Nonagon
A Neuneck ( Nonagon, rare: Enneagon ) is a geometric figure. It belongs to the group of polygons (polygons ). It is defined by nine points. This article deals hereinafter exclusively with regular nine corners ( see picture); a polygon is called regular if it is convex, all sides are equal and its vertices lie on a common circle.
- 2.1 Exact construction
- 2.2 Proximity constructions 2.2.1 Initial construction
- 2.2.2 Second construction
- 2.2.3 Dürer's construction
Mathematical relationships
Formula for angle calculations
The angle which the two adjacent side edges in the plane, regular nonagon with each other, is according to a general formula for regular polygons, must be used in the variable n for the number of vertices of the polygon (in this case: n = 9):
The acute angle of the nine sub-triangles is 360 ° / 9 = 40 °. The sum of the angles is 140 ° x 9 = 1260 °.
Formula for the area A
A nonagon has a clearly identifiable area, which can be calculated by decomposing into triangles always. The area of a regular nonagon is nine times the area of one of those triangles that are spanned by its center and two adjacent vertices.
Or the radius radius:
Formula for the side length s
Diagonals
There are three types of diagonals, which include two, three or four sides. Their lengths are as follows:
The difference between the lengths of the longest and the shortest diagonal is equal to the side length.
Constructions
Exact construction
Only with compass and ruler ( Euclidean tools) can not be constructed a regular nonagon. If we extend the tools so that a general trisection of an angle is possible, eg an absorbed. Tomahawk or the Archimedes method, one can obtain the required angle of 40 ° by trisection of constructible with compass and straightedge angle of 120 °.
Proximity structures
There are some sufficiently accurate in practice, possible with tools Euclidean proximity constructions.
First construction
In the simplest approximation construction a right triangle with short sides 6 and 5 will be used.
This triangle gives an angle of approximately 39.80557 °.
A much better result is achieved with a triangle of the side lengths 87 and 73, which provides an angle of approximately 39.99936 °:
Since the angle should be 40 °, in both cases, the result of the side length is less than the true value. When 6:5 triangle is the rel. Error F:
So these are 0.319 percent. At 87:73 Triangle he is about -0.0000106, so 0.00106 percent, or 1/1000 percent. In other words, nonagons, which are constructed with a large auxiliary triangle must have a radius greater than a radius of 94.591 meters, so that the error of the side length is greater than 1 millimeter. Such Neuneck would be greater than three and a half football fields. The construction of the large triangle is in practice hardly feasible, because each circle is erroneous, and the removal of 87 routes, the error in the worst case can reproduce.
Second construction
A much more practical design is carried out as follows:
The distance s has a length of
With this construction, the error amounts to
This corresponds to a radius of 150.3 cm of a deviation of 1 mm. The site is a little bit too short.
Dürer's construction
An even more elegant, but also less accurate approximation construction has already used Albrecht Dürer ( 1471-1528 ).
As mentioned above, this construction is relatively inaccurate. The triangle MEF has an acute angle of approximately 39.59407 ° instead of 40 °. The line EF is therefore approximately 0.97 % less than the true value of the side length.
Use of the nonagon in practice
The fortress town of Palmanova is built on a Neuneck. The annually published 5 Euro silver coins from Austria in the form of a nonagon. In addition, based the architecture of the houses of worship ( the sacred Baha'i ) on a Neuneck. Radial engines were mostly 5 -, 7 - or 9- built zylindrig.