Programming Cellular Automata

What is Cellular Automata?

Cellular Automata are a sequence of states that can change itself according to nearby states. Here is an example what a "rule" can be:

The above is a 2D cellular automata, meaning the world is 2D. A pixel is affected by $3 \times 3 - 1$ many pixels. In 1D CA, a pixel is affected by only $2$ pixels. There are many more rules with 2D CA than Conway's Life. If we permute the rule, there are about $2^{(3^2)!}$ many rules.

Some rules produce boring behavior, while other produce complex behavior that no mathematician knows how to predict without simulating it completely. This is heavyly studied in complexy theory.

Game of life is proven to be complex: you can play with it below, try draw some images can see how the rule behaves.

These are discrete CA, people also study continuous step CAs:

And people use machine learning to train cellular automata rules for specific behaviors:

Why Do I Have This Idea?

I was thinking about how to send animation to the future or aliens who can no longer interpret our message due to language barrier. This is a hard problem since making sense of message need to establisha a common "language". Further more, this "language" need to have physical linkage to the world we live in. For example, a mathematical language is perfectly fine for aliens to understand, but it cannot mean anything outside of the mathematical context: you can't tell aliens "I hate you" using math. This is similar to the idea of a time and space capsule:

and long-term nuclear waste warning messages:

Almost all time capsule are either still life or non-visual. This might due to the physical hardness of any intricate device that could produce controlled animation. This project shows how it can be done through cellular automata.

Cellular Automata executes with simple rules and therefore very easy to interpretate. It assumes no other prior knowledge, such as language, signs, and mathematics. In fact, an alien society can guess the rule sets with high certainty just by providing a few example runs or generations. Although it is computational heavy, one could reproduce the entire animation on pieces of paper. Being proved turing complete, it can produce animations with little constaints. All these properties made CA a good candidate for sending animated time-based artwork to the distant future.

Make Some Contraptions

So, my explaination perhaps makes you confused. How can this "game" transmit any meaningful messages? And if it can transmit messages, how can people on earth or aliens understand it?

To answer this question, let me walk you though. If you have played my interactive panel above, you might notice something called a glider.

A glider is a useful thing because it can go to one direction permanently. But we have a more powerful thing: a glider gun that can generate gliders out of nowhere.

This is conterintuitive. You will think that Game of Life rule should prevent this behavior because it is based on modeling of natural selection.

If we assemble these glider guns and many more "device" such as reflector, duplicator together in a correct way, we can create a what I called a printer:

Time to Do Art

If we have many of these printers: we can finally draw animations!

Beyond Pixel Art

With pinters we can do beyond pixel art. We can design a program to translate any image into flowing animations. This is especially true for Nyan Cat.

Observe that each printer has a head and a tail. We need to encode our image into the middle where the glider bullets are. So in order to generate a big printer, we need to write some code.

Below is a Run-length Encoding (RLE) of the initial state of Game of Life.

bottom = u'''8b2o$8b2o$39bo$37b2o$38b2o34b3o$b3o3b3o66bo$o2bo3bo2bo64bo$4bobo$2o7b
2o$2o7b2o$3bo3bo$3b2ob2o$b3o3b3o$b2o5b2o$bo7bo16bobo$26b2o35b2o$27bo
34bobo$64bo10$51b3o$18bo34bo$19b2o31bo$18b2o5$9bo7bo$9b2o5b2o$9b3o3b3o
$11b2ob2o24b2o$11bo3bo23bobo$8b2o7b2o10bobo9bo$8b2o7b2o11b2o$12bobo15b
o$8bo2bo3bo2bo$9b3o3b3o$28b2o$28b2o2$16b2o$16b2o6$28b3o3b3o44bo$28bo2b
obo2bo43b3o$28bo7bo42bo3bo3b2o$78bo5bo2b2o$29bo5bo42bo5bo$30b2ob2o44bo
3bo2$79bo3bo$78bo5bo$78bo5bo$29bo49bo3bo$28bobo49b3o$28bobo50bo$27bo3b
o$27b2ob2o$27bo3bo$28b3o$28b3o4b2o43bo7bo$35b2o41b2ob2o3b2ob2o2$81bobo
bobo$79bo3bobo3bo$79bo2b2ob2o2bo$80b3o3b3o$81bo5bo5$87b2o$87b2o!''' # at (16, 216)
middleA = u'''2bo$2o$b2o34b3o$39bo$38bo!''' # 2 black first at (53, 218) -> add (x=11, y=-12) for middleA
middleB = u'''obo$2o35b2o$bo34bobo$38bo!''' # 1 white first at (65, 207) -> add (x=49, y=-9) for bottom
top = u'''30bo3b3o$15b2o12bobo2b5o3b2o$15b2o12bo3b2o3b2o2b2o$30bo2bo3b2o$31b3o3b
o2$31b3o3bo$30bo2bo3b2o$29bo3b2o3b2o$29bobo2b5o$30bo3b3o$23bobo$23b2ob
o$24bo20b3o4b2o$25bo18bo3bo3b2o$44b2ob2o$45bobo2$45bobo$44b2ob2o$44bo
3bo$45b3o$13bo$11b2o23b2o$12b2o22bobo$36bo5$44b3o5b3o$44bo2b2ob2o2bo$
45b3o3b3o$46bo5bo$obo$2o35b2o$bo34bobo$38bo3$45b2o$45b2o!''' # at (65, 173) -> (x=12, y=-45) for middle A

We need to write a parser to be able to convert these non-sense message to matrix:

def decode_line(l):
    rows = []
    line = []
    number_stack = ''
    for char in l:
        if char == 'b':
            line += [False] * int('1' if number_stack == '' else number_stack)
            number_stack = ''
        elif char == 'o':
            line += [True] * int('1' if number_stack == '' else number_stack)
            number_stack = ''
        elif char == '!':
            rows.append(line)
            break
        elif char == '\n':
            continue
        elif char == '$':
            num = int('1' if number_stack == '' else number_stack)
            number_stack = ''
            rows.append(line)
            line = []
            num -= 1
            if num > 0: rows += [[]] * num
        else:
            number_stack = number_stack + char
    return rows

def fix_length(mat, length):
    for l in mat:
        assert(length >= len(l))
        l += [False] * (length - len(l))
    return mat

def print_mat(mat):
    for l in mat:
        print(''.join(map(lambda c: '■' if c else '□', l)))

def decode(string):
    # strings = string.split('$')
    mat = decode_line(string)

    width = max(map(lambda x: len(x), mat))
    mat = fix_length(mat, width)
    return mat

We now can have some testing output

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□□□■□□□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□■■□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□■■■□□□■■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□■■□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□■□□□□□□□■□□□□□□□□□□□□□□□□■□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□■□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□■■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
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□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
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□□□□□□□□□■■□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□■■■□□□■■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□■■□■■□□□□□□□□□□□□□□□□□□□□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□■□□□■□□□□□□□□□□□□□□□□□□□□□□□■□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□■■□□□□□□□■■□□□□□□□□□□■□■□□□□□□□□□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□■■□□□□□□□■■□□□□□□□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□■□■□□□□□□□□□□□□□□□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□■□□■□□□■□□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□■■■□□□■■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□■■■□□□■■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□■□□■□■□□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□■■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□■□□□□□□□■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□■□□□■□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□■□□□□□■□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
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□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□

Because we need to constantly transform positions, matrix is good for printing our data to debug, but is not good at manipulating images. This is because most of our image is blank, and adding two images together has O(n^2) complexity where n is $\text{width}+\text{height}$. But if we use tuple, we no longer need to store black pixel. Also, adding two images together is free by concat the list.

def mat2tuples(mat):
    d = []
    for i in range(len(mat)):
        for j in range(len(mat[0])):
            if mat[i][j]:
                d.append((j, i)) # switch for (x, y)
    return d

def normalize_tuples(tuples):
    xs = map(lambda x: x[0], tuples)
    ys = map(lambda x: x[1], tuples)
    minx = min(xs)
    miny = min(ys)
    return list(map(lambda x: (x[0] - minx, x[1] - miny), tuples))

def tuples2mat(tuples):
    assert(tuples == normalize_tuples(tuples))
    xs = map(lambda x: x[0], tuples)
    ys = map(lambda x: x[1], tuples)
    maxx = max(xs)
    maxy = max(ys)
    mat = [ [False]*(maxx + 1) for i in range(maxy + 1)]

    assert(len(mat) == maxy + 1)
    assert(len(mat[0]) == maxx + 1)

    for x in tuples:
        mat[x[1]][x[0]] = True

    return mat

def tuples_add(base, sauce, translation):
    return base + list(map(lambda x: (x[0]+translation[0], x[1]+translation[1]), sauce))

Now we build our printer:

def build_printer(size):
    bottom_t = mat2tuples(decode(bottom))
    A_t = mat2tuples(decode(middleA))
    B_t = mat2tuples(decode(middleB))
    top_t = mat2tuples(decode(top))

    C_t = normalize_tuples(tuples_add(B_t, A_t, (11, -12)))
    for i in range(size):
        # 2i is size of adding
        bottom_t = normalize_tuples(tuples_add(bottom_t, C_t, (23*i + 49, -2*i -21)))

    return tuples2mat(bottom_t)

Believe me, this part is painful. I just can't count the pixels correctly. It f*cking took me more than 3 hours writing this stupid function.

Animate It! Well... At Least To the Current Viewers

The journey is not yet ended. If you tint them with color... IT IS A REAL ANIMATION!

And More Possibiliies