**Cardioids**

Excel files: Cardioids (adjust n and k manually) Cardioids (n as a linear function of k) Cardioids-with Loops

**A General Overview**: A cardioid can be obtained by connecting vertices of the polygon (1, 2), (2, 4), (3, 6) … (** j**, 2*j) for

**= 1, …,**

*j***-1. Of course, many of the 2**

*n***second vertices will be larger than**

*j***in size … these points are simply the MOD**

*n**values of 2**

**n****(MOD is the remainder upon division by**

*j***). For example, if**

*n***= 6 and**

*n***= 4 then the rule is to connect point 4 to 8, but 8 is the same as 2 because 2 = MOD(8, 6). Note that this is a second time to have drawn that particular line since (2, 4) was already drawn. Similarly, (5, 10) becomes (5, 4) because 4 = MOD(10, 6).**

*j*We can generalize the doubling rule to obtain multiple gathering points (or cusps) around the circle by changing the rule from (** j**, 2

*) to (*

**j****,**

*j****

**k***). This will produce*

**j****-1 gathering points or cusps.**

*k*The cardioid images are created much like **File 9**, which creates general similar triangles using parallel lines, in that each of the lines is set independently from others, rather than as a single closed circuit as was the case in** PART I**. Interestingly, some of the relations between ** n** and

**produce images which are amenable to analysis using the counting rules discussed in**

*k***PART II**.

*Explainers*:

- Cardioid basics: 11.1a Cardioids are all about cusps
- Take 2: 11.1b Lines in an Image
- Take 3: 11.1c Three Special Cases
- Take 4: 11.1d It helps to start at vertex 1

- Symmetry 11.2a Vertical Symmetry
- Take 2: 11.2b Rotational Symmetry

- Measuring angles, Take 1: 11.3a Angles in Cardioids
- Take 2: 11.3b Right Angles

- Why the image appears curved 11.4 How the Image Curves
- Adjacent interior angles 11.5a Adjacent Interior Angles
- Implied exterior angles: 11.5b Implied Exterior Angles

- Vertex Loops: 11.6a Vertex Loops
- Take 2: 11.6b Identity and Paired Vertices
- Take 3: 11.6c Ribbons and Strips
- Take 4a: 11.6_4a Ribbons everywhere given
= 361,*n*= 18*k*- MA. 4bi: 11.6_4bi Deconstructing Loops using
*Excel*- MA. 4bii:
*Excel*file: 11.6_4bii 361_18 Loop Explainer

- MA. 4bii:

- MA. 4bi: 11.6_4bi Deconstructing Loops using

- The
=**n**single Circle Fan: 11.7a Vertices as Directions*k*- Multiple Circle Fans: 11.7b Circle Fans show Remainders

- MA. Searching for patterns using equations in Excel: 118a Examining
as a function of*n*,**k**(**n**)**k** - MA. Deconstructing an image: 11.9 Deconstructing a Clock-face
- Some interesting images

**EXTRA MATERIALS**

*FOR USERS
*

Challenge Questions

11.Cardioid Counting Questions

11.Circle Fans and Cusps (no hints)

*FOR INSTRUCTORS*