 TOPICS  The regular polygon of 17 sides is called the heptadecagon, or sometimes the heptakaidecagon. Gauss proved in 1796 (when he was 19 years old) that the heptadecagon is constructible with a compass and straightedge. Gauss's proof appears in his monumental work Disquisitiones Arithmeticae. The proof relies on the property of irreducible polynomial equations that roots composed of a finite number of square root extractions only exist when the order of the equation is a product of the form , where the are distinct primes of the form (1)

known as Fermat primes. Constructions for the regular triangle ( ), square ( ), pentagon ( ), hexagon ( ), etc., had been given by Euclid, but constructions based on the Fermat primes were unknown to the ancients. The first explicit construction of a heptadecagon was given by Erchinger in about 1800.

The trigonometric functions and are both algebraic numbers of degree 8 given respectively by   (2)   (3)   (4)   (5)

both of which can be expressed in terms of finite root extractions. They can also be expressed as the largest roots of the algebraic equations (6) (7)

The area of the regular heptadecagon is given by (8)

where can be expressed as the largest root of (9) The following elegant construction for the heptadecagon (Yates 1949, Coxeter 1969, Stewart 1977, Wells 1991) was first given by Richmond (1893).

1. Given an arbitrary point , draw a circle centered on and a diameter drawn through .

2. Call the right end of the diameter dividing the circle into a semicircle .

3. Construct the diameter perpendicular to the original diameter by finding the perpendicular bisector .

4. Construct a quarter of the way up .

5. Join and find so that is a quarter of .

6. Find so that is .

7. Construct the semicircle with diameter .

8. This semicircle cuts at .

9. Draw a semicircle with center and radius .

10. This cuts the line segment at .

11. Construct a line perpendicular to through .

12. This line meets the original semicircle at .

13. You now have points and of a heptadecagon.

14. Use and to get the remaining 15 points of the heptadecagon around the original circle by constructing , , , , , [filled circles], , , , , , [single-ringed filled circles], , , , , and [double-ringed filled circles].

15. Connect the adjacent points for to 17, forming the heptadecagon.

This construction, when suitably streamlined, has simplicity 53. The construction of Smith (1920) has a greater simplicity of 58. Another construction due to Tietze (1965) and reproduced in Hall (1970) has a simplicity of 50. However, neither Tietze (1965) nor Hall (1970) provides a proof that this construction is correct. Both Richmond's and Tietze's constructions require extensive calculations to prove their validity. DeTemple (1991) gives an elegant construction involving the Carlyle circles which has geometrography symbol and simplicity 45. The construction problem has now been automated to some extent (Bishop 1978).

257-gon, 65537-gon, Compass, Constructible Polygon, Fermat Number, Fermat Prime, Regular Polygon, Straightedge, Trigonometry Angles, Trigonometry Angles--pi/17

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## References

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