Patterns {compact pattern} can have only points, which connect horizontally or vertically to at least one other point.
value
Patterns have numerical values.
boundary
Unique patterns have unique boundaries and surfaces. If surface has value, pattern has value, and vice versa.
differences
Changing one pattern to another adds or subtracts one point.
number
Same number of points has same number of patterns. 1 point has 1 possible pattern. 2 points have 1 possible pattern. 3 points have 2 possible patterns. 4 points have 5 possible patterns. 5 points have 12 possible patterns. 6 points have 35 possible patterns. 7 points have 108 possible patterns. 8 points have 369 possible patterns. 9 points have 1285 possible patterns. 10 points have 4655 possible patterns. 11 points have 17072 possible patterns. 12 points have 63565 possible patterns. 13 points have 238299 possible patterns.
number: multiples
Multiple for each step is 1, 2.00, 2.50, 2.40, 2.92, 3.09, 3.41, 3.48, 3.62, 3.67, 3.72, and 3.75. Multiple for odd steps is 2, 6, 9, 11.89, 13.29, and 13.96. Multiple for even steps is 5, 7, 10.54, 12.61, and 13.66.
factoring
Pattern-number factors are 1*1, 2*1, 5*1, 3*2*2, 7*5, 3*3*3*2*2, 41*3*3, 257*5, 19*7*7*5, 97*11*2*2*2*2, 12713*5, and 79433*3.
point types
Patterns can have different point types, such as colors. For three point types, 1 point has 3 possible patterns. 2 points have 6 possible pattern. 3 points have 36 possible patterns. 4 points have 246 possible patterns. 5 points have 2115 possible patterns. Multiple for each step is 2.00, 6.00, 6.83, and 8.60. Multiple for odd steps is 12 or 58.75. Multiple for even steps is 41. Pattern-number factors are 3*1, 3*2, 3*2*2, 41*3*2, and 47*5*3*3.
transformations
Pattern translation, reflection, rotation, and inversion make same pattern.
diagonals
If points can connect diagonally, patterns are not fundamentally different, only less compact. Patterns with points connected diagonally can transform to connect only horizontally or vertically. For example, V is L rotated 45 degrees.
To be unique, pattern representations {unique pattern representation} must use pattern center and pairwise relations between points.
center
The center is x, y, and z coordinate means plus unit distance along higher dimension. The extra dimension avoids false equivalences that can happen if center lies near point.
vectors
Vectors go from center to pattern points. For vector pairs, find something like cross product. Calculate each pair only once, not again for different order. Ignore unit vectors. Square difference, such as x1*y2 - x2*y1, assigned to unit vectors. Sum squares. Alternatively, use sum square root.
value
Add all cross products. Resulting sum is unique for compact pattern.
program 1
In array, in first coordinate, 0 is for y-coordinate, and 1 is for x-coordinate. k = pattern size. p = pattern-point number. max(p) = k. m = pattern number. n(#, k, p, m) are point coordinates. v = 0. v1 = 0. v2 = 0.
For p = 1 To k + 1. v = v + n(0, k, p, m). v1 = v1 + n(1, k, p, m). Next p. v = v / (k + 1). v1 = v1 / (k + 1). For p = 1 To k. For p1 = p + 1 To k + 1. x1 = n(0, k, p, m) - v. x2 = n(1, k, p, m) - v1. x3 = 1. y1 = n(0, k, p1, m) - v. y2 = n(1, k, p1, m) - v1. y3 = 1. w2 = ((x2 * y3 - y2 * x3) ^ 2 + (x1 * y3 - y1 * x3) ^ 2 + (x1 * y2 - y1 * x2) ^ 2) ^ 0.5. v2 = v2 + w2. Next p1. Next p.
Program compares patterns rapidly.
eye
Eye can perform this computation, because it is pattern center, and all points are in front of it. Eye can compare patterns and judge distances.
program 2
Patterns with different colors or point types can have pattern representations. In first coordinate, 0 is for y-coordinate, 1 is for x-coordinate, and 2 is for color or point type. k = pattern size. p = pattern point number; max(p) = k. m = pattern number. n(#, k, p, m) are coordinates and point type. v = 0. v1 = 0. v2 = 0.
For p = 1 To k + 1. v = v + n(0, k, p, m). v1 = v1 + n(1, k, p, m). Next p. v = v / (k + 1). v1 = v1 / (k + 1). For p = 1 To k. For p1 = p + 1 To k + 1. x1 = n(0, k, p, m) - v. x2 = n(1, k, p, m) - v1. x3 = 1. y1 = n(0, k, p1, m) - v. y2 = n(1, k, p1, m) - v1. y3 = 1. w2 = ((n(2, k, p, m) + 1) * (n(2, k, p1, m) + 1) + (x2 * y3 - y2 * x3) ^ 2 + (x1 * y3 - y1 * x3) ^ 2 + (x1 * y2 - y1 * x2) ^ 2) ^ 0.5. v2 = v2 + w2. Next p1. Next p.
Sets can have symbols in sequence {order group}, such as pattern or k-tuple.
subsets
The set can have subsets. Subsets are symbol sets in sequences and patterns.
group
All subsets form order groups. For example, pattern "acg" has subsets NULL, "a", "c", "g", "ac", "cg", and "acg". Order groups contain null set and pattern. If sets can be circular, the set can have subsets "ga", "cga", and "gac".
equivalence
Rules can be that patterns are equivalent over gaps and insertions, so "acg" = "ac gX". Gaps or insertion size or number can have restrictions.
alignment
Two patterns share largest subset. Two patterns share two largest equivalent subsets at optimum alignment.
process
To compare patterns, add or remove gaps and insertions from both patterns to find largest subset. If symbols are dimensions, spaces have maximum number of shared dimensions and minimum number of new dimensions.
index
Indexes are part of, and have position in, patterns. Pattern symbols have one or more indices. In pattern "acgta", symbol "a" is at position 1 and 5. Pattern subsets start at one or more indices. In pattern "acgta", subset "ac" starts at position 1.
union
Combining patterns results in new symbol sequences and patterns. Start with first pattern and add new symbols in sequence. Discard symbols that are the same in sequence, for example, "abc" and "ag" nets "abcg". It is like union of sets but with order in elements.
Combining is associative but not commutative. Null pattern combines with pattern to give same pattern. The same pattern combines with itself to give same pattern. Inverse pattern combines with pattern to give null pattern, but there can be no inverse pattern.
intersection
Finding largest aligned subset is like set intersection. Aligning "abc" and "ag" nets "a".
Aligning is associative and commutative. Null pattern aligns with pattern to give null pattern. The same pattern aligns with itself to give same pattern. Inverse pattern combines with pattern to give null pattern.
conversion
Natural or artificial objects, events, lines, surfaces, solids, n-dimensional figures, geometric points, figures, or images can be linear single-symbol series and so can be patterns. Patterns have order groups, and so all things can align. For example, letter "a" can stand for angle of 45 degrees and letter "L" can stand for angle of 90 degrees, so pattern "aLa" can stand for right triangle with two 45-degree angles.
Symbol sequences can transform into symbol group sequences. For example, pattern "acgta" has three-symbol subsets, "acg", "cgt", and "gta", rather than a five-symbol sequence. Subsets can align rather than single symbols.
Two objects or events can transform into linear RNA-base sequences. They can align by hybridization.
length vs. symbol number
Things can use patterns with few symbols and long sequences or shorter sequences with more symbols.
brain
Perhaps, brain can compare patterns using order groups.
3-Computer Science-System Analysis
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Date Modified: 2022.0225