Star polygons are constructed by connecting evenly-spaced dots on a circle. The segments formed must all be the same length. If the segments do not cross each other, the figure formed is a regular polygon. If the segments cross each other, the polygon formed is a star polygon.
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A 5-pointed star can be formed by connecting every second dot on a circle with 5
evenly-spaced dots, as is shown at the right. The code for this star polygon is
given below the diagram. If every dot is connected in order, the resulting
figure is a regular pentagon. If every third dot were connected, the result
would be the same as if every second dot were connected. There are only two
figures that can be shown on a 5-dot circle: the 5-point star and a regular
pentagon.
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Use the 7-dot circles below to find the two different star polygons and write the code below each.
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Use the 12-dot circle below to find the 12-point star polygon and write its code next to it.
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How many regular polygons can be drawn on a circle with n evenly-spaced dots? Fill in the information in the table below.
| Dots | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| Regular Polygons | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ |
| Dots | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| Regular Polygons | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ |
To find a pattern it is necessary to examine the prime factorization of the number of dots on the circle. This can be found by a branching process. Each number n can be factored into a unique product of primes and may be expressed in the form:
The number of divisors of n, d(n), can then be found by the expression below.
The formula for the number of regular polygons that can be drawn on a circle with n evenly-spaced dots will include the value d(n).
Write the formula: _________________
How many star polygons can be drawn on a circle with n evenly-spaced dots? Fill in the information in the table below.
| Dots | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| Star Polygons | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ |
| Dots | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| Star Polygons | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ |
The formula for the number of star polygons that can be drawn on a circle with n evenly-spaced dots will also include d(n).
Write the formula: _________________
How many star polygons with n points are there? Fill in the table below.
| Dots | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| Star Polygons | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ |
| Dots | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| Star Polygons | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ | _____ |
In order to find a formula for the number of n-point stars, it is necessary to use the Eüler Phi Function to find the number of numbers that have no factors in common with n. The Eüler Phi Function is:
where n is written as a product of powers of primes and each pi is one of the primes.
Write the formula for the number of n-point stars using the Eüler Phi Function. _________________
Determine the measure of the angle at each point of an n-point star. The inscribed angle theorem will be of use.
An angle inscribed in a circle has a measure that is one-half of the arc that is cut off by the angle.
Find the angle measure at a point for each star polygon listed below.
What is the angle at the point of a star that is formed by connecting every pth point on a circle with n evenly-spaced points?_______________
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