# Geometry Through Art

### Inscribed Polygon Grids, Part I

#### What Children Can Learn About Art and Geometry

A simple straight edge and a percentage circle offer children access to profound and fundamental learnings in both art and geometry. Making grids from scratch can exercise children's powers of visualization: the abilities developed through interpreting configurations for their 'hidden' geometric shapes, patterns, symmetries, and other attributes.

When I am teaching, these constructions grow out of children making chords. Counting clockwise by 25 they construct a square; counting by 20, a pentagon; by 30, a 20-pointed star. They learn that an inscribed polygon is made of chords, that a regular polygon is the result of counting correctly, that the chords and the inscribed angles are congruent. They know these things because they have MADE them from scratch!

Let's begin our grid with a set of inscribed polygons. First, a pair of squares at eight points:

Here we should be perceiving at least 8 triangles, 2 squares, 1 eight-pointed star, and 1 octagon, all inside 1 circle:

Below, you will see more sets of polygons. There are 2 octagons, not 1, and more sets of different triangles, some congruent, some similar. All rotate around the center of the circle.

When I present polygons, inscribed polygons, or the diameter of a circle, students (children or teachers) seem to know what these are. We don't go into formal definitions, especially in the context of my introduction, where I talk about two distinct and contrasting kinds of patterns: regular and random.

Regular patterns have motifs that are 'units of repeat'. Though the units themselves are the same, we can vary the pattern by counting them with different numbers, i.e., repetitions: (1212121) (112112112) (112221122211222).

Random patterns are like camouflage: there is not only a specific or single motif, but the viewer perceives an overall similarity of shapes and colors which, like the transcendental-numbers-after-the-decimal-sign, cannot be predicted. The motif (unit of repeat) and the pattern (the numbered intervals) are 'random'.

Highlighting the polygons with lines in colors helps children to visualize, to 'seek and find' . Given the opportunity to draw this grid from the initial 8 points on the circumference, using a straight edge, learners discover elemental geometric properties as did ancient geometers long ago. They begin to experience the quintessential beauty of geometry.

Inscribed Grids, continued