65537-corner

Figure 1: 65537 corner or circle?

The 65537-gon is a geometric figure from the group of polygons ( polygons ). It is defined by 65,537 points , which are connected by as many edges to form a closed figure.

This article deals exclusively with the regular 65537 corner, where all sides are the same length and whose corner points lie on a common perimeter . In a graphical representation, the 65537 corner can only be visually distinguished from a circle with large radii (see Figure 1).

construction

The special thing about the 65537-Eck is the fact that it can theoretically be constructed by using the Euclidean tools of compasses and ruler . In practice, however, the construction is impossible to carry out. The number 65,537 is the largest known Fermat prime number :

{\ displaystyle 65,537 = 2 ^ {16} +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 Fermatian prime numbers.

In 1894, after more than ten years of effort , Johann Gustav Hermes found a design specification for the regular 65537 corner and described it in a manuscript of more than 200 pages, which is today in a specially made case in the mathematical library of the Georg-August- University of Göttingen is located.

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 . ${\ displaystyle x ^ {65537} -1 = 0}$ ${\ displaystyle g = 3 \}$

Proportions

The graphic shows a seventeenth corner. It has clearly recognizable corners, but comes very close to the circular shape.

Already a seventeenth corner is very close to a circle.

angle

The central angle has the value . ${\ displaystyle {\ tfrac {360 ^ {\ circ}} {65 {.} 537}} \ approx 0 ^ {\ circ} 0'19 {,} 8 '' = 0 {,} 0055 ^ {\ circ} }$

The interior angle has the value . ${\ displaystyle {\ tfrac {65 {.} 537-2} {65 {.} 537}} \ cdot 180 ^ {\ circ} \ approx 179 ^ {\ circ} 59'40 {,} 2 '' = 179 {,} 9945 ^ {\ circ} = 180 ^ {\ circ} -0 {,} 0055 ^ {\ circ}}$

Side length

The side length has the value in the unit circle ${\ displaystyle 2 \ cdot \ sin \ left ({\ tfrac {180 ^ {\ circ}} {65,537}} \ right) \ approx 9 {,} 5872 \ cdot 10 ^ {- 5} \; [LE]}$

illustration

To illustrate the proportions of this practically unrepresentable figure, the following considerations may serve:

Whether a tower clock or an alarm clock: half a day has 43200 seconds. The tip of the slow hour hand points to the next 65537 corner point on the 12-hour dial about every two thirds of a second.
What the interior angle at 180 ° lacks is exactly the central angle of one of the 65537 sides: about 0.0055 °. If you lift a 10 m long, ideally rigid rod at one end 1 mm from the ideally flat floor, you create the practically identical angle of 1/10000 (rad) radians.
If you want to draw a 65537 corner with a side length of 1 cm, it has a diameter of more than 200 m.
Conversely, if you draw a 65537 corner with a diameter of 20 cm on a drawing sheet, the side length is about 1/100 mm, a fraction of the diameter of the thinnest human hair .
If you circumscribe the globe with a 65537 corner, its sides get a length of about 600 m; its corners then protrude only 7.3 mm from the earth's surface, its inscribed circle .

literature

Johann Gustav Hermes: About the division of the circle into 65537 equal parts . In: News from the Society of Science in Göttingen, Mathematical-Physical Class . Göttingen, 1894, pp. 170-186 ( online ).
Article on the work of Johann Gustav Hermes. In: Die Zeit , No. 34/27. August 2012. The print edition, which is available online as an ePaper , contains a picture of the suitcase.
Heidi Niemann: The "Göttinger suitcase" of Osnabrück Rector Hermes , Osnabrücker Zeitung, January 25, 2018, online , report with a picture of the suitcase and its contents.

Web links

Wikibooks: Approximate construction of the first side of the 65537 corner in two main steps - learning and teaching materials

65537-Eck, exact construction of the 1st side
Eric W. Weisstein : 65537-gon . In: MathWorld (English).
65537-Eck from mathematik-olympiaden.de, with pictures of the documentation according to HERMES; accessed on July 16, 2016