Mathematical Topics Encountered in Playing with Polygons

This mathematical appendix is provided up-front rather than at-the-end because many of the mathematical topics encountered occur in a number of venues and I wanted to make sure that these are a bit more visible than if I had placed them all at the end of the book. These topics are discussed via explainers because, like other parts of PwP, the underlying mathematics need not be understood to enjoy playing with the files. Like elsewhere within PwP, those marked MA (Mathematical Approach) may not be readily approachable for those in primary grades.

Note: In PwP, the identifier E# identifies this material as being in Chapter # of ESA. Material that was created after ESA are at the end in their own section, even when they amplify something from an earlier section of ESA.

E21. Basic properties of numbers. Because of the continuously-drawn nature of the images in many parts of PwP, the images “collapse” when there is commonality between various factors in the models presented here. Therefore it is worth reminding you about primes, composites, and relatively prime numbers.

  1. Prime and composite numbers: Prime and Composite
    1. Finding primes: Primes and the Sieve of Eratosthenes
      1. MA. Eratosthenes Sieve in Excel: Finding Primes
  2. Comparing numbers: Commonality between Numbers
    1. Relatively prime numbers: Relatively Prime
    2. MA. Finding the GCD (Greatest Common Divisor) without knowing factors: Euclid’s Algorithm
      1. MA. Euclid’s Algorithm in Excel
  3. Using the Difference between Squares formula: 21.3. A Times Table Number Pattern

E22. Angles in polygons and stars. The regularity of the polygonal vertices allows us to say lots of things about the angles created in images especially when those images are NOT based on subdivisions so this is of greatest interest in the Introduction, and Parts II and III of PwP. Almost all angular discussion leans heavily on a theorem from geometry called the Inscribed Angle Theorem which says that the measure of an inscribed angle is half the size of the central angle (arc of the circle). The starting point for this discussion is a pair of explainers in File 1, angles in polygons and stars and the other poses a couple of questions about those angles. A more detailed discussion of inscribed and interior angles occurs in File 11 which also includes a discussion of implied exterior angles. File 11 has a number of challenge questions dealing with angular issues of various types. Right angles are explored in a variety of places including the Brunes star challenge question in File 2, all of File 7,  and in the discussion of right angles in cardioids in File 11.

  1.  About Inscribed Angles
    1. Why the inscribed Angle Theorem Works
  2. Interior Angles and Parallel Lines
  3. About Implied Exterior Angles

Modular arithmetic. These images are based on modular arithmetic and that is not something that is systematically taught in K-12. Nonetheless, you know something about it because you know clock-arithmetic. Modular arithmetic focuses attention on remainders. In terms of clocks, we all know that if it is now 11 o’clock and you want to know what time it is 3 hours from now, it is 2 o’clock not 14 o’clock (unless you are in the military!). In modular terms,   2 = 14 MOD 12.

E23. Modular basics

    1. Introduction To MOD
    2. MA. Properties of modular arithmetic MOD Take 2
    3. MA. Counting Backwards (MOD take 3)

E24. Modular Multiplicative Inverses

    1. Introduction to Modular Multiplicative Inverses, MMI
    2. MA. MMI and Negative MMI
    3. MA. Backtracking Euclid to find the modular multiplicative inverse
    4. MA. Finding MMI from Euclid Excel file

Using Excel to explore mathematical relations. The files that form the backbone of PwP are Excel files and you have to have Excel installed on your computer to use the files. But readers need not have experience using Excel to enjoy playing with the files since all you need to do is click arrows up and down to manipulate the image. (And, of course, the web-version does not use Excel at all.)

Those wishing to explore the mathematics beneath the images will find Excel to be a powerful tool in that exploration. This type of Excel exploration is modeled in various helping files for explainers such as the Finding Primes, Euclid’s Algorithm, and Finding MMI from Euclid Excel files referenced above. For those wanting to learn a bit more about using Excel, there are better starting points such as the Creating Angles document and Excel file in File 1, the tracking lines in the first cycle and the level jump pattern Excel files in File 2, or the deconstructing loops with Excel document and Excel file in File 11.

E25. Using the Web version of String Art

1 Introduction to the Web Version

2 Changing Numbers in the Web Version

3 Web Jumps

4 Drawing Mode

5.1. For Younger Users I Changing n, S, and P, and Visualizing SCF

5.2. For Younger Users II, Changing J and Visualizing VCF

5.3. For Younger Users III, Drawing Modes, Simplifying Values, and an Exercise

POST ESA Mathematical Materials.