 2nd power tree,
+here's the trees for the 3rd, 4th, 5th, and 6th powers (each image is a link to
+a larger version):
+
+3rd power:
+[![Third attempt, 3rd power](/imgs/happy-tree/happy-tree-atmp3-pow3-small.png)](/imgs/happy-tree/happy-tree-atmp3-pow3.png)
+
+4th power:
+[![Third attempt, 4th power](/imgs/happy-tree/happy-tree-atmp3-pow4-small.png)](/imgs/happy-tree/happy-tree-atmp3-pow4.png)
+
+5th power:
+[![Third attempt, 5th power](/imgs/happy-tree/happy-tree-atmp3-pow5-small.png)](/imgs/happy-tree/happy-tree-atmp3-pow5.png)
+
+6th power:
+[![Third attempt, 6th power](/imgs/happy-tree/happy-tree-atmp3-pow6-small.png)](/imgs/happy-tree/happy-tree-atmp3-pow6.png)
+
+A couple things to note:
+
+* 3-5 are still very blue. It's not till the 6th power that the distribution
+  becomes random enough to become very colorful.
+
+* Some powers have more coils than others. Power of 3 has a lot, and actually a
+  lot of them aren't coils, but single narcissistic numbers. Narcissistic
+  numbers are those which happify to themselves. `000000` and `000001` are
+  narcissistic numbers in all powers, power of 3 has quite a few more.
+
+* 4 looks super cool.
+
+Using unsigned 64-bit integers I could theoretically go up to the power of 15.
+But I hit a roadblock at power of 7, in that there's actually a melancoil which
+occurs whose members are all greater than `FFFFFF`. This means that my strategy
+of repeating happifying until I get under `FFFFFF` doesn't work for any numbers
+which lead into that coil.
+
+ All images linked to in this post are licensed under the [Do what the fuck you
+ want to public license](http://www.wtfpl.net/txt/copying/).