There are multiple versions of file 4 and they are loaded up separately. 4.1 is the general version.

Excel file 1: 4.1_4 to 25_2Jump Polygons

Excel file 3: 4.3_4to25_Both Jumps by Equation

One-page handout explaining jump patterns: 4.Calculating Lines with Multiple Jumps

Porcupine-like images given jump sets: 4.Needle Fans and Fan Blades

Challenge questions: 4.Challenge Questions

To understand the underlying frame given by a repeated pairs of vertex jumps, if is worthwhile to set S = P = 1, then adjust the two jumps as well as n and see what happens. Note in this situation that, if n is even and one jump is 1 and the other is n/2 then a frame that looks like fractions of a pie result. In this instance, set P near but not at half of S and flower images result. All petals are visible when n is divisible by 4 but half are missing if n is divisible by 2 but not 4. (6 is the smallest n version with missing petals, the resulting 3 petals are oriented kind of like a radioactive materials sign. Since n = 6 you can think of a clock face with only even numbers showing. In this instance, the frame goes from 12 o’clock to 2 to 8 to 10 to 4 to 6 and back to 12 o’clock where the cycle is completed.) Such values of n may be written as n = 4k+2 where k is a whole number.

The second version focuses strictly on even-petaled flowers from 4 to 24. It is initially set with Jump 1 = 1 and with Jump 2 = n/2 but you can adjust Jump 1 via scroll arrows and Jump 2 by entering a number or an equation in place of its initial equation. There is a click box that allows you to fill in the other petals when n = 4k+2 by counting the last half of jumps counterclockwise rather than clockwise via a click box. (It is worth noting that when this box is checked and VCF = 2 (or a multiple of 2), the resulting image will appear as if it were VCF = 1 (or half as large), See this handout for further details.) Many images beyond flowers are possible in this setting by varying the parameters under control via scroll arrows. All images from 4.2 can be obtained using 4.1 except the images where n = 4k+2 and the counterclockwise click box is checked.

The third version returns to clockwise counting of jumps by adjusting the first jump to 2 when n = 4k+2, otherwise the first jump is 1. The second jump is set to be the closest whole number to n/2. These provide the ability to have 3 to 25 petal flowers just by changing n as long as P is near but not half of S. This version allows you to override the scroll arrows controlling S and P by putting equations in place of those numbers. For example, by putting the equation =B1*L1-1 in place of P in cell C1 you obtain maximally sharp star burst if n is even (especially when n = 4k) but more complex images when n is odd. The reason these are maximally sharp is that there are 2nS subdivision points in this instance (there are nS subdivision points but each point is counted twice, once as part of the first jump and the second time as part of the second jump). If P = nS  is put in cell C1, a single line results, but if P = nS-1 a 2nS image emerges (the same image as when P = nS+1).

There is no paper written at this point for these files.