Heptagon
The heptagon (also Heptagon from Greek heptagon of hepta = seven ) is a geometric figure. It belongs to the group of polygons (polygons ). It is defined by seven points. Unless otherwise indicated, is from a flat, regular heptagon the speech (see picture), whose seven sides are of equal length and its seven vertices lie on a common circle.
- 2.1 First approximation
- 2.2 Second approximation 2.2.1 Coordinate System
- 2.2.2 With a given radius
Mathematical relationships
Formula for angle calculations
The sum of the interior angles of the heptagon is always 900 °, and results from a general formula for polygons in which the variable is the number of vertices of the polygon has to be used ( in this case ):
The angle which the two adjacent side edges in the plane, regular heptagon with one another is (again, according to a general formula for regular polygons ):
Formula for the area A
A heptagon has a clearly identifiable area, which can be calculated by decomposing into triangles always. The area of a regular heptagon is seven 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
Proximity structures
A regular heptagon can not be constructed exactly by ruler and compass, because it is not konstruierbares polygon.
In practice there are some constructions sufficiently accurate approximation.
It's about to get a distance which is as exactly 0.86776747823 times a given radius.
First approximation
A very simple approximation construction is shown in the following diagram:
Exactly the same route length can be constructed as follows:
In this form it was already known to the acting in Baghdad in the 10th century scholar Abu al - Wafa.
From the right triangle AHM is calculated:
With
With this construction, the error is
The page length obtained with this construction is a bit too short and is 99.8 percent of the true value. Or to put it differently: from a beam radius of about 57.4 cm of error in the page length is more than a millimeter.
Second approximation
With the coordinate system
A slightly more complex but amazingly accurate approximation construction is shown in the following diagram:
Denoting the radius radius, the distance of the substituted with of and, it follows, in this construction:
And with the values
Results in:
The page length obtained with this construction is a little bit too long, the error is approximately 0.00057821133, ie 0.0578 per cent. Or in other words: If a radius radius of about 199.3 cm, the error in the side length of one millimeter.
With a given radius
A disadvantage of the above-mentioned Construction is that it is not expected from a directly given radius. If you want to go out on the radius, so the task is, belonging to the given radius distance between the straight line and the center (which is the unit length of the construction to non- coordinate system) to find.
From the construction with coordinate system and the drawing can be read:
This is true
In addition, according to the Pythagorean theorem nor
In the right triangle MZP applies after Kathetensatz
The quotient is as depicted above
And thus
Wherein p and q are the hypotenuse. Their lengths are 4/5 and 1/5 of the radius. This allows the point construct Z and thus define the distance d.
- Construct the radius a Thales circle.
- Construct the perpendicular distance of M. The so- obtained point on the valley circle is the point Z of the right triangle MZP (equivalent to item (2 / 0) in the construction of the coordinate system ).
- Construct M by the parallel with the longer cathetus. The point of intersection with the circle is the point A.
- Carrying the track on the straight AM by M of in the opposite direction from (Z ' corresponds to point ( 0/-2 ) in the construction of coordinate system).
- The distance d = is obtained by halving distance
- Construct to N. by the points of intersection with the circle are perpendicular straight points C and F
- The rest will follow as in the construction of the coordinate system.
Precise design by trisection of an angle
Referring to the classical ( Euclidean ) tools ruler and compass come with an extra tool for the trisection of an angle, such as a tomahawk so, however, the heptagon can be accurately constructed.
- Construct a circle - the later perimeter of the heptagon - around a center (O ) on a baseline (AZ). One of the points of intersection with the circle is the first corner (A) of the subsequent heptagon.
- Halving the two radii of the first diameter ( points Q and R )
- Build on the line thus obtained two equilateral triangles with sides of length equal to the radius of the circle. ( You get points K and L).
- Carrying on the baseline ( AZ) from the center 1/6 of the radius in the direction which lies on the baseline corner opposite from ( point P).
- Drawing to the point obtained through an auxiliary circuit to the two non- baseline lying corners of the equilateral triangles.
- Draw a circle in these two radii on these two points.
- Parts of the angle formed by these radii using the extra tool in three parts (eg Tomahawk, in the drawing shown in red ) and draw a straight lines thus obtained. Cut the auxiliary circuit in two other points ( points S and T).
- The line through these points - it is perpendicular to the base line - cuts the periphery of the heptagon to the first corner point (A) adjacent corners of said heptagon ( points B and G).
- Fill in the missing corners by removing the sides.
Use of the heptagon in practice
The 20 euro cent piece has seven notches to facilitate the blind distinguish it from other coins ( Spanish flower ). The old Spanish 200 -peseta coin shows on both sides of a heptagon, just as the British 20 pence and 50 pence pieces have a heptagonal shape.
In the architecture of the heptagon is very rare use. The concert hall " Hegelsaal " at the Culture and Congress Centre Hall has songs in Stuttgart as well as its glass dome plan in the form of a regular heptagon. Other examples are the royal mausoleum in the city of Hagen, the bell tower of the Church of Mary on the Strand in Vienna or the ship of the village church Ketzür.
The diagonals of the regular heptagon form the Heptagramm ( seven -pointed star ), which is popular as a symbol of esotericism.
Radial engines were mostly as 5 -, 7 - or 9- built cylinder.
Occurrence
There are fullerenes ( carbon molecules ) which have a hexagonal substructures; the chemical compound azulene and the groups of substances Tropolone, benzodiazepines and other cyclic compounds containing seven-membered rings.