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The photon/XOR system
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Posted by: Gabriel Nivasch

Here is a one-dimensional CA-like system, in which a particle (a "photon") bounces back and forth among cells which are otherwise ON of OFF.

_ _ _ _ _ _ * L * _ _ _ _ _ _
:_ _ _ _ _ * R * * _ _ _ _ _
_ _ _ _ _ * * L _ * _ _ _ _ _
:_ _ _ _ * _ R _ * * _ _ _ _
_ _ _ _ * * _ R * _ * _ _ _ _
:_ _ _ * _ * _ L * * * _ _ _
_ _ _ * * * * L * _ _ * _ _ _
:_ _ * _ _ _ R * * _ * * _ _
_ _ * * _ _ _ L _ * * _ * _ _
:_ * _ * _ _ L _ * _ * * * _
_ * * * * _ L _ * * * _ _ * _
:* _ _ _ * L _ * _ _ * _ * *
* * _ _ * R _ * * _ * * * _ *

The photon is denoted by L or R, depending on whether it is facing left or right. ON cells are deonoted *, and OFF cells are denoted _.

The cells in each row lie midway between the cells of the previous row. The regular ON and OFF cells follow the XOR rule, meaning that a cell is ON if and only if exactly one of the two cells above it is ON. The photon moves in the direction it is facing until it hits an ON cell, in which case it switches direction. The photon counts as an OFF cell for XOR purposes.

The picture attached below shows the system with the same initial condition, run to 60 generations.

The amazing thing about this system is that the photon seems to describe a pseudorandom walk. Meaning, the sequence of states of the photon seems statistically indistinguishable from a random sequence of L's and R's, as far as I have have tested.

For example, for the starting condition shown above, we get the following sequence:

LRLRR LLRLL LLRRL RLRLR LRLLL RRRLL RRLLL LRLRR LRLLR RRLLR RRLRR LRLLL RRRRR RLRRR RRRRR LLRRL LLRLR LLRRL RRLLR LLLRL RRRRR RLLRL RLRRL RRRRR RLRRL LLLLR LLLRR LRLLL LLLRL RRLLR RRRRR LRRRR RLRRL RLLLR RLLLR RRLLL RRRRL RRRLL LRLRR LRLLL LLRLL RLRRR RRRRL RRLLL RLLRR RRRLR RLRRL RLRLL LLRLR LLLLR RLLLR RRRRL RLRRL RLLLR RLRRR RRLRL LLRRL LLRRL RLRLL RRRRR RLRLR RRLRR RRRLL RRLRR LLRRR RLRLR RLLLR RLLRR LRLRL RRRRR RRRLR LRLRR LRLRL RLRRL LLLLL RRRRL RRRLR RRLLL LRRRL RRRLR RRRRL LLLLL RLRLR LLLRR RLRRR RRRRL RLLRR RLLLL LLRLR RRLLR LRLLL LLRRL LRRRR RLRLL LRLLR LRRRL LRRLR LLLLR LLLLR LRRRR RRRRR RLRLL LRLLL LRLRL RLRLL LLLRL RRRRR RRRRL RLRRL LRLLR LRLLL RLRLL RRLLL LRLRL RRLRL LRRLL RLRRL RRLRL LLLRR RRRLR LLRLR RLRLR RLRLL LLLRR LLLLR RRLLR RLRRR LLLLR LRRRR LRLLL LLLLL LLLLR LLRLR RLLRL LRRRR LRRLR LRLLL RRRRR LRLLL LRLRL LRLLL RLLLL LRLRL RLRRL LRLLR RLLLR RRRRR RRLRR LLRRL RLLRR ...

Similarly, other starting configurations produce other pseudorandom sequences.

For more information on the photon/XOR system, follow this link:

http://yucs.org/~gnivasch/life/photonXOR/index.html

Gabriel

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