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!AWMar92/Goodies/ArcAut/Info/Table3
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File contents
TABLE 3: The Neighbourhoods
ArcAut currently permits the use of two distinct neighbourhood schemes, called Moore & Margolus.
To select a particular neighbourhood for an automaton, set the system variable
called 'neig' to either 'moore' or 'margolus' within the initialisation (see table 4).
Before any neighbour can be referred to within the rules defining an automaton, the command READ_NEIG must be included.
All automatons that I have come across in magazines use the Moore neighbourhood, so i'll describe it first.
The Moore neighbourhood: each cell is surrounded by eight diagonally or orthogonally adjacent neighbours.
Within automaton code these are given the following names;
TL TM TR
ML CELL MR
BL BM BR
T meaning top, M middle, B bottom, L left & R right, with respect to the central cell
under consideration (which is called CELL).
Many of the more colourful automatons use the Moore neighbourhood.
The Margolus neighbourhood is rather more complex to describe; it is well suited to the programming of particle based
automatons such as DLA, Brownian Motion & Sound, to name a few. It was originally devised by Norman Margolus to investigate the
possibility of a reversible 'ballistic computer' automaton.
The Margolus neighbourhood: we ensure both the width & height of the CA universe are even. A partition is then imposed,
dividing the universe up by a grid into blocks, each made up from 4 cells in a 2x2 arrangement
for example in an 4x4 universe;
ab cd We say that a grid aligned with the universe in this way has
ef gh even phase
ij kl
mn op
Each cell has 3 other neighbours, these being the other three cells within the same block.
All 4 cell within a particular block can be referred to by the names;
UL UR
LL LR
UL meaning upper left, UR upper right, LL lower left & LR lower right
More usefully we can refer to all four cells via their location relative to the cell under
consideration, with the names CELL, CW, CCW & OPP
CW meaning clockwise, CCW counter clockwise & OPP opposite
The four possible actual orientations are;
CELL CW CCW CELL OPP CCW CW OPP
CCW OPP OPP CW CW CELL CELL CCW
If we were to leave the partition positioned as above, then the cells in different blocks could
never communicate & we couldn't do anything interesting. Instead, we allow the phase of the grid
to alternate between even & odd, as the generations progress. An even partition is shown above.
The odd partition is obtained by shifting the even grid by one cell diagonally.
For the 4x4 universe this would give us;
a bc d
e fg h
i jk l
m no p
where, using wrap around, the cells form blocks as follows; pm no he fg
da bc li jk
It is often useful within the automaton code to know what the phase is, or which position within
the block CELL has. This information can be passed with the register that is referred to by
the name FLAG. Either the phase, the orientation, both or neither are held in FLAG depending
on the setting of the system variable flag; see table 4
The Margolus scheme is quite different from the Moore, & at first may seem unnecessarily
complicated & of no obvious use. To dispell this idea consider the following trivial example.
We define a simple automaton which starts from a single white point on a black background, & then
applies the rule (OPP ==). Imagine this yourself, acting on a small universe. It soon becomes
clear that this results in the white point appearing to move along a diagonal - the direction it
goes in depends on it's position within the 2x2 block containing it. Just try and design an
automaton using the Moore neighbourhood which does this; even if you can work out how, it will
be very cumbersome & difficult to follow (although I ought to point out that anything you can do
with the Margolus neighbourhood can also be done with the Moore). Take a look at any of the
automatons provided which use the margolus neighbourhood; see if you can work out what they do
before you run them.
While an automaton is running, you can force it to use the same phase on two succesive generations
rather than automatically alternating, by clicking menu & then selecting the Phase option from the
menu; either resume with Cont. or watch the action a frame at a time by clicking on Next. In some
cases this can have a dramatic effect, even causing the automaton to run backwards! (As would be
the case with the trivial example given above).
00000000 0a 20 20 54 41 42 4c 45 20 33 3a 20 54 68 65 20 |. TABLE 3: The | 00000010 4e 65 69 67 68 62 6f 75 72 68 6f 6f 64 73 0a 0a |Neighbourhoods..| 00000020 0a 20 20 41 72 63 41 75 74 20 63 75 72 72 65 6e |. ArcAut curren| 00000030 74 6c 79 20 70 65 72 6d 69 74 73 20 74 68 65 20 |tly permits the | 00000040 75 73 65 20 6f 66 20 74 77 6f 20 64 69 73 74 69 |use of two disti| 00000050 6e 63 74 20 6e 65 69 67 68 62 6f 75 72 68 6f 6f |nct neighbourhoo| 00000060 64 20 73 63 68 65 6d 65 73 2c 20 63 61 6c 6c 65 |d schemes, calle| 00000070 64 20 4d 6f 6f 72 65 20 26 20 4d 61 72 67 6f 6c |d Moore & Margol| 00000080 75 73 2e 0a 20 20 54 6f 20 73 65 6c 65 63 74 20 |us.. To select | 00000090 61 20 70 61 72 74 69 63 75 6c 61 72 20 6e 65 69 |a particular nei| 000000a0 67 68 62 6f 75 72 68 6f 6f 64 20 66 6f 72 20 61 |ghbourhood for a| 000000b0 6e 20 61 75 74 6f 6d 61 74 6f 6e 2c 20 73 65 74 |n automaton, set| 000000c0 20 74 68 65 20 73 79 73 74 65 6d 20 76 61 72 69 | the system vari| 000000d0 61 62 6c 65 0a 20 20 63 61 6c 6c 65 64 20 27 6e |able. called 'n| 000000e0 65 69 67 27 20 74 6f 20 65 69 74 68 65 72 20 27 |eig' to either '| 000000f0 6d 6f 6f 72 65 27 20 6f 72 20 27 6d 61 72 67 6f |moore' or 'margo| 00000100 6c 75 73 27 20 77 69 74 68 69 6e 20 74 68 65 20 |lus' within the | 00000110 69 6e 69 74 69 61 6c 69 73 61 74 69 6f 6e 20 28 |initialisation (| 00000120 73 65 65 20 74 61 62 6c 65 20 34 29 2e 0a 20 20 |see table 4).. | 00000130 42 65 66 6f 72 65 20 61 6e 79 20 6e 65 69 67 68 |Before any neigh| 00000140 62 6f 75 72 20 63 61 6e 20 62 65 20 72 65 66 65 |bour can be refe| 00000150 72 72 65 64 20 74 6f 20 77 69 74 68 69 6e 20 74 |rred to within t| 00000160 68 65 20 72 75 6c 65 73 20 64 65 66 69 6e 69 6e |he rules definin| 00000170 67 20 61 6e 20 61 75 74 6f 6d 61 74 6f 6e 2c 20 |g an automaton, | 00000180 74 68 65 20 63 6f 6d 6d 61 6e 64 20 52 45 41 44 |the command READ| 00000190 5f 4e 45 49 47 20 6d 75 73 74 20 62 65 20 69 6e |_NEIG must be in| 000001a0 63 6c 75 64 65 64 2e 0a 20 20 41 6c 6c 20 61 75 |cluded.. All au| 000001b0 74 6f 6d 61 74 6f 6e 73 20 74 68 61 74 20 49 20 |tomatons that I | 000001c0 68 61 76 65 20 63 6f 6d 65 20 61 63 72 6f 73 73 |have come across| 000001d0 20 69 6e 20 6d 61 67 61 7a 69 6e 65 73 20 75 73 | in magazines us| 000001e0 65 20 74 68 65 20 4d 6f 6f 72 65 20 6e 65 69 67 |e the Moore neig| 000001f0 68 62 6f 75 72 68 6f 6f 64 2c 20 73 6f 20 69 27 |hbourhood, so i'| 00000200 6c 6c 20 64 65 73 63 72 69 62 65 20 69 74 20 66 |ll describe it f| 00000210 69 72 73 74 2e 0a 0a 20 20 54 68 65 20 4d 6f 6f |irst... The Moo| 00000220 72 65 20 6e 65 69 67 68 62 6f 75 72 68 6f 6f 64 |re neighbourhood| 00000230 3a 20 20 65 61 63 68 20 63 65 6c 6c 20 69 73 20 |: each cell is | 00000240 73 75 72 72 6f 75 6e 64 65 64 20 62 79 20 65 69 |surrounded by ei| 00000250 67 68 74 20 64 69 61 67 6f 6e 61 6c 6c 79 20 6f |ght diagonally o| 00000260 72 20 6f 72 74 68 6f 67 6f 6e 61 6c 6c 79 20 61 |r orthogonally a| 00000270 64 6a 61 63 65 6e 74 20 6e 65 69 67 68 62 6f 75 |djacent neighbou| 00000280 72 73 2e 0a 20 20 20 20 20 20 20 20 20 20 20 20 |rs.. | 00000290 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000002a0 57 69 74 68 69 6e 20 61 75 74 6f 6d 61 74 6f 6e |Within automaton| 000002b0 20 63 6f 64 65 20 74 68 65 73 65 20 61 72 65 20 | code these are | 000002c0 67 69 76 65 6e 20 74 68 65 20 66 6f 6c 6c 6f 77 |given the follow| 000002d0 69 6e 67 20 6e 61 6d 65 73 3b 0a 0a 20 20 20 20 |ing names;.. | 000002e0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000002f0 20 20 20 20 20 20 20 20 20 20 54 4c 20 20 20 54 | TL T| 00000300 4d 20 20 20 54 52 0a 20 20 20 20 20 20 20 20 20 |M TR. | 00000310 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000320 20 20 20 20 20 4d 4c 20 20 43 45 4c 4c 20 20 4d | ML CELL M| 00000330 52 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |R. | 00000340 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000350 42 4c 20 20 20 42 4d 20 20 20 42 52 0a 0a 20 20 |BL BM BR.. | 00000360 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000370 20 20 20 20 20 20 20 20 20 20 54 20 6d 65 61 6e | T mean| 00000380 69 6e 67 20 74 6f 70 2c 20 4d 20 6d 69 64 64 6c |ing top, M middl| 00000390 65 2c 20 42 20 62 6f 74 74 6f 6d 2c 20 4c 20 6c |e, B bottom, L l| 000003a0 65 66 74 20 26 20 52 20 72 69 67 68 74 2c 20 77 |eft & R right, w| 000003b0 69 74 68 20 72 65 73 70 65 63 74 20 74 6f 20 74 |ith respect to t| 000003c0 68 65 20 63 65 6e 74 72 61 6c 20 63 65 6c 6c 0a |he central cell.| 000003d0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000003e0 20 20 20 20 20 20 20 20 20 20 20 20 75 6e 64 65 | unde| 000003f0 72 20 63 6f 6e 73 69 64 65 72 61 74 69 6f 6e 20 |r consideration | 00000400 28 77 68 69 63 68 20 69 73 20 63 61 6c 6c 65 64 |(which is called| 00000410 20 43 45 4c 4c 29 2e 0a 20 20 20 20 20 20 20 20 | CELL).. | 00000420 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000430 20 20 20 20 4d 61 6e 79 20 6f 66 20 74 68 65 20 | Many of the | 00000440 6d 6f 72 65 20 63 6f 6c 6f 75 72 66 75 6c 20 61 |more colourful a| 00000450 75 74 6f 6d 61 74 6f 6e 73 20 75 73 65 20 74 68 |utomatons use th| 00000460 65 20 4d 6f 6f 72 65 20 6e 65 69 67 68 62 6f 75 |e Moore neighbou| 00000470 72 68 6f 6f 64 2e 0a 0a 20 20 54 68 65 20 4d 61 |rhood... The Ma| 00000480 72 67 6f 6c 75 73 20 6e 65 69 67 68 62 6f 75 72 |rgolus neighbour| 00000490 68 6f 6f 64 20 69 73 20 72 61 74 68 65 72 20 6d |hood is rather m| 000004a0 6f 72 65 20 63 6f 6d 70 6c 65 78 20 74 6f 20 64 |ore complex to d| 000004b0 65 73 63 72 69 62 65 3b 20 69 74 20 69 73 20 77 |escribe; it is w| 000004c0 65 6c 6c 20 73 75 69 74 65 64 20 74 6f 20 74 68 |ell suited to th| 000004d0 65 20 70 72 6f 67 72 61 6d 6d 69 6e 67 20 6f 66 |e programming of| 000004e0 20 70 61 72 74 69 63 6c 65 20 62 61 73 65 64 0a | particle based.| 000004f0 61 75 74 6f 6d 61 74 6f 6e 73 20 73 75 63 68 20 |automatons such | 00000500 61 73 20 44 4c 41 2c 20 42 72 6f 77 6e 69 61 6e |as DLA, Brownian| 00000510 20 4d 6f 74 69 6f 6e 20 26 20 53 6f 75 6e 64 2c | Motion & Sound,| 00000520 20 74 6f 20 6e 61 6d 65 20 61 20 66 65 77 2e 20 | to name a few. | 00000530 49 74 20 77 61 73 20 6f 72 69 67 69 6e 61 6c 6c |It was originall| 00000540 79 20 64 65 76 69 73 65 64 20 62 79 20 4e 6f 72 |y devised by Nor| 00000550 6d 61 6e 20 4d 61 72 67 6f 6c 75 73 20 74 6f 20 |man Margolus to | 00000560 69 6e 76 65 73 74 69 67 61 74 65 20 74 68 65 0a |investigate the.| 00000570 70 6f 73 73 69 62 69 6c 69 74 79 20 6f 66 20 61 |possibility of a| 00000580 20 72 65 76 65 72 73 69 62 6c 65 20 27 62 61 6c | reversible 'bal| 00000590 6c 69 73 74 69 63 20 63 6f 6d 70 75 74 65 72 27 |listic computer'| 000005a0 20 61 75 74 6f 6d 61 74 6f 6e 2e 0a 0a 20 20 54 | automaton... T| 000005b0 68 65 20 4d 61 72 67 6f 6c 75 73 20 6e 65 69 67 |he Margolus neig| 000005c0 68 62 6f 75 72 68 6f 6f 64 3a 20 20 77 65 20 65 |hbourhood: we e| 000005d0 6e 73 75 72 65 20 62 6f 74 68 20 74 68 65 20 77 |nsure both the w| 000005e0 69 64 74 68 20 26 20 68 65 69 67 68 74 20 6f 66 |idth & height of| 000005f0 20 74 68 65 20 43 41 20 75 6e 69 76 65 72 73 65 | the CA universe| 00000600 20 61 72 65 20 65 76 65 6e 2e 20 41 20 70 61 72 | are even. A par| 00000610 74 69 74 69 6f 6e 20 69 73 20 74 68 65 6e 20 69 |tition is then i| 00000620 6d 70 6f 73 65 64 2c 0a 20 20 20 20 20 20 20 20 |mposed,. | 00000630 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000640 20 20 20 20 20 20 20 64 69 76 69 64 69 6e 67 20 | dividing | 00000650 74 68 65 20 75 6e 69 76 65 72 73 65 20 75 70 20 |the universe up | 00000660 62 79 20 61 20 67 72 69 64 20 69 6e 74 6f 20 62 |by a grid into b| 00000670 6c 6f 63 6b 73 2c 20 65 61 63 68 20 6d 61 64 65 |locks, each made| 00000680 20 75 70 20 66 72 6f 6d 20 34 20 63 65 6c 6c 73 | up from 4 cells| 00000690 20 69 6e 20 61 20 32 78 32 20 61 72 72 61 6e 67 | in a 2x2 arrang| 000006a0 65 6d 65 6e 74 0a 20 20 20 20 20 20 20 20 20 20 |ement. | 000006b0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000006c0 20 20 20 20 20 66 6f 72 20 65 78 61 6d 70 6c 65 | for example| 000006d0 20 69 6e 20 61 6e 20 34 78 34 20 75 6e 69 76 65 | in an 4x4 unive| 000006e0 72 73 65 3b 0a 0a 20 20 20 20 20 20 20 20 20 20 |rse;.. | 000006f0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000700 20 20 20 20 20 20 20 61 62 20 20 63 64 20 20 20 | ab cd | 00000710 20 20 20 57 65 20 73 61 79 20 74 68 61 74 20 61 | We say that a| 00000720 20 67 72 69 64 20 61 6c 69 67 6e 65 64 20 77 69 | grid aligned wi| 00000730 74 68 20 74 68 65 20 75 6e 69 76 65 72 73 65 20 |th the universe | 00000740 69 6e 20 74 68 69 73 20 77 61 79 20 68 61 73 0a |in this way has.| 00000750 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | * 00000770 20 65 66 20 20 67 68 20 20 20 20 20 20 65 76 65 | ef gh eve| 00000780 6e 20 70 68 61 73 65 0a 0a 20 20 20 20 20 20 20 |n phase.. | 00000790 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000007a0 20 20 20 20 20 20 20 20 20 20 69 6a 20 20 6b 6c | ij kl| 000007b0 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |. | 000007c0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000007d0 20 20 6d 6e 20 20 6f 70 0a 0a 20 20 20 20 20 20 | mn op.. | 000007e0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000007f0 20 20 20 20 20 20 20 20 20 45 61 63 68 20 63 65 | Each ce| 00000800 6c 6c 20 68 61 73 20 33 20 6f 74 68 65 72 20 6e |ll has 3 other n| 00000810 65 69 67 68 62 6f 75 72 73 2c 20 74 68 65 73 65 |eighbours, these| 00000820 20 62 65 69 6e 67 20 74 68 65 20 6f 74 68 65 72 | being the other| 00000830 20 74 68 72 65 65 20 63 65 6c 6c 73 20 77 69 74 | three cells wit| 00000840 68 69 6e 20 74 68 65 20 73 61 6d 65 20 62 6c 6f |hin the same blo| 00000850 63 6b 2e 0a 20 20 20 20 20 20 20 20 20 20 20 20 |ck.. | 00000860 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000870 20 20 20 41 6c 6c 20 34 20 63 65 6c 6c 20 77 69 | All 4 cell wi| 00000880 74 68 69 6e 20 61 20 70 61 72 74 69 63 75 6c 61 |thin a particula| 00000890 72 20 62 6c 6f 63 6b 20 63 61 6e 20 62 65 20 72 |r block can be r| 000008a0 65 66 65 72 72 65 64 20 74 6f 20 62 79 20 74 68 |eferred to by th| 000008b0 65 20 6e 61 6d 65 73 3b 0a 0a 20 20 20 20 20 20 |e names;.. | 000008c0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000008d0 20 20 20 20 20 20 20 20 20 20 20 55 4c 20 55 52 | UL UR| 000008e0 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |. | 000008f0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000900 20 20 4c 4c 20 4c 52 0a 0a 20 20 20 20 20 20 20 | LL LR.. | 00000910 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000920 20 20 20 20 20 20 20 20 55 4c 20 6d 65 61 6e 69 | UL meani| 00000930 6e 67 20 75 70 70 65 72 20 6c 65 66 74 2c 20 55 |ng upper left, U| 00000940 52 20 75 70 70 65 72 20 72 69 67 68 74 2c 20 4c |R upper right, L| 00000950 4c 20 6c 6f 77 65 72 20 6c 65 66 74 20 26 20 4c |L lower left & L| 00000960 52 20 6c 6f 77 65 72 20 72 69 67 68 74 0a 20 20 |R lower right. | 00000970 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000980 20 20 20 20 20 20 20 20 20 20 20 20 20 4d 6f 72 | Mor| 00000990 65 20 75 73 65 66 75 6c 6c 79 20 77 65 20 63 61 |e usefully we ca| 000009a0 6e 20 72 65 66 65 72 20 74 6f 20 61 6c 6c 20 66 |n refer to all f| 000009b0 6f 75 72 20 63 65 6c 6c 73 20 76 69 61 20 74 68 |our cells via th| 000009c0 65 69 72 20 6c 6f 63 61 74 69 6f 6e 20 72 65 6c |eir location rel| 000009d0 61 74 69 76 65 20 74 6f 20 74 68 65 20 63 65 6c |ative to the cel| 000009e0 6c 20 75 6e 64 65 72 0a 20 20 20 20 20 20 20 20 |l under. | 000009f0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000a00 20 20 20 20 20 20 20 63 6f 6e 73 69 64 65 72 61 | considera| 00000a10 74 69 6f 6e 2c 20 77 69 74 68 20 74 68 65 20 6e |tion, with the n| 00000a20 61 6d 65 73 20 43 45 4c 4c 2c 20 43 57 2c 20 43 |ames CELL, CW, C| 00000a30 43 57 20 26 20 4f 50 50 0a 20 20 20 20 20 20 20 |CW & OPP. | 00000a40 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000a50 20 20 20 20 20 20 20 20 43 57 20 6d 65 61 6e 69 | CW meani| 00000a60 6e 67 20 63 6c 6f 63 6b 77 69 73 65 2c 20 43 43 |ng clockwise, CC| 00000a70 57 20 63 6f 75 6e 74 65 72 20 63 6c 6f 63 6b 77 |W counter clockw| 00000a80 69 73 65 20 26 20 4f 50 50 20 6f 70 70 6f 73 69 |ise & OPP opposi| 00000a90 74 65 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 |te. | 00000aa0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000ab0 20 20 54 68 65 20 66 6f 75 72 20 70 6f 73 73 69 | The four possi| 00000ac0 62 6c 65 20 61 63 74 75 61 6c 20 6f 72 69 65 6e |ble actual orien| 00000ad0 74 61 74 69 6f 6e 73 20 61 72 65 3b 0a 0a 20 20 |tations are;.. | 00000ae0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000af0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 43 | C| 00000b00 45 4c 4c 20 20 20 43 57 20 20 20 20 20 20 20 20 |ELL CW | 00000b10 20 20 20 43 43 57 20 20 43 45 4c 4c 20 20 20 20 | CCW CELL | 00000b20 20 20 20 20 20 20 20 4f 50 50 20 20 20 43 43 57 | OPP CCW| 00000b30 20 20 20 20 20 20 20 20 20 20 20 20 43 57 20 20 | CW | 00000b40 4f 50 50 0a 20 20 20 20 20 20 20 20 20 20 20 20 |OPP. | 00000b50 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000b60 20 20 20 20 20 20 43 43 57 20 20 4f 50 50 20 20 | CCW OPP | 00000b70 20 20 20 20 20 20 20 20 20 4f 50 50 20 20 20 20 | OPP | 00000b80 43 57 20 20 20 20 20 20 20 20 20 20 20 20 43 57 |CW CW| 00000b90 20 20 43 45 4c 4c 20 20 20 20 20 20 20 20 20 20 | CELL | 00000ba0 43 45 4c 4c 20 20 43 43 57 0a 0a 20 20 20 20 20 |CELL CCW.. | 00000bb0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000bc0 20 20 20 20 20 20 20 20 20 20 49 66 20 77 65 20 | If we | 00000bd0 77 65 72 65 20 74 6f 20 6c 65 61 76 65 20 74 68 |were to leave th| 00000be0 65 20 70 61 72 74 69 74 69 6f 6e 20 70 6f 73 69 |e partition posi| 00000bf0 74 69 6f 6e 65 64 20 61 73 20 61 62 6f 76 65 2c |tioned as above,| 00000c00 20 74 68 65 6e 20 74 68 65 20 63 65 6c 6c 73 20 | then the cells | 00000c10 69 6e 20 64 69 66 66 65 72 65 6e 74 20 62 6c 6f |in different blo| 00000c20 63 6b 73 20 63 6f 75 6c 64 0a 20 20 20 20 20 20 |cks could. | 00000c30 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000c40 20 20 20 20 20 20 20 20 20 6e 65 76 65 72 20 63 | never c| 00000c50 6f 6d 6d 75 6e 69 63 61 74 65 20 26 20 77 65 20 |ommunicate & we | 00000c60 63 6f 75 6c 64 6e 27 74 20 64 6f 20 61 6e 79 74 |couldn't do anyt| 00000c70 68 69 6e 67 20 69 6e 74 65 72 65 73 74 69 6e 67 |hing interesting| 00000c80 2e 20 49 6e 73 74 65 61 64 2c 20 77 65 20 61 6c |. Instead, we al| 00000c90 6c 6f 77 20 74 68 65 20 70 68 61 73 65 20 6f 66 |low the phase of| 00000ca0 20 74 68 65 20 67 72 69 64 0a 20 20 20 20 20 20 | the grid. | 00000cb0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000cc0 20 20 20 20 20 20 20 20 20 74 6f 20 61 6c 74 65 | to alte| 00000cd0 72 6e 61 74 65 20 62 65 74 77 65 65 6e 20 65 76 |rnate between ev| 00000ce0 65 6e 20 26 20 6f 64 64 2c 20 61 73 20 74 68 65 |en & odd, as the| 00000cf0 20 67 65 6e 65 72 61 74 69 6f 6e 73 20 70 72 6f | generations pro| 00000d00 67 72 65 73 73 2e 20 41 6e 20 65 76 65 6e 20 70 |gress. An even p| 00000d10 61 72 74 69 74 69 6f 6e 20 69 73 20 73 68 6f 77 |artition is show| 00000d20 6e 20 61 62 6f 76 65 2e 0a 20 20 20 20 20 20 20 |n above.. | 00000d30 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000d40 20 20 20 20 20 20 20 20 54 68 65 20 6f 64 64 20 | The odd | 00000d50 70 61 72 74 69 74 69 6f 6e 20 69 73 20 6f 62 74 |partition is obt| 00000d60 61 69 6e 65 64 20 62 79 20 73 68 69 66 74 69 6e |ained by shiftin| 00000d70 67 20 74 68 65 20 65 76 65 6e 20 67 72 69 64 20 |g the even grid | 00000d80 62 79 20 6f 6e 65 20 63 65 6c 6c 20 64 69 61 67 |by one cell diag| 00000d90 6f 6e 61 6c 6c 79 2e 0a 20 20 20 20 20 20 20 20 |onally.. | 00000da0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000db0 20 20 20 20 20 20 20 46 6f 72 20 74 68 65 20 34 | For the 4| 00000dc0 78 34 20 75 6e 69 76 65 72 73 65 20 74 68 69 73 |x4 universe this| 00000dd0 20 77 6f 75 6c 64 20 67 69 76 65 20 75 73 3b 0a | would give us;.| 00000de0 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |. | 00000df0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000e00 20 20 61 20 20 62 63 20 20 64 0a 20 20 20 20 20 | a bc d. | 00000e10 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000e20 20 20 20 20 20 20 20 20 20 20 20 20 0a 20 20 20 | . | 00000e30 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000e40 20 20 20 20 20 20 20 20 20 20 20 20 20 20 65 20 | e | 00000e50 20 66 67 20 20 68 0a 20 20 20 20 20 20 20 20 20 | fg h. | 00000e60 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000e70 20 20 20 20 20 20 20 20 69 20 20 6a 6b 20 20 6c | i jk l| 00000e80 0a 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |.. | 00000e90 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000ea0 20 20 20 6d 20 20 6e 6f 20 20 70 0a 0a 20 20 20 | m no p.. | 00000eb0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000ec0 20 20 20 20 20 20 20 20 20 20 20 20 77 68 65 72 | wher| 00000ed0 65 2c 20 75 73 69 6e 67 20 77 72 61 70 20 61 72 |e, using wrap ar| 00000ee0 6f 75 6e 64 2c 20 74 68 65 20 63 65 6c 6c 73 20 |ound, the cells | 00000ef0 66 6f 72 6d 20 62 6c 6f 63 6b 73 20 61 73 20 66 |form blocks as f| 00000f00 6f 6c 6c 6f 77 73 3b 20 20 70 6d 20 20 6e 6f 20 |ollows; pm no | 00000f10 20 68 65 20 20 66 67 0a 20 20 20 20 20 20 20 20 | he fg. | 00000f20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | * 00000f70 20 20 20 20 64 61 20 20 62 63 20 20 6c 69 20 20 | da bc li | 00000f80 6a 6b 0a 0a 20 20 20 20 20 20 20 20 20 20 20 20 |jk.. | 00000f90 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000fa0 20 20 20 49 74 20 69 73 20 6f 66 74 65 6e 20 75 | It is often u| 00000fb0 73 65 66 75 6c 20 77 69 74 68 69 6e 20 74 68 65 |seful within the| 00000fc0 20 61 75 74 6f 6d 61 74 6f 6e 20 63 6f 64 65 20 | automaton code | 00000fd0 74 6f 20 6b 6e 6f 77 20 77 68 61 74 20 74 68 65 |to know what the| 00000fe0 20 70 68 61 73 65 20 69 73 2c 20 6f 72 20 77 68 | phase is, or wh| 00000ff0 69 63 68 20 70 6f 73 69 74 69 6f 6e 20 77 69 74 |ich position wit| 00001000 68 69 6e 0a 20 20 20 20 20 20 20 20 20 20 20 20 |hin. | 00001010 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00001020 20 20 20 74 68 65 20 62 6c 6f 63 6b 20 43 45 4c | the block CEL| 00001030 4c 20 68 61 73 2e 20 54 68 69 73 20 69 6e 66 6f |L has. This info| 00001040 72 6d 61 74 69 6f 6e 20 63 61 6e 20 62 65 20 70 |rmation can be p| 00001050 61 73 73 65 64 20 77 69 74 68 20 74 68 65 20 72 |assed with the r| 00001060 65 67 69 73 74 65 72 20 74 68 61 74 20 69 73 20 |egister that is | 00001070 72 65 66 65 72 72 65 64 20 74 6f 20 62 79 0a 20 |referred to by. | 00001080 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00001090 20 20 20 20 20 20 20 20 20 20 20 20 20 20 74 68 | th| 000010a0 65 20 6e 61 6d 65 20 46 4c 41 47 2e 20 45 69 74 |e name FLAG. Eit| 000010b0 68 65 72 20 74 68 65 20 70 68 61 73 65 2c 20 74 |her the phase, t| 000010c0 68 65 20 6f 72 69 65 6e 74 61 74 69 6f 6e 2c 20 |he orientation, | 000010d0 62 6f 74 68 20 6f 72 20 6e 65 69 74 68 65 72 20 |both or neither | 000010e0 61 72 65 20 68 65 6c 64 20 69 6e 20 46 4c 41 47 |are held in FLAG| 000010f0 20 64 65 70 65 6e 64 69 6e 67 0a 20 20 20 20 20 | depending. | 00001100 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00001110 20 20 20 20 20 20 20 20 20 20 6f 6e 20 74 68 65 | on the| 00001120 20 73 65 74 74 69 6e 67 20 6f 66 20 74 68 65 20 | setting of the | 00001130 73 79 73 74 65 6d 20 76 61 72 69 61 62 6c 65 20 |system variable | 00001140 66 6c 61 67 3b 20 73 65 65 20 74 61 62 6c 65 20 |flag; see table | 00001150 34 0a 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |4. | 00001160 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00001170 20 54 68 65 20 4d 61 72 67 6f 6c 75 73 20 73 63 | The Margolus sc| 00001180 68 65 6d 65 20 69 73 20 71 75 69 74 65 20 64 69 |heme is quite di| 00001190 66 66 65 72 65 6e 74 20 66 72 6f 6d 20 74 68 65 |fferent from the| 000011a0 20 4d 6f 6f 72 65 2c 20 26 20 61 74 20 66 69 72 | Moore, & at fir| 000011b0 73 74 20 6d 61 79 20 73 65 65 6d 20 75 6e 6e 65 |st may seem unne| 000011c0 63 65 73 73 61 72 69 6c 79 0a 20 20 20 20 20 20 |cessarily. | 000011d0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000011e0 20 20 20 20 20 20 20 20 20 63 6f 6d 70 6c 69 63 | complic| 000011f0 61 74 65 64 20 26 20 6f 66 20 6e 6f 20 6f 62 76 |ated & of no obv| 00001200 69 6f 75 73 20 75 73 65 2e 20 54 6f 20 64 69 73 |ious use. 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T| 000015a0 61 6b 65 20 61 20 6c 6f 6f 6b 20 61 74 20 61 6e |ake a look at an| 000015b0 79 20 6f 66 20 74 68 65 0a 20 20 20 20 20 20 20 |y of the. | 000015c0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000015d0 20 20 20 20 20 20 20 20 61 75 74 6f 6d 61 74 6f | automato| 000015e0 6e 73 20 70 72 6f 76 69 64 65 64 20 77 68 69 63 |ns provided whic| 000015f0 68 20 75 73 65 20 74 68 65 20 6d 61 72 67 6f 6c |h use the margol| 00001600 75 73 20 6e 65 69 67 68 62 6f 75 72 68 6f 6f 64 |us neighbourhood| 00001610 3b 20 73 65 65 20 69 66 20 79 6f 75 20 63 61 6e |; see if you can| 00001620 20 77 6f 72 6b 20 6f 75 74 20 77 68 61 74 20 74 | work out what t| 00001630 68 65 79 20 64 6f 0a 20 20 20 20 20 20 20 20 20 |hey do. | 00001640 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00001650 20 20 20 20 20 20 62 65 66 6f 72 65 20 79 6f 75 | before you| 00001660 20 72 75 6e 20 74 68 65 6d 2e 0a 20 20 20 20 20 | run them.. | 00001670 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00001680 20 20 20 20 20 20 20 20 20 20 57 68 69 6c 65 20 | While | 00001690 61 6e 20 61 75 74 6f 6d 61 74 6f 6e 20 69 73 20 |an automaton is | 000016a0 72 75 6e 6e 69 6e 67 2c 20 79 6f 75 20 63 61 6e |running, you can| 000016b0 20 66 6f 72 63 65 20 69 74 20 74 6f 20 75 73 65 | force it to use| 000016c0 20 74 68 65 20 73 61 6d 65 20 70 68 61 73 65 20 | the same phase | 000016d0 6f 6e 20 74 77 6f 20 73 75 63 63 65 73 69 76 65 |on two succesive| 000016e0 20 67 65 6e 65 72 61 74 69 6f 6e 73 0a 20 20 20 | generations. | 000016f0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00001700 20 20 20 20 20 20 20 20 20 20 20 20 72 61 74 68 | rath| 00001710 65 72 20 74 68 61 6e 20 61 75 74 6f 6d 61 74 69 |er than automati| 00001720 63 61 6c 6c 79 20 61 6c 74 65 72 6e 61 74 69 6e |cally alternatin| 00001730 67 2c 20 62 79 20 63 6c 69 63 6b 69 6e 67 20 6d |g, by clicking m| 00001740 65 6e 75 20 26 20 74 68 65 6e 20 73 65 6c 65 63 |enu & then selec| 00001750 74 69 6e 67 20 74 68 65 20 50 68 61 73 65 20 6f |ting the Phase o| 00001760 70 74 69 6f 6e 20 66 72 6f 6d 20 74 68 65 0a 20 |ption from the. | 00001770 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00001780 20 20 20 20 20 20 20 20 20 20 20 20 20 20 6d 65 | me| 00001790 6e 75 3b 20 65 69 74 68 65 72 20 72 65 73 75 6d |nu; either resum| 000017a0 65 20 77 69 74 68 20 43 6f 6e 74 2e 20 6f 72 20 |e with Cont. or | 000017b0 77 61 74 63 68 20 74 68 65 20 61 63 74 69 6f 6e |watch the action| 000017c0 20 61 20 66 72 61 6d 65 20 61 74 20 61 20 74 69 | a frame at a ti| 000017d0 6d 65 20 62 79 20 63 6c 69 63 6b 69 6e 67 20 6f |me by clicking o| 000017e0 6e 20 4e 65 78 74 2e 20 49 6e 20 73 6f 6d 65 0a |n Next. In some.| 000017f0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00001800 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 63 | c| 00001810 61 73 65 73 20 74 68 69 73 20 63 61 6e 20 68 61 |ases this can ha| 00001820 76 65 20 61 20 64 72 61 6d 61 74 69 63 20 65 66 |ve a dramatic ef| 00001830 66 65 63 74 2c 20 65 76 65 6e 20 63 61 75 73 69 |fect, even causi| 00001840 6e 67 20 74 68 65 20 61 75 74 6f 6d 61 74 6f 6e |ng the automaton| 00001850 20 74 6f 20 72 75 6e 20 62 61 63 6b 77 61 72 64 | to run backward| 00001860 73 21 20 28 41 73 20 77 6f 75 6c 64 20 62 65 0a |s! (As would be.| 00001870 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00001880 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 74 | t| 00001890 68 65 20 63 61 73 65 20 77 69 74 68 20 74 68 65 |he case with the| 000018a0 20 74 72 69 76 69 61 6c 20 65 78 61 6d 70 6c 65 | trivial example| 000018b0 20 67 69 76 65 6e 20 61 62 6f 76 65 29 2e 0a | given above)..| 000018bf
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