Neuneck

This article deals with the geometric figure Neunagon. For other meanings of the word see Neuneck (disambiguation)

Regular neunagon

A nonagon or Nonagon (rare: Enneagon ) is a geometric figure . It belongs to the group of polygons ( polygons ). It is defined by nine points . A polygon is called regular if it is convex , all sides are the same length and its corner points lie on a common perimeter . In the following, this article deals exclusively with regular nine-corners (see picture) and regular overturned nine-corners.

Mathematical relationships

Formula for angle calculations

According to a general formula for regular polygons, the angle that two adjacent side edges enclose with each other in the flat, regular hexagon, in which the number of corner points of the polygon must be inserted for the variable n (in this case: n = 9):

${\displaystyle \alpha ={\frac {(n-2)}{n}}\cdot 180^{\circ }={\frac {7}{9}}\cdot 180^{\circ }=140^{\circ }}$

The acute angle of one of the nine sub-triangles is 360 ° / 9 = 40 °. The sum of the angles is 140 ° 9 = 1260 °.

Formula for area A

A nine-corner has a clearly determinable area , which can always be calculated by breaking it down into triangles. The area of ​​the regular triangle is nine times the area of ​​one of those triangles that are spanned by its center and two adjacent corner points.

${\displaystyle A={\frac {9}{4}}\cdot s^{2}\cdot {\frac {\cos 20^{\circ }}{\sin 20^{\circ }}}}$

or with the perimeter radius:

${\displaystyle A={\frac {9}{2}}\cdot r_{u}^{2}\cdot \sin 40^{\circ }}$

Formula for the side length s

${\displaystyle s=2\cdot r_{u}\cdot \sin {20^{\circ }}}$

${\displaystyle s\approx r_{u}\cdot 0{,}684\,040\,286\,651\,337\,466\,088\,199\,229\,364\,52}$

Diagonals

There are three types of diagonals that encompass two, three and four sides, respectively. Their lengths are:

${\displaystyle d_{2}=2r_{u}\sin 40^{\circ }}$

${\displaystyle d_{3}=2r_{u}\sin 60^{\circ }}$

${\displaystyle d_{4}=2r_{u}\sin 80^{\circ }}$

The difference between the lengths of the longest and the shortest diagonal is equal to the side length . ${\displaystyle d_{4}-d_{2}}$ ${\displaystyle s}$

Approximate constructions

A regular hexagon cannot be constructed only with compasses and ruler (Euclidean tools) . However, there are some approximate constructions that are sufficiently accurate for practice and that are possible with Euclidean tools.

Dürer's construction

Approximate construction for regular neunagon according to Dürer

Albrecht Dürer (1471–1528) used an elegant, but also imprecise approximation construction:

1. The corner point A is marked on the circumference of the neon with center M and radius r.
2. Then you draw a circle with the same radius r around the opposite circle point N and you get the two corner points D and G. (Note: These two corner points are exact because the diagonals of the triangle between A, D and G result in an equilateral triangle .)
3. Now one again draws two circles with the radius r around the points D and G.
4. Next, the route MN is divided into three parts. A perpendicular is drawn on the straight line MN through the dividing point, which is closer to the center of the triangle.
5. The points of intersection of this perpendicular with the circular lines around D and G result in points P and Q.
6. Finally, the straight lines MP and MQ are extended until they intersect the circumference. These intersection points are a good approximation for the corner points E and F. The segment EF is a good approximation for the side length of the nine-corner.
7. The corner points B, C, H and I are obtained by cutting off the side length obtained in this way on the circular line.

calculation

If one imagines a coordinate system with M as the zero point for the Dürer construction, the following coordinates result first:

${\displaystyle M=(0,0)}$

${\displaystyle A=(0,-r)}$

${\displaystyle N=(0,r)}$

${\displaystyle G=(-r{\tfrac {\sqrt {3}}{2}},{\tfrac {r}{2}})}$

${\displaystyle D=(+r{\tfrac {\sqrt {3}}{2}},{\tfrac {r}{2}})}$

We are now looking for point Q. The circle around D through M and N is given by the equation

${\displaystyle \left(x-r\cdot {\tfrac {1}{2}}{\sqrt {3}}\right)^{2}+\left(y-r\cdot {\tfrac {1}{2}}\right)^{2}=r^{2}}$

described. The coordinates of the intersection point Q with the straight line thus satisfy both equations. By inserting the straight line equation into the circle equation one obtains: ${\displaystyle y=r/3}$

${\displaystyle \left(x-r\cdot {\tfrac {\sqrt {3}}{2}}\right)^{2}+\left(r\cdot {\tfrac {1}{3}}-r\cdot {\tfrac {1}{2}}\right)^{2}=r^{2}}$    or     ${\displaystyle \left({\tfrac {x}{r}}-{\tfrac {\sqrt {3}}{2}}\right)^{2}+\left({\tfrac {1}{3}}-{\tfrac {1}{2}}\right)^{2}=1}$

The solution of this equation gives the X-coordinates of the two intersection points of the circle with the straight line, of which the one belongs to point Q (the other is outside the representation). ${\displaystyle x<0}$

${\displaystyle \left(x-r\cdot {\frac {\sqrt {3}}{2}}\right)^{2}+{\frac {r^{2}}{36}}=r^{2}}$    or     ${\displaystyle \left(x-r\cdot {\frac {\sqrt {3}}{2}}\right)^{2}=r^{2}\cdot {\frac {35}{36}}}$

This has the solutions

${\displaystyle \left|x-r\cdot {\frac {\sqrt {3}}{2}}\right|={\frac {\sqrt {35}}{6}}}$

with so ${\displaystyle x<0}$

${\displaystyle x=r\cdot \left({\frac {\sqrt {3}}{2}}-{\frac {\sqrt {35}}{6}}\right)}$

This applies to point Q

${\displaystyle Q=\left(r\cdot \left({\frac {\sqrt {3}}{2}}-{\frac {\sqrt {35}}{6}}\right),{\frac {r}{3}}\right)}$

The central angle thus results in ${\displaystyle \varphi =FME}$

${\displaystyle \varphi =2\cdot \arctan \left({\frac {{\sqrt {35}}-3\cdot {\sqrt {3}}}{2}}\right)}$

${\displaystyle \varphi \approx 39{,}594\,068\,226\,860\,461\,444\,438\,527\,840\,699^{\circ }}$

This results in a distance that is approx. 0.974% shorter than the true value of the side length. With a radius of 150 mm, the side is 1 mm too short.

Second construction

Approximate construction for a regular hexagon
(second construction)

In the simplest approximation construction, a right triangle with cathets 6 and 5 is used.

${\displaystyle \arctan {\frac {5}{6}}\approx 39{,}80557^{\circ }}$

With this triangle you get an angle of approx. 39.80557 °. The relative error F is:

${\displaystyle f={\frac {\sin \left({\tfrac {1}{2}}\cdot \arctan \left({\tfrac {5}{6}}\right)\right)}{\sin(20^{\circ })}}-1}$

${\displaystyle f=-0,00466311591224554414418296383376}$

With a radius of approx. 313.5 mm, the side is 1 mm too short.

Third construction

Approximate construction for a neunagon (third construction)

A much more practical construction is carried out as follows:

1. Draw around a point M the circumference of the nine-corner ( k ₁ ).
2. Draw a diameter AN and extend the distance three times.
3. Mark off four more radii on this straight line. From point A for a total of six radii to point south .
4. Draw a Thales circle over AS ( k ₂ )
5. With an arc ( k ₃ ) around point A, mark a distance of 5 radii on the Thales circle (point T ).
6. With an arc ( k ₄ ) around point S, plot the distance TS on the straight line (point U ).
7. The NU = s is a good approximation for the side of the nine-corner.

The segment s has a length of

${\displaystyle r\cdot \left(4-{\sqrt {11}}\right)\approx r\cdot 0{,}68337521}$

With this construction the relative error is

${\displaystyle {\frac {0{,}68337521-0{,}68404029}{0{,}68404029}}\approx -0{,}000972=-0{,}097}$%

With a radius of 150.3 cm, this corresponds to a deviation of −1 mm. So the page is a little too short.

Exact constructions

If you expand the tools so that a general three-way division of the angle is possible, e.g. B. to a so-called. Tomahawk or with the Archimedes method , the required angle of 40 ° can be obtained by dividing the angle of 120 °, which can be constructed with a compass and ruler, into three.

With a given radius

Circumference given, three-way division of the angle 120 ° with the help of the tomahawk , animation at the end of a 10-second break

Circumference given, base is a hexagon with a three-way division of the angle 120 ° according to Archimedes, animation at the end of 10 s pause

For a given side length

Neunagon with a given side length, right angle hook method according to Ludwig Bieberbach

If the side length of a regular triangle is given, for the required three-way division of the angular width z. B. the right angle hook method according to Ludwig Bieberbach can be used. ${\displaystyle 60^{\circ }}$

In the following the steps of the adjacent construction are described.

1. Plot the given side length a on the straight line g ₁ and denote its ends with E ₁ and E _2, respectively .
2. Build a perpendicular on g ₁ in E ₁
3. Choose point A on g ₁ for the following three-quarter circle around E ₁ with radius r = E ₁ A ; yields the intersection points B and C .
4. Draw a semicircle around B with radius r ; yields the intersection points D and F .
5. Drag the arc to B from A .
6. Draw a line from B through F until it intersects the arc in G. This results in the angle GBA with an angular width of 60 ° .
7. To now the angular width of 60 ° to thirds put z. B. Place a set square on the drawing as follows:

The apex of the 90 ° angle of the triangle determines point H on the angle leg BG , one leg of the triangle runs through point D and the other is tangent to the three-quarter circle around E ₁ . After connecting point D with H and drawing the tangent from H on the three-quarter circle around E ₁ , the above-mentioned right - angle hook appears .

8. From B, draw a parallel to DH until it intersects the semicircle around B in I. The angle FBI with its 20 ° is the third part of the angle GBA .
9. Halve a in J and make a vertical line in J.
10. Transfer the tendon FI on the three-quarter circle from C with intersection K .
11. Draw a line from E ₁ through K until it intersects the perpendicular to a in M ; thus the perimeter radius r _u = ME _{1 is} found.
12. Connect M to E ₂ ; This results in the central angle E ₁ ME ₂ = μ = 40 ° of the resulting triangle.
13. Draw the perimeter around M with r _u = ME ₁ .
14. Plot the length of the side a seven times counterclockwise onto the circumference and connect the corner points to form a regular nine-corner.

Regular overturned nine corners

A regular overturned neagon results when at least one is skipped over each time when connecting the nine corner points and the chords thus created are of the same length. Such regular stars are noted with Schläfli symbols , indicating the number of corner points and connecting every -th point. ${\displaystyle \left\{n/k\right\}}$ ${\displaystyle n}$ ${\displaystyle k}$

There are only two regular nine-ray stars, also called enneagrams .

The "stars" with the symbols {9/3} and {9/6} are equilateral triangles .

• Regular nine-ray stars
• 

  ${\displaystyle \left\{9/2\right\}{,}\ \left\{9/7\right\}}$

• 

  ${\displaystyle \left\{9/4\right\}{,}\ \left\{9/5\right\}}$

Use of the hexagon in practice

The fortress city of Palmanova is built on a nine-corner. The annual 5 euro silver coins from Austria are in the shape of a nine-point. In addition, the architecture of the Houses of Worship (the sacred buildings of the Baha'i ) is based on a nine-corner. Radial engines were mostly built with 5, 7 or 9 cylinders. The main shape of the Liberation Hall in Kelheim is an eighteenth-corner plan, which cannot be constructed either because the nine-corner cannot be constructed.

Web links

Wiktionary: Neuneck  - explanations of meanings, word origins, synonyms, translations

Commons : Neuneck  - Collection of images, videos and audio files

Wikibooks: Neuneck  - learning and teaching materials

• More mathematical details on the Neunagon

Individual evidence

1. ^ Emil Artin : Galois theory. Verlag Harri Deutsch, Zurich 1973, ISBN 3-87144-167-8 , p. 85.
2. ↑ Ernst Bindel Helmut von Kügelgen: CLASSICAL PROBLEMS OF GREEK AGE IN THE MATHEMATICS CLASSROOM. (PDF) In: EDUCATIONAL ART. Association of Free Waldorf Schools in Germany, August 1965, pp. 234–237 , accessed on July 14, 2019 . 
3. ↑ Ludwig Bieberbach: On the theory of cubic constructions, journal for pure and applied mathematics. H. Hasse and L. Schlesinger, Volume 167, Walter de Gruyter, Berlin 1932, pp. 142–146, DigiZeitschriften , image on p. 144, accessed on August 8, 2020.
4. ↑ Oesterreichische Nationalbank : Münzbroschüre 2006 issue ( Memento of 5 March 2007 at the Internet Archive ) (pdf, 1.0 MB)