**CLOCK ARITHMETIC**

*Excel* file 5.1: 5.1_4Color_ClockArithmetic

*Excel* file 5.2: 5.2_4Color_ClockArithmetic_2Jumps

*Excel* file 5.3: 5.3_4Color_ClockArithmetic_3Jumps

**E17. ***Explainers*

D.1 Comparing 12_5-Stars across Jump Sets

D.2 A Line Analysis of a * J* = 0 Image

E. Venturing beyond Curved-tip Stars

**E20.*** Challenge Questions:*

4CCA VF ID 2, Comparing Double and Triple Jump Sets

If you’ve read the* explainers*, you need not read the introductory material below. However, I would like to suggest two additional string art points of discussion.

- File 12, Stacked Stars, deals with larger jump sets. There are
*explainers*there that not only deal with the vertex frame (no subdivisions) but also bring in string art with larger jump sets and form the basis for**E18**. - If you have gotten this far in the book you may well be ready to break out of the polygonal vertices we have used thus far in the book. You could certainly simply read the Bridges paper, but it seems more efficient to provide a few more
*explainers*that describe what changes once you expand your horizon beyond placing vertices at equal distances around the circle.

**E19. Non-Polygonal Models. ***Excel* file from **Bridges**, 2020: 4ColorAestheometry **Bridges** File

Graph paper with axes labelled on both sides (*Excel*): Blank Graph Page with numbered edges

3: Listing vertices twice in a row NP Creating Asymmetry

4 NP The Four-Color Model, Exploring* Inside the Box*

5 NP Moving Beyond Inside the Box

6 NP Creating Initials, *ESA* in Four-Color

**E20. **NP Challenge Questions

*Below is an introduction written when there were no explainers written.*

This material starts from the file written for a Bridges 2020 workshop paper but ends with a much simpler to use file for two reasons. First, the user need not worry about placing vertices in the coordinate plane. Second, this file is restricted to a 12-sided polygon, a dodecagon, to take advantage of the familiarity people have with time, and reading hours on a clock-face. We start at 12 o’clock and each jump is simply a number of hours forward from there. For example, the vertices in a 5 hour jump pattern would be:: 12 ⃕ 5 ⃕ 10 ⃕ 3 ⃕ 8 ⃕ 1 ⃕ 6 ⃕ 11 ⃕ 4 ⃕ 9 ⃕ 2 ⃕ 7 ⃕ 12.

All repeated jumps of less than 12 hours except 1, 5, 7, and 11 end up missing half or more vertices before returning to 12 o’clock and completing the cycle. Excel file 5.1 examines such single-jump patterns.

The next two files allow repeated jump patterns. It may help when initially setting up these patterns to only work with a single color (red is the base here) and to set ** S** =

**so that only the frame shows.**

*P*File 5.2 allows a repeated double-jump pattern, much like the double jump patterns examined in file 4, except that here they are more readily analyzable due to being able to tie them to the hours on a clock-face, and you can do them in four colors.

File 5.3 allows a repeated triple-jump pattern. This version also allows you to set a jump to zero … doing this creates a vertex without a curve (if * P* <

**). Set the jumps at (5, 6, 0) to see a 12 point star with each point bisected through the center of the circle but only half of the angles created will shows a curve. For example consider the 11 o’clock vertex: the angle from 5-11-4 is open but the angle from 5-11-6 shows a curve. This built in asymmetry is discussed in the Bridges 2020 paper on p. 553 in a section entitled:**

*S**A trick that Introduces Asymmetry*.