Live Pixels Algorithm Analysis

Let us examine the various options offered by the algorithm of the Live Pixels - different combinations of active and blank pixels, one-dimensional and two-dimensional bounded and unbounded matrix. All of these examples can be found in the program:

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The Universe. Big Bang.

"Live Pixels" is  a system of pixels changing where each pixel of the active state changes of neighboring pixels to the opposite: active becomes empty, and empty becomes active. A simple rule creates a very complex structure. One pixel grow on an infinite field simulates the Big Bang. After 21 moves the picture is a square, consisting of 8 squares, each of them also consists of 8 squares, each of them consists of 8 pixels. This similarity is fractal, our universe also has a fractal structure: galaxies rotate around the center of the universe, stars rotate around the centers of galaxies, planets around stars, electrons around the nucleus of atoms. Alteration of the number of active pixels at each step forms a very interesting mathematical series:

1 8 8 24 8 64 24 112 8 64 192 24 192 112 416 8 64 64 192 64 512 192 896 24 192 192 576 112 896 416 1728 8

On the one hand, the number of pixels is constantly increasing, the other - their number is constantly returns to 8! On the image this treatment seems magical: from fancy green mist comes a newborn - the same 8 pixels, but at a greater distance from each other. This is a typical “shift from quantity to quality". We see exactly 8 pixels on the screen at times corresponding to round numbers in the binary system (decimal - binary representation): 1-1, 2-10, 4-100, 8-1000, 16-10000, etc. . The same number represented in a binary form:

1 1000 1000 11000 1000 1000000 11000 1110000 1000 1000000 11000000 1110000 11000 11000000 110100000 1000 1000000 1000000 11000000 1000000 11000000 1000000000 1110000000 1001000000 11000 11000000 11000000 110100000 1110000 1110000000 11011000000 1000

Most numbers have first ones, then zeros. Another series consisting of the number of ones in the above series:

1 1 1 2 1 1 2 3 1 1 2 2 3 3 1 1 1 2 1 1 2 3 2 2 2 2 3 3 3 4 1

If we apply it to the keys of a piano, it turns harmonious music ...

Reproduction.

Reproduction is a very interesting feature of live pixels. The LP sign ​​ after 16 turns into 8 signs with the same parameters. On an infinite field any initial pattern would be replicated in this way:

This effect has previously been described by professor Edward Fredkin. Calculations show that the "reproduction period" depends on the size of the initial pattern. It consist of 8 steps to a pattern that is "included" in the square at 8 pixels, 16 steps to a pattern with a maximum size of 9-up to 16 and so on. "Reproduction period" can be calculated as: 2^x, where x > 2 and the maximum width or height of the pattern is less than or equal to x.

1 Row

Let us consider the simplest cases of the Live Pixels.

One cell:

Empty pixel - nothing happens. Active pixel becomes empty at the first move.

Two cells:

In the first case we observe the pendulum - the consistent movement of live pixels to the right and to the left.

In the second case two active pixels are the simplest form of the Live Pixels static figure.

The three cells:

All the patterns disappear.

The four cells:

All patterns return to the initial state. This is the simplest case of conservation of heritage. See also DNA.

2 Rows

Analysis of Live Pixels in two rows.

In this case, the elements are either static or flashing lights as a signal with period two.

3 Rows

The figures are divided and run from right to left. An example can be found in the program.

Squares

Squares or square matrix are fields with the same number of columns and rows. Some of the squares have the unique possibility of a complete revolution of the pattern to the initial state.

The square 2x2 contains two "blinking" options, passing from one to another, and two static.

4x4

There are some patterns which have reversibility. Most of these figures are irreversible, and they turn to the reversible patterns with a period of 6. Some patterns have the period of 3, due to symmetry.

3x3 - all the variants disappear

5x5 - a pixel in the corner turns into a structure with a period of four

6x6 - 1 active pixel in the corner

Reversibility in 14 moves

8x8 - 1 active pixel in the corner

Reversibility in 14 moves

Reversibility table of even squares:

* “/” means that for some patterns - initial figures - reversibility does not exist or there are static unchanging figures, with the reversibili ty of 0, or there is a special case of reversibility in fewer steps, what happens to some symmetrical patterns.

Here we see another mathematical series:

2 6 14 14 62 126 30 30 1022 126 4094 2046 1022 32766 62 62 8190 5242 86 8190 2046

All the numbers belong to the set of 2^x - 2, where x is an even number, starting with 2.

Even squares can be used in practice for encryption. To find the "inverse" algorithm which determines the previous state of the pattern "Live Pixels" is very difficult, it is possible that this is unprofitable in practice. The "direct run" of any pattern in a reversible even square gives an initial pattern in a a certain number of iterations. I will not go into details of the theory of encryption and leave the question of specific techniques how “Live Pixels” can be used for these purposes  to be open.

DNA

Analyzing the one-dimensional or a linear field, we can simulate the copying of DNA and the preservation of genetic information. Any sequence of live pixels repeated and reproduced in some time.

In this case, the number 19 is written in binary: 1011. In the fourth step the figure is repeated, at 8^th it is repeated once more, plus a mirror formed by the figure on the left. In the 24^th move we see two figures from the 8th move. The figure  1011 is visible in the right half of each of them,  thus we observe the inheritance of parental traits, like as it occurs in DNA.