"Typical patterns"
Posted: May 9th, 2013, 12:34 pm
I posted previously some observations based on the concept of "typical Life patterns".
I'm thinking of how to quantify "typicalness". An approach that seems promising might be a simple bounding box approach:
• If the smallest parent of an M×N pattern is of size M'×N', it has a typicality rank of (M+N)–(M'+N')-4. This can be obviously broken into distinct horizontal and vertical components, and moreover NWES components.
• a 0-typical pattern has a parent that fits within (M–2)×(N–2) (the minimum typicality possible). The smallest such pattern is generation 5 of T-tetromino, the 5×5 diamond. Other common examples include gen 8 (4 crosses) and gen 10 (traffic lights in the "+" phase) of the same; gens 18, 20, 22, 24, 27 (= finished honeyfarm) of a bun; gen 9 of pi-heptomino.
• the smallest 1-typical pattern is 4×5, ie. the diamond minus one corner cell, which has a 3×3 parent.
• the smallest 2-typical pattern is the beehive (3×4 ← 3×2)
• the smallest 3-typical pattern is the V spark (3×3 ← 3×2)
• a 4-typical pattern either has a parent of the same size, or one that's wider but shorter. Vacuum is trivially 4-typical; the smallest non-zero such pattern is the blinker.
• from 5-typical on, patterns only have parents larger than themselves.
• a domino is 6-typical
• a dot is 8-typical
• 9+-typical examples are not obvious to come up with but I hypothesize they dominate at large pattern sizes
• more generally, I hypothesize that at least for the lower values, let's say x < y ≤ 8, if y-typical patterns of a given size exist, they are more numerous than x-typical ones
• there is probably a maximum typicality, around maybe 12-16?
We could also speak of the typicality of a particular pattern transition: the first steps of a B's evolution are 4, 2, 3, 3, 3, 2, 4, 2, 4, 4, 3, 4. Perhaps more illuminating though would be 4 – typicality (we could call this "growth rate" or something): this sequence then becomes 0, 2, 1, 1, 1, 2, 0, 2, 0, 0, 1, 0 and it is simple to see the pattern is growing.
A more finer-granulated measure of typicality might take all parents of a pattern into account, with smaller parents very strongly weighed compared to larger ones (after all, any given pattern has arbitrarily large parents) — but this seems difficult to calculate. Considering the bounding octagon rather than rectangle might also produce some further insight.
I'm thinking of how to quantify "typicalness". An approach that seems promising might be a simple bounding box approach:
• If the smallest parent of an M×N pattern is of size M'×N', it has a typicality rank of (M+N)–(M'+N')-4. This can be obviously broken into distinct horizontal and vertical components, and moreover NWES components.
• a 0-typical pattern has a parent that fits within (M–2)×(N–2) (the minimum typicality possible). The smallest such pattern is generation 5 of T-tetromino, the 5×5 diamond. Other common examples include gen 8 (4 crosses) and gen 10 (traffic lights in the "+" phase) of the same; gens 18, 20, 22, 24, 27 (= finished honeyfarm) of a bun; gen 9 of pi-heptomino.
• the smallest 1-typical pattern is 4×5, ie. the diamond minus one corner cell, which has a 3×3 parent.
• the smallest 2-typical pattern is the beehive (3×4 ← 3×2)
• the smallest 3-typical pattern is the V spark (3×3 ← 3×2)
• a 4-typical pattern either has a parent of the same size, or one that's wider but shorter. Vacuum is trivially 4-typical; the smallest non-zero such pattern is the blinker.
• from 5-typical on, patterns only have parents larger than themselves.
• a domino is 6-typical
• a dot is 8-typical
• 9+-typical examples are not obvious to come up with but I hypothesize they dominate at large pattern sizes
• more generally, I hypothesize that at least for the lower values, let's say x < y ≤ 8, if y-typical patterns of a given size exist, they are more numerous than x-typical ones
• there is probably a maximum typicality, around maybe 12-16?
We could also speak of the typicality of a particular pattern transition: the first steps of a B's evolution are 4, 2, 3, 3, 3, 2, 4, 2, 4, 4, 3, 4. Perhaps more illuminating though would be 4 – typicality (we could call this "growth rate" or something): this sequence then becomes 0, 2, 1, 1, 1, 2, 0, 2, 0, 0, 1, 0 and it is simple to see the pattern is growing.
A more finer-granulated measure of typicality might take all parents of a pattern into account, with smaller parents very strongly weighed compared to larger ones (after all, any given pattern has arbitrarily large parents) — but this seems difficult to calculate. Considering the bounding octagon rather than rectangle might also produce some further insight.