51-Eck - 51-Eck
The 51-Eck or Pentakontahenagon is a geometric figure , and a polygon ( polygon ). It is determined by fifty-one points and their fifty-one connections called lines , sides or edges.
Regular 51-sided
According to Carl Friedrich Gauß and Pierre-Laurent Wantzel, the regular 51-gon is a constructible polygon , since the number of its sides is the product of a power of two with pairwise different Fermat's prime numbers () can be displayed.
Sizes
Sizes of a regular 51-sided | ||
---|---|---|
Interior angle | | |
Central angle (Center angle) | ||
Side length | ||
Perimeter radius | ||
Inscribed radius | ||
height | ||
Area |
Mathematical relationships
Interior angle
The interior angle is enclosed by two adjacent side edges. The variable is in the general formula for regular polygons for the number of corner points of the polygon. In this case the variable is number to use.
Central angle
The central angle or central angle is made up of two neighboring perimeter radii locked in. In the general formula is for the variable the number to use.
Side length and perimeter radius
The 51-gon is divisible into fifty-one isosceles triangles, so-called partial triangles . From half of such a partial triangle, i.e. from a right-angled triangle with the cathetus (half the side length), the hypotenuse (perimeter radius) and half the central angle one obtains the side length with the help of the trigonometry in the right triangle as follows
the circumferential radius is obtained by forming
Inscribed radius
The inscribed radius is the height of a partial triangle, perpendicular to the length of the side of the 51-sided. If the same right-angled triangle is used for the calculation as for the side length, the inscribed radius applies
height
The height of a regular 51-sided result from the sum of the incircle radius and radius .
Area
The area of a triangle is generally calculated . The results of the side length are used to calculate the 51 gon and the incircle radius used what for the height is used.
- from this it follows for the area of a partial triangle
- summarized it results
and for the area of the entire 51 gon
construction
As described above in the regular 51-square , the 51-square can be represented as a construction with compass and ruler . Since the number of its vertices results from the multiplication of the two Fermat prime numbers and results, the regular 51-sided can be found by an extension of an already known construction of the 17- sided. The two polygons triangle and seventeen-sided (their number of sides corresponds to the Fermat prime numbers or. ) are placed on top of each other symmetrically with regard to their central angles in the common circumference, as is the case with z. B. Johannes Kepler in his work WORLD HARMONIK in the construction of Fünfzehnecks shows ^{[1]} .
In principle, one of the three methods described in Siebzehneck can be selected as the basis for the construction . For reasons of the very little effort required, the method of Duane W. DeTemple , ^{[2]} from 1991, is used.
Preliminary considerations
In the drawing of the seventeenth-corner by Duane W. DeTemple (Fig. 1) it is easy to see the perpendicular from the center not only cuts the arc of a circle but also the periphery. If this intersection is called marked, it is right next to the corner point This results in the central angle with the angular width of an equilateral triangle , which is added geometrically clockwise to the central angle of the seventeenth corner.
Hence for
- Central angle of the district sector
- Central angle of the district sector
- because of
- Central angle des 51-Ecks
- also applies
Thus the route is one side length and a corner point of the 51 corner you are looking for.
The position of the corner point of the 51-sided result from the number of side lengths those in the central angle are included
- it follows
- starting from the corner point not counted clockwise the 17th corner corresponds to the corner which is counted counterclockwise.
The 17th corner point of the 51 corner is therefore based on the central axis , exactly opposite the 34th corner point.
Construction description
The identifiers in Figure 2, which have been changed compared to the original, correspond to those used today.
- Draw a straight line (analytically an X-axis) and determine a point on the later center point of the polygon (analytically a coordinate origin).
- Drawing a circle as a perimeter (analytically a unit circle) one . There are two points of intersection , the corner point of the polygon and the counterpoint .
- Establishing the vertical (analytically a Y-axis) on the straight line in . The point of intersection results.
- Halving the route in .
- Establishing the vertical on the straight line in . The two intersections with are the cornerstones and des 51-Ecks.
- Draw the circular arc one with the radius . The intersection with the vertical is.
- Now is around the first Carlyle circle through the point drawn; the intersections are and .
- The distance is halved. You get.
- Draw a second Carlyle circle one by . The points of intersection with x are the points and (the latter is not shown because it is no longer needed).
- The distance is halved. You get.
- Draw a third Carlyle circle one by . The points of intersection with x are the points and (The latter is also not shown, as it is no longer needed).
- Removal of the route on of off off. You get point
- Connect the dots and with a route.
- Halve the route . You get point.
- Draw a fourth Carlyle circle one by . The points of intersection with x are the points and (the latter not labeled as it is no longer needed).
- Redraw an arc of a circle with the perimeter radius . The intersections with the perimeter are the two too adjacent points of the 17-sided and thus the points and des 51-Ecks.
- By repeatedly removing the route on the perimeter , starting with , you get the missing points of a regular 17-sided. Up to this point the construction corresponds to that of the 17-sided.
- By repeatedly removing the route on the perimeter , starting from the points (blue) and (red), you get all the missing corner points of the 51 corner, which can be connected to each other to form the 51 corner.
Occurrence
architecture
The cross-section of the RWE tower in Essen is a regular 51-sided.
literature
- H. Maser: The division of the circle ..., Article 365. , in Carl Friedrich Gauss' studies on higher arithmetic , published by Julius Springer, Berlin 1889; Göttingen Digitization Center, University of Göttingen; accessed on March 15, 2018.
Individual evidence
- ↑ Johannes Kepler: WORLD HARMONICS. XLIV. Sentence., Page of the Fifteenth, Page 44. In: Google Books. R. OLDENBURG VERLAG 2006, translated and introduced by MAX CASPAR 1939, p. 401 , accessed on February 21, 2018 .
- ↑ Duane W. DeTemple: Carlyle Circles and the Lemoine Simplicity of Polygon Constructions. (Memento vom 11. August 2011 im Internet Archive). The American Mathematical Monthly, Vol. 98, No. 2 (Feb., 1991), S. 101–104 (JSTOR 2323939) aufgerufen am 16. Februar 2018.