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 2^aF_aF_b...F_s, where the F_n are distinct primes of the form


known as Fermat primes. Constructions for the regular triangle (3^1), square (2^2), pentagon (2^(2^1)+1), hexagon (2^13^1), etc., had been given by Euclid, but constructions based on the Fermat primes >=17 were unknown to the ancients. The first explicit construction of a heptadecagon was given by Erchinger in about 1800.

The trigonometric functions cos(pi/17) and cos(2pi/17) are both algebraic numbers of degree 8 given respectively by


The heptadecagon with unit edge lengths has inradius and circumradius given by


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


The area of the regular heptadecagon is given by


where A can be expressed as the largest root of


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 O, draw a circle centered on O and a diameter drawn through O.

2. Call the right end of the diameter dividing the circle into a semicircle P_1.

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

4. Construct J a quarter of the way up OB.

5. Join JP_1 and find E so that ∠OJE is a quarter of ∠OJP_1.

6. Find F so that ∠EJF is 45 degrees.

7. Construct the semicircle with diameter FP_1.

8. This semicircle cuts OB at K.

9. Draw a semicircle with center E and radius EK.

10. This cuts the line segment OP_1 at N_4.

11. Construct a line perpendicular to OP_1 through N_4.

12. This line meets the original semicircle at P_4.

13. You now have points P_1 and P_4 of a heptadecagon.

14. Use P_1 and P_4 to get the remaining 15 points of the heptadecagon around the original circle by constructing P_1, P_4, P_7, P_(10), P_(13), P_(16) [filled circles], P_2, P_5, P_8, P_(11), P_(14), P_(17) [single-ringed filled circles], P_3, P_6, P_9, P_(12), and P_(15) [double-ringed filled circles].

15. Connect the adjacent points P_i for i=1 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 8S_1+4S_2+22C_1+11C_3 and simplicity 45. The construction problem has now been automated to some extent (Bishop 1978).

See also

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

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