Point arrangements can represent patterns. Forexample, a square has points at four corners. A cube has points at eightcorners. A circle has a center and evenly spaced points around circumference. Asphere has a center and evenly spaced points around surface. A leaf has pointsat stem beginning and end, leaf tips and indentations, and along veins.
Patterns have a center point. Patterns havepoints at salient features, such as corners, borders, and contrasts.
Pattern points have relative distances. A cubediffers from a rectangle by having points with different relative distances.However, patterns differing only in size are the same pattern, because relativedistances are the same for patterns of the same shape.
Point combinations and their relative distancesmake features. Two points make a line segment. Three points make a triangle.Four points make a square, diamond, rectangle, T, or pyramid.
A unique pattern representation can use apattern center, vectors, and pairwise relations between points. The first twopoints have distance one unit.
Using Cartesian coordinates, pattern center isat three-dimensional coordinate origin and one distance unit along fourthdimension (0,0,0,1). The extra dimension avoids false equivalences that canhappen if pattern center lies near a point.
Vectors go from pattern center to patternpoints. Pattern points have 0 for fourth dimension (x,y,z,0). Different pointshave different vectors.
To represent relation between two points, fromtheir two vectors, calculate their vector cross product and square to use onlypositive numbers. Different pairs have unique cross products.
After finding cross product for all vectorpairs, multiply all cross products. Same-shape patterns have the same uniquepattern representation.
Points can have different colors and/or types byadding a fifth dimension (x,y,z,0,n).
Patterns are multiples of pattern features.Removing a point factors out its pairwise factors and leaves the remainingpattern features. Adding a point adds pairwise features and keeps previousfeatures.
Pairwise features already account for alltriple-point, quadruple-point, and higher features.
Pattern translation, reflection, rotation, andinversion have same pattern representation, so orientation and position do notmatter.
Copyright © 2011 John Franklin Moore. All rights reserved.