257 corner

Regular 257-corner or circle?

The 257 corner is a geometrical figure , more precisely a polygon . It is defined by 257 corners that are connected to a closed figure by the same number of edges.

As a rule, this designation means the regular 257-corner , which is convex, in which all sides are of the same length and whose corner points lie on a common circumference .

construction

The special feature of the regular 257-gon is the fact that it is under restriction to the auxiliary circle and ruler (the Euclidean tools ) constructed can be. The number 257 is one of the five well-known Fermat prime numbers :

{\ displaystyle 257 = 2 ^ {8} +1}

.

Carl Friedrich Gauß proved in 1796 that a regular polygon can be constructed with a compass and ruler if and only if the number of its vertices, apart from any power of two, is equal to a product of various Fermat's prime numbers.

A construction manual for the regular 257-Eck was first presented in 1822 by Magnus Georg Paucker and again in 1832 by Friedrich Julius Richelot . In 1991 Duane W. DeTemple published a construction method using 150 auxiliary circles, and in 1999 Christian Gottlieb published another construction specification (see literature).

The practical implementation of the construction is hardly possible by hand, since the requirements for precision are very difficult to meet with the necessary size.

The GIF animation shows the construction process in numerous individual steps.

Construction of the 257 gon using the Carlyle circle.

Math background

The construction is based on a solution of the circular division equation using nested square roots . This resolution takes place analogously to the way described for the seventeen- sided, whereby as there can be chosen again as a primitive root . ${\ displaystyle x ^ {257} -1 = 0}$ ${\ displaystyle g = 3 \}$

properties

Illustrative image of the 257 corner

The central angle has the value . ${\ displaystyle {\ frac {360 ^ {\ circ}} {257}} \ approx 1 {,} 4 ^ {\ circ}}$

The interior angle has the value . ${\ displaystyle {\ frac {257-2} {257}} \ cdot 180 ^ {\ circ} \ approx 178 {,} 6 ^ {\ circ}}$

literature

Christian Gottlieb: The Simple and Straightforward Construction of the Regular 257-gon . In: Mathematical Intelligencer . Vol. 21, No. 1, 1999, pp. 31-37, doi : 10.1007 / BF03024829 .

Individual evidence

↑ Magnus Georg Paucker: The regular two hundred and fifty-seven-corner in a circle. . In: Annual Negotiations of the Courland Society for Literature and Art . 2, 1822, pp. 160-219. Retrieved December 10, 2015.
↑ ^a ^b Friedrich Julius Richelot: De resolutione algebraica aequationis x ²⁵⁷ = 1, ... . In: Source: Journal for Pure and Applied Mathematics . 9, 1832, pp. 1-26, 146-161, 209-230, 337-358. Retrieved December 10, 2015.
↑ Duane W. DeTemple: Carlyle circles and Lemoine simplicity of polygon constructions Archived from the original on December 21, 2015. In: The American Mathematical Monthly . 98, No. 2, Feb 1991, pp. 104-107. Retrieved July 16, 2016.

Web links

Wikibooks: 257-Eck, exact construction of the 1st page - learning and teaching materials

The 257-Eck at www.mathworld.com (English)
257-Eck on mathematik-olympiaden.de, with video; accessed on August 14, 2018

[1] Magnus Georg Paucker: The regular two hundred and fifty-seven-corner in a circle. . In: Annual Negotiations of the Courland Society for Literature and Art . 2, 1822, pp. 160-219. Retrieved December 10, 2015.

[Richelot-2] Friedrich Julius Richelot: De resolutione algebraica aequationis x ²⁵⁷ = 1, ... . In: Source: Journal for Pure and Applied Mathematics . 9, 1832, pp. 1-26, 146-161, 209-230, 337-358. Retrieved December 10, 2015.

[DeTemple-3] Duane W. DeTemple: Carlyle circles and Lemoine simplicity of polygon constructions Archived from the original on December 21, 2015. In: The American Mathematical Monthly . 98, No. 2, Feb 1991, pp. 104-107. Retrieved July 16, 2016.