## 11. Cardioids

**Cardioids**

Excel file: 11. Cardioids

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 cardioids around the circle by changing the rule from (** j**, 2j) to (

**,**

*j****

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

**j****-1 gathering points.**

*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**.

There is no paper written at this point for this file.