## File 6

Excel file: 6.Spirals

Paper: 6.SpiralsPaper

This is a great file for thinking about fractions.

This comes right after File 1 in terms of difficulty because a simple addition to that file, *the radius reduction factor r*, produces spirals. Specifically, when

**> 0, the radius is reduced by 1/**

*r***each time a point is created. (A static sheet,**

*r**Annotated r*, is included for teaching purposes. It shows these fractional reductions for

**= 2, 3, 4, and 5 given a square parent polygon,**

*r***= 4.) By contrast, when**

*n***= 0, there is no radius reduction and polygons or stars result.**

*r*Unlike Files 1 through 5, these images are not closed circuits when ** r** > 0. The starting point (at the top of the parent polygon) is not connected to the ending point which, after

**jumps, is the center of the circle.**

*r*If teaching from **PwP**, it may be useful to cover this right after File 1. I have not placed it there because I do not want users to think that we will be doing aestheometry on spirals. As a result, I have placed it at the end of Part I.