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Heptadecagon

Heptadecagon

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

F_n=2^(2^n)+1,

(1)

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

1-544r^2+38080r^4-792064r^6+6223360r^8-19914752r^(10)+25346048r^(12)-11141120r^(14)+1114112r^(16)

(6)

1-17R^2+119R^4-442R^6+935R^8-1122R^(10)+714R^(12)-204R^(14)+17R^(16).

(7)

The area of the regular heptadecagon is given by

A=(17)/4cot(pi/(17)),

(8)

where A can be expressed as the largest root of

2862423051509815793-21552361799603318912A^2+20881180982314634240A^4-6011468019822067712A^6+653743432704327680A^8-28954726431195136A^(10)+510054948143104A^(12)-3103113871360A^(14)+4294967296A^(16).

(9)

17-gonConstruction

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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References

Archibald, R. C. "The History of the Construction of the Regular Polygon of Seventeen Sides." Bull. Amer. Math. Soc. 22, 239-246, 1916.Archibald, R. C. "Gauss and the Regular Polygon of Seventeen Sides." Amer. Math. Monthly 27, 323-326, 1920.Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 95-96, 1987.Bishop, W. "How to Construct a Regular Polygon." Amer. Math. Monthly 85, 186-188, 1978.Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 63-69, 1982.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 201 and 229-230, 1996.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 26-28, 1969.DeTemple, D. W. "Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions." Amer. Math. Monthly 98, 97-108, 1991.Dickson, L. E. "Construction of the Regular Polygon of 17 Sides." §8.20 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 372-373, 1955.Dixon, R. "Gauss Extends Euclid." §1.4 in Mathographics. New York: Dover, pp. 52-54, 1991.Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998.Gauss, C. F. §365 and 366 in Disquisitiones Arithmeticae. Leipzig, Germany: Fleischer, 1801. Reprinted in New Haven, CT: Yale University Press, 1965.Hall, T. Carl Friedrich Gauss: A Biography. Cambridge, MA: MIT Press, 1970.Hardy, G. H. and Wright, E. M. "Construction of the Regular Polygon of 17 Sides." §5.8 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 57-62, 1979.Klein, F. Famous Problems of Elementary Geometry and Other Monographs. New York: Chelsea, 1956.Ore, Ø. Number Theory and Its History. New York: Dover, 1988.Rademacher, H. Lectures on Elementary Number Theory. New York: Blaisdell, 1964.Richmond, H. W. "A Construction for a Regular Polygon of Seventeen Sides." Quart. J. Pure Appl. Math. 26, 206-207, 1893.Smith, L. L. "A Construction of the Regular Polygon of Seventeen Sides." Amer. Math. Monthly 27, 322-323, 1920.Stewart, I. "Gauss." Sci. Amer. 237, 122-131, 1977.Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Modern Times. New York: Graylock Press, 1965.Trott, M. "cos(2pi/257) à la Gauss." Mathematica Educ. Res. 4, 31-36, 1995.Trott, M. "cos(2pi/257) à la Gauss." §1.10.2 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, pp. 312-321, 2006. http://www.mathematicaguidebooks.org/.Vélez, P. and Luis, O. "A Chord Approach for an Alternative Ruler and Compasses Construction of the 17-Side Regular Polygon." Geom. Dedicata 52, 209-213, 1994.Update a linkWang, P. "The Regular Heptadecagon." http://www.ugcs.caltech.edu/~peterw/studies/17gon/Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin, pp. 212-213, 1991.Yates, R. C. Geometrical Tools: A Mathematical Sketch and Model Book. St. Louis, MO: Educational Publishers, 1949.

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Heptadecagon

Cite this as:

Weisstein, Eric W. "Heptadecagon." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Heptadecagon.html

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