10. Spirals

Archimedean Spirals

Excel file: 10. Spirals v3

Paper: 10.Bridges2021SpiralsWorkshop

This is a great file for thinking about fractions. This is an extension of the file that was the centerpiece of the paper “Using Archimedean Spirals to Explore Fractions” presented at a Bridges 2021 workshop in August, 2021. If you like math and art you should know about Bridges. (An Archimedean spiral is one where the radius expands or contracts a constant amount for a given angular change (like a watch spring). A true Archimedean spirals would be obtained if the points found were connected by a curve rather than a series of straight lines.)

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 r > 0, the radius is reduced by 1/r each time a point is created. A static sheet, Annotated r, is included in the Excel file for teaching purposes and as explainer 10.1a below. It shows these fractional reductions for r = 2, 3, 4, and 5 given a square parent polygon, n = 4.) By contrast, when r = 0, there is no radius reduction and polygons or stars result.

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 r jumps, is the center of the circle.

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 string art on spirals using subdivisions between vertices. Instead, the internal points which form the connecting points for the image are created from r equally spaced subdivisions of the line from a vertex to the center of the circle.

Explainers

1. Spiral basics: 10.1a Annotated radius reduction, r
   • Drawing the spiral: 10.1b Teaching Spirals with Dot Plots
   • How spirals curl inward: 10.1c Clockwise and Counterclockwise Spirals
2. Are the images polygons? 10.2a Almost polygons
   • 10.2b Are Almost Polygon Lines Parallel, Take 1
   • 10.2c Are Almost Polygons and Star Lines Parallel, Take 2
   • 10.2d None of the Almost Square Angles are Right Angles
3. Creating swirls from almost polygons:
4. Mirror images based on n–J:
5. An alternative mirror based on 2r:
6. Where are the two mirrors located?
7. When do the two mirrors coincide?
8. Flower power:
9. You Can Teach An Old Dog New Tricks: Zayla Introduces Teachers to The Twilight Zone

10. GLOSSARY

FOR USERS

Pencil and Ruler Exercises

10. Elementary version (four pages, 2 per page)

10. Varying n

10. Varying r

10. Varying J