# Patterns and Factoring

## 1. Patterns and Factoring

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.

## 2. Point Figures

Patterns have a center point. Patterns havepoints at salient features, such as corners, borders, and contrasts.

## 3. Relative Point Distances

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.

## 4. Features

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.

## 5. Pattern Representation

A unique pattern representation can use apattern center, vectors, and pairwise relations between points. The first twopoints have distance one unit.

### 5.1. Pattern Center

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.

### 5.2. Vectors

Vectors go from pattern center to patternpoints. Pattern points have 0 for fourth dimension (x,y,z,0). Different pointshave different vectors.

### 5.3. Vector Pairs

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.

### 5.4. Multiply Factors

After finding cross product for all vectorpairs, multiply all cross products. Same-shape patterns have the same uniquepattern representation.

### 5.5. Colors or Point Types

Points can have different colors and/or types byadding a fifth dimension (x,y,z,0,n).

## 6. Patterns and Factors

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.

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Copyright © 2011 John Franklin Moore. All rights reserved.