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 ...

, a 257-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 to ...

with 257 sides. The sum of the interior angles of any non- self-intersecting 257-gon is 45,900°.

Regular 257-gon

The area of a regular 257-gon is (with ) :

A = \frac t^2 \cot \frac\approx 5255.751t^2.

A whole regular 257-gon 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 con ...

, and its perimeter 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 24

parts per million 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, th ...

Construction

The regular 257-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 257 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, 429496 ...

, being of the form 2^{2^''n''} + 1 (in this case ''n'' = 3). Thus, the values

\cos \frac

and

\cos \frac

are 128-degree algebraic numbers, and like all

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 cons ...

s they can be written using

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 . ...

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 257-gon was constructible, the first explicit constructions of a regular 257-gon were given by

Magnus Georg Paucker Magnus Georg von Paucker (russian: Магнус-Георг Андреевич Паукер, translit=Magnus-Georg Andreevič Pauker; – ) was a Baltic German astronomer and mathematician and the first Demidov Prize winner in 1832 for his wor ...

(1822) and

Friedrich Julius Richelot Friedrich Julius Richelot (6 November 1808 – 31 March 1875) was a German mathematician, born in Königsberg. He was a student of Carl Gustav Jacob Jacobi. He was promoted in 1831 at the Philosophical Faculty of the University of Königsberg wit ...

(1832). Retrieved 8. December 2015. Another method involves the use of 150 circles, 24 being

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: this method is pictured below. 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 value, and , , and represent known numbers, where . (If and then the equation is linear, not q ...

''x''² + ''x'' − 64 = 0. 257-gon-step-1.png, Step 1 257-gon-step-2.png, Step 2 257-gon-step-3.png, Step 3 257-gon-step-4.png, Step 4 257-gon-step-5.png, Step 5 257-gon-step-6.png, Step 6 257-gon-step-7.png, Step 7 257-gon-step-8.png, Step 8 257-gon-step-9.png, Step 9

Symmetry

The ''regular 257-gon'' has Dih₂₅₇ symmetry, order 514. Since 257 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: Z₂₅₇, and Z₁.

257-gram

A 257-gram is a 257-sided star polygon. As 257 is prime, there are 127 regular forms generated by Schläfli symbols 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 languag ...

s 2 ≤ ''n'' ≤ 128 as

\left\lfloor \frac \right\rfloor = 128

. Below is a view of , with 257 nearly radial edges, with its star vertex

internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...

s 180°/257 (~0.7°). :

References

External links

* * 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 ''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 reme ...