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 j = 1, …, n-1. Of course, many of the 2j second vertices will be larger than n in size … these points are simply the MOD n values of 2*j (MOD is the remainder upon division by n). For example, if n = 6 and j = 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).

We can generalize the doubling rule to obtain multiple gathering points (or cusps) around the circle by changing the rule from (j, 2j) to (j, k*j). This will produce k-1 gathering points or cusps.

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 k produce images which are amenable to analysis using the counting rules discussed in PART II.


  1. Cardioid basics: 11.1a Cardioids are all about cusps
  2. Symmetry 11.2a Vertical Symmetry
  3. Measuring angles, Take 1: 11.3a Angles in Cardioids
  4. Why the image appears curved 11.4 How the Image Curves
  5. Adjacent interior angles 11.5a Adjacent Interior Angles
  6. Vertex Loops: 11.6a Vertex Loops
  7. The n = k single Circle Fan: 11.7a Vertices as Directions
  8. MA. Searching for patterns using equations in Excel: 118a Examining n as a function of k, n(k)
  9. MA. Deconstructing an image: 11.9 Deconstructing a Clock-face
  10. Some interesting images