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Finally I would like to present another prime number pattern which I discovered some years ago. You make this constructions out of the residues you get when dividing **1** with a prime number. This division results in a series of decimals which repeats itself indefinitely - but you also get a series of residues. If you for instance divide **1** with the prime number **7** you get the decimals **142857** and the residues **132645**. If you draw a circle, mark the circumference from *1* to *6* and connect the different points in the residue order you get a symmetric pattern. Here you can see the result for the prime numbers 7, 19, 29 and 61.

Now, the real interesting thing begins when you reach higher numbers. Then a beuatiful pattern appears with nine arched curves (nine, because you work in the decimal system), like a mandela or a church window. How can the curves be formed from these straight lines, and what is creating this pattern? Here you can take a look at the circle patterns from the prime numbers **181** and **709**.

It seems like the nine curves have got the form of a hypercycloid, which is a cycloid whose circle travels around the outside of another circle centered at the origin. But of course you have to investigate this more carefully to be shure. **Click here** to see an illustration.

Finnaly we shall see what happens when you try to get a circle pattern from a prime number where you have two series of residues. **359** is such a number, and to get a symmetric pattern you have to convert the numeric values in the two series to order values.

The first residue series begins with:* 1 10 100 282 307 198 185 55 191 115...*

The converted series instead becomes:* 1 9 62 151 165 111 104 39 107 69...*

The two series consist of 179 residues each and are of course put into the same circle, divided into 178 parts. And here you have the result. Does this picture remind you of anything? **Take a look here** and be amused.

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© *Gunnar Appelqvist* 2009

Latest update 2009-01-29