Constructed Patterns

This challenge will be to construct patterns inscribed in a circle. All of the examples shown below can be constructed within circles using lines, arcs, and polygonal figures. The student challenge would be to do a construction either using standard compass and straightedge or with GSP. For example, the circular window problem has the design at the right.

The pattern is clearly built by arcs from three mutually tangent small circles within the large circle. So the first challenge is to determine where the centers of these three circles would be. There may be several alternatives.

One alternative would be to find a triangle for which one of the small circles would be an incircle:

In part this is because we have ways of contrucing circles tangent to lines but not tangent to circles except by taking the common tangent line at the point of tangency.

THE CHALLENGE

Here are 42 challenge patterns. Some of them are relative easy and some are quite demanding.

GSP presents an additional challenge to develop colors for the various regions. Without the limitations of GSP, coloring of regions in a ruler and compass construction is rather easy.

See PatternsM.htm.

Pictures from Clarke Middle School

The following three figures are from (1) the original pattern from the side of a purse, (2) the skeletal image of the structure using GSP, and (3) some coloring using PaintBucket software.

Pattern in 3D from PostIt notes:  (From Sonyita Ghosh Hajra).    Click on the picture to see in full size.

Simple Patterns

Pentagonal Patterns

Hexagonal Patterns

Octagonal Patterns

Miscellaneous Patterns

Challenge 2.

Create additional patterns that can be constructed with straightedge and compas or with GSP.