geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a 65537-gon is a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

with 65,537 (2¹⁶ + 1) sides. The sum of the interior angles of any non– self-intersecting is 11796300°.

Regular 65537-gon

The area of a ''regular '' is (with ) :

A = \frac t^2 \cot \frac

A whole regular is not visually discernible from a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

, and its

perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pract ...

differs from that of the

circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

by about 15

parts per billion In science and engineering, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they ...

Construction

The regular 65537-gon (one with all sides equal and all angles equal) is of interest for being a

constructible polygon In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinite ...

: that is, it can be constructed using a compass and an unmarked straightedge. This is because 65,537 is a

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 4294967 ...

, being of the form 2^{2^''n''} + 1 (in this case ''n'' = 4). Thus, the values

\cos \frac

and

\cos \frac

are 32768-

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...

s, and like any

constructible number In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is con ...

s, they can be written in terms of

square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...

s and no higher-order roots. Although it was known to

Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

by 1801 that the regular 65537-gon was constructible, the first explicit construction of a regular 65537-gon was given by

Johann Gustav Hermes Johann Gustav Hermes (20 June 1846 – 8 June 1912) was a German mathematician. Hermes is known for completion of a polygon with 65,537 sides. Early life On 20 June 1846, Hermes was born in Königsberg, a former German city (presently Kalining ...

(1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript. Another method involves the use of at most 1332

Carlyle circle In mathematics, a Carlyle circle (named for Thomas Carlyle) is a certain circle in a coordinate plane associated with a quadratic equation. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of ...

s, and the first stages of this method are pictured below. This method faces practical problems, as one of these Carlyle circles solves the

quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...

''x''² + ''x'' − 16384 = 0 (16384 being 2¹⁴).

Symmetry

The ''regular 65537-gon'' has Dih₆₅₅₃₇ symmetry, order 131074. Since 65,537 is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

there is one subgroup with dihedral symmetry: Dih₁, and 2

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

symmetries: Z₆₅₅₃₇, and Z₁.

65537-gram

A 65537-gram is a 65,537-sided

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...

. As 65,537 is prime, there are 32,767 regular forms generated by

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

s for all

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s 2 ≤ ''n'' ≤ 32768 as

\left\lfloor \frac \right\rfloor = 32768

References

Bibliography

* * Robert Dixon ''Mathographics''. New York: Dover, p. 53, 1991. *Benjamin Bold, ''Famous Problems of Geometry and How to Solve Them'' New York: Dover, p. 70, 1982. *

H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

''Introduction to Geometry'', 2nd ed. New York: Wiley, 1969. Chapter 2, Regular polygons *

Leonard Eugene Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remem ...

''Constructions with Ruler and Compasses; Regular Polygons'' Ch. 8 in Monographs on Topics of Modern Mathematics *Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352–386, 1955.

External links

65537-gon
mathematik-olympiaden.de (German), with images of the documentation HERMES; retrieved on July 9, 2018
Wikibooks 65573-Eck (German) Approximate construction of the first side in two main steps
* 65537-gon, exact construction for the 1st side, using the

Quadratrix of Hippias The quadratrix or trisectrix of Hippias (also quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is one of the oldest examples for a kinematic curve (a curve created through motion). Its discovery is attributed to the G ...

and

GeoGebra GeoGebra (a portmanteau of ''geometry'' and ''algebra'') is an interactive geometry, algebra, statistics and calculus application, intended for learning and teaching mathematics and science from primary school to university level. GeoGebra is av ...

as additional aids, with brief description (German) {{Polygons Constructible polygons Polygons by the number of sides Euclidean plane geometry Carl Friedrich Gauss