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Re: Consturction of 17-sided regular polygon
Posted:
Jan 31, 1999 6:37 PM
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Peter Hung wrote:
>I have trying to find out the procedure for constructing a regular 17-sided
>polygon. I know it was Gauss who discovered the method and is based on the
>formula
>
> 2^2^k + 1
>
>I have tried to find it from the WWW but I was unable to find anything.
^^^^^^^^^^^^^^^^^^^^^
Do you know why?
Because you were not aware of the polygon's name (to use it as keyword
in your searches).
John H. Conway has already named it: HEPTAKAIDECAGON (many authors, however,
use the type HEPTADECAGON, which is incorrect)
>
>Anyone cares to to show me or recommend any reading materials, books or
>online WWW.
Well... a web page on HEPTAKAIDECAGON is:
http://www.astro.virginia.edu/~eww6n/math/Heptakaidecagon.html
the author refers to HEPTADECAGON page:
http://www.astro.virginia.edu/~eww6n/math/Heptadecagon.html
I quote the last paragraph and the References:
<quote>
This construction, when suitably streamlined, has Simplicity 53. The
Richmond's (1893)
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. De
Temple (1991) gives an elegant construction involving the Carlyle Circles
which has Geometrography symbol 8S1+4S2+22C1+11C3 and Simplicity 45.
The construction problem has now been automated to some extent (Bishop
1978).
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.
De Temple, 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."
paragraph 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."
paragraph 1.4 in Mathographics. New York: Dover, pp. 52-54, 1991.
Gauss, C. F. paragraph365 and 366 in Disquisitiones Arithmeticae.
Leipzig, Germany, 1801. 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." paragraph 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, O.. 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. New York: Graylock Press, 1965.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry.
New York: Penguin, pp. 212-213, 1991.
Yates, R. C. Geometrical Tools. St. Louis, MO: Educational Publishers, 1949.
</quote>
Antreas
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