{"id":397,"date":"2021-05-21T14:35:47","date_gmt":"2021-05-21T14:35:47","guid":{"rendered":"https:\/\/blogs.dickinson.edu\/playing-with-polygons\/?page_id=397"},"modified":"2024-11-18T13:15:17","modified_gmt":"2024-11-18T13:15:17","slug":"11-cardioids","status":"publish","type":"page","link":"https:\/\/blogs.dickinson.edu\/playing-with-polygons\/11-cardioids\/","title":{"rendered":"11. Cardioids"},"content":{"rendered":"
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Cardioids<\/strong><\/p>\n

Excel files: Cardioids (adjust n and k manually)<\/a> \u00a0\u00a0\u00a0\u00a0 Cardioids (n as a linear function of k)<\/a> \u00a0\u00a0\u00a0\u00a0 Cardioids-with Loops<\/a><\/p>\n

A General Overview<\/strong>: A cardioid\u00a0 can be obtained by connecting vertices of the polygon (1, 2), (2, 4), (3, 6) \u2026 (j<\/em><\/strong>, 2*j) for j<\/em><\/strong> = 1, \u2026, n<\/em><\/strong>-1. Of course, many of the 2j<\/em><\/strong> second vertices will be larger than n<\/em><\/strong> in size \u2026 these points are simply the MOD n<\/strong><\/em> values of 2*j<\/em><\/strong> (MOD is the remainder upon division by n<\/em><\/strong>). For example, if n<\/em><\/strong> = 6 and j<\/em><\/strong> = 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).<\/p>\n

We can generalize the doubling rule to obtain multiple gathering points (or cusps) around the circle by changing the rule from (j<\/em><\/strong>, 2j<\/strong><\/em>) to (j<\/em><\/strong>, k<\/strong><\/em>*j<\/strong><\/em>). This will produce k<\/em><\/strong>-1 gathering points or cusps.<\/p>\n

The cardioid images are created much like File 9<\/strong>, 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<\/strong>. Interestingly, some of the relations between n<\/em><\/strong> and k<\/em><\/strong> produce images which are amenable to analysis using the counting rules discussed in PART II<\/strong>.<\/p>\n

P15. Cardioids
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  1. Cardioid basics: 11.1a Cardioids are all about cusps<\/a>\n