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## More Stars, Honeycombs, and Snowflakes

### Skinny Pentagrams and Fat Pentagrams

So far, in our study of pentagrams—that is, five-pointed stars—we have considered only the quite familiar pentagram that appears on the flag of the United States and in so many other places. We shall say that the five points that appear on this particular pentagram are normal or “regular” in shape.

In contrast, our next step is to examine the geometry of pentagrams with points that range in shape from “long and skinny” to “short and fat.” Let figure 2.1 serve as our definition sketch.

The main geometrical variable in our problem is the ratio r/R, in which R is the radius of the external circumscribing circle and r is the radius of the internal circumscribing circle. ...

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