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Simplicityand its obverse,elaboration, are crucibles for validating grids and designs visualized from them. The goal is always to strive for "Beauty without inconsistency or redundancy."

In mathematics, the whole is the sum of its parts. In art, it could be said that the whole isgreater. The most severely simpleunit of repeatcan be translated into astonishingly gorgeous and colorful patterns; however, the slightest inconsistency can mar the result.Learners seek to discover some principle (the

algorithm) that governs each grid's internal order. This may have to do with the number of points in any set

or the directions and number of line segments at each vertex.

Many ideas will unfold. Students will be motivated as they compare and discuss them to learn the vocabulary associated with their properties and attributes.## Sequence of Stages

Drawing lines in color helps to differentiate the sequence of stages, revealing its sequence of steps leading to the final version of a grid. Teachers should ask:"What makes my grid unique?"

"What is the underlying algorithm?"

Analysis of student work promotes learning and employing terminology in an authentic way. It provides a setting and foundation for the later, more formal stages of learning mathematics. It is inductive thinking leading to deductive logic at more rigorous levels.

With every new grid that is colored, inspiration is instilled to explore yet another. It can be addictive. After an initial free-style exploration, set a limit to the number of colors: two colors, then three, and four. This sets the stage for more formal logical and/or mathematical discussion of ideas.

Coloring engages both sides of the brain. Learners preview and compare alternative shape and color visualizations. The nuances and subtleties of each rendering highlight part-to-part and part-to-whole relationships, giving a holistic view that lines alone cannot demonstrate. Every investigation leads inevitably to analysis, and even to algorithmic generalizations.

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