## Neighborhood Theory

For general discussion about Conway's Game of Life.
erictom333
Posts: 136
Joined: January 9th, 2019, 2:44 am

### Neighborhood Theory

Neighborhood theory is a new mathematical theory I just invented.
Basics
A neighborhood is a set of ordered integer pairs (x,y), each corresponding to a cell on a grid. (x = right, y = up) (0,0) is defined as the center; a neighborhood containing the center is called closed, otherwise it is open. The empty neighborhood is denoted 0. The neighborhood containing only the center is denoted 1. A neighborhood containing only one cell is called singular.
Operators
Basic operators: Set union, subtraction, intersection, and XOR, expressed as X + Y, X - Y, X & Y, X @ Y.

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```x = 27, y = 25, rule = LifeHistory .2A3.F10.3F4.2A\$AEA2.3F2.2AE2A7.2AE2A\$2A4.F10.3F3.2A5\$.2A14.3F4.2A\$AE A2.3F2.2AE2A9.D\$2A15.3F3.2A4\$6.F\$.2A2.F.F9.3F\$AEA3.2F2.2AE2A8.AEA\$2A 3.F.F9.3F\$6.2F3\$6.F\$.2A2.F.F9.3F4.2A\$AEA2.F.F2.2AE2A7.A.D.A\$2A3.F11. 3F3.2A\$6.2F! ```
Count: The number of cells in a neighborhood X, expressed |X|.
Multiplication: For all pairs of cells (a,b) and (c,d) in neighborhoods X and Y respectively, the product of X and Y (expressed X * Y) contains cell (a+c,b+d).

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```x = 56, y = 25, rule = LifeHistory 53.2A\$5.F.F3.A3.3F6.A6.2A2.F.F3.A3.3F4.2A\$.DA3.F4.D11.D6.AE4.F4.D11.E \$5.F.F7.3F13.A3.F.F7.3F7\$53.2A\$5.F.F3.A3.3F5.2A6.2A2.F.F3.A3.3F4.3A\$. EA3.F4.E11.EA5.AE4.F4.E10.AE\$5.F.F7.3F13.A3.F.F7.3F5.A7\$22.3A27.4A\$.A 3.F.F2.3A2.3F3.5A5.2A2.F.F2.3A2.3F3.5A\$AEA3.F3.AEA8.2AE2A4.AE4.F3.AEA 8.2AE2A\$.A3.F.F2.3A2.3F3.5A5.A3.F.F2.3A2.3F3.4A\$22.3A27.3A! ```
Exponents: Defined as repeated multiplication.

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```x = 55, y = 26, rule = LifeHistory 4.F.F\$4.F.F24.A8.3A7.A\$5.F5.DA8.EA7.AEA7.AEA8.EA\$4.F.F24.A8.3A7.2A\$4. F.F5\$4.2F\$F.F3.F24.A7.5A5.A\$F.F2.F24.3A6.5A6.2A\$.F2.3F4.D.A7.E2A5.2AE 2A5.2AE2A5.2AE2A\$F.F27.3A6.5A6.3A\$F.F28.A7.5A5.3A5\$4.2F25.A6.7A3.A\$F. F3.F23.3A5.7A4.2A\$F.F2.2F22.5A4.7A3.5A\$.F2.2F5.D2.A6.E3A3.3AE3A3.3AE 3A4.2AE3A\$F.F26.5A4.7A3.6A\$F.F27.3A5.7A4.4A\$31.A6.7A3.4A! ```
Explosions: For all pairs of cells (a,b) and (c,d) in neighborhoods X and Y respectively, the explosion of X and Y (expressed X # Y) contains cell (ac-bd,ad+bc).

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``````x = 56, y = 26, rule = LifeHistory
5.F.F28.F.F\$.2A2.3F3.A3.3F4.A9.2A2.3F7.3F5.2A\$AE3.F.F3.D10.AEA6.AE3.F
F15.A\$.2A2.3F4.A2.3F4.A9.2A2.3F4.A2.3F4.3A\$AE3.F.F3.D11.E7.AE3.F.F3.D
A9.AE\$.A3.3F7.3F4.A.A7.A3.3F7.3F4.3A\$5.F.F28.F.F5\$55.A\$5.F.F15.A.A10.
F.F14.A\$.2A2.3F7.3F14.2A2.3F5.A.3F\$AE3.F.F3.D.A7.A.E7.AE3.F.F3.D11.E\$
.A3.3F7.3F14.A3.3F7.3F3.A\$5.F.F15.A12.F.F\$55.A!
``````
Identities
For any neighborhoods X and Y and any singular neighborhood S = {(x,y)}:
X + Y = Y + X
X & Y = Y & X
X @ Y = Y @ X
X + 0 = X - 0 = X @ 0 = X
X & 0 = 0
0 - X = 0
X + X = X & X = X
X - X = X @ X = 0
|X + Y| <= |X| + |Y|
|0| = 0
|1| = |S| = 1
X * Y = Y * X
X * 0 = 0
X * 1 = X
X * S = X translated right x and up y
X * Y is closed if both X and Y are closed
|X * Y| <= |X| * |Y|
|X * S| = |X|
0 ^ n = 0
1 ^ n = 1
{(x,y)} ^ n = {(nx,ny)}
X ^ n is closed if X is closed
|X ^ n| <= |X| * n^2 (unproven)
X # Y = Y # X
X # 0 = 0
X # 1 = 1
X # r = X
X # Y is closed iff either X or Y are closed
|X # Y| <= |X| * |Y|
|X # S| = |X|
Notable neighborhoods
The elementary singular neighborhoods: r = {(1,0)}, u = {(0,1)}, l = {(-1,0)}, d = {(0,-1)} and their closed forms: R = {(0,0),(1,0)}, U = {(0,0),(0,1)}, L = {(0,0),(-1,0)}, D = {(0,0),(0,-1)}. Can be multiplied and added to form all neighborhoods.
Symmetric neighborhoods: h = l + r, H = L + R, v = u + d, V = U + D, n = u + d + l + r, N = n + 1, m = ul + u + ur + l + r + dl + d + dr, M = m + 1

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```x = 43, y = 19, rule = LifeHistory .F3.F.F2.2F3.F4.2F3.F.F2.F.F2.2F3.F.F\$2F3.F.F2.F.F2.F4.F.F2.F.F2.F.F 2.F.F2.3F\$.F3.F.F2.F.F2.F4.2F3.3F2.F.F2.F.F2.F.F\$.F3.F.F2.F.F2.F4.F.F 2.F.F2.F.F2.F.F2.F.F\$3F3.2F2.2F3.3F2.F.F2.F.F3.F3.F.F2.F.F2\$6.A24.A4. A3.3A\$.E4.E4.E3.AE4.EA2.AEA3.E3.AEA2.AEA\$11.A19.A4.A3.3A2\$.2F9.F2.F9. F\$F.F9.F2.F9.F\$F.F2.F.F3.2F2.F5.2F2.2F3.F.F2.2F3.2F\$F.F2.F.F2.F.F2.F 4.F4.F.F2.F.F2.F.F2.3F\$2F4.2F3.2F3.2F2.F4.F.F3.F3.F.F2.F.F2\$6.A24.A4. A3.3A\$.D4.D4.D3.AD4.DA2.ADA3.D3.ADA2.ADA\$11.A19.A4.A3.3A! ```