The seventeen- ( Heptadekagon ) is a geometric figure that belongs to the group of polygons (polygons ). It is defined by seventeen points which are connected by lines to a seventeen closed line. This is about the regular seventeen- which has seventeen sides of equal length and whose vertices lie on a common circle.
The central angle α has a value of.
The ratio of the length of one side to the radius of curvature is:
The special feature of a regular seventeen- is the fact that it is constructible, that is, it can be drawn ( the Euclidean tools ) using only ruler and compass. This was proved by Carl Friedrich Gauss in 1796. He showed that the cosine of the central angle of the formula
Equivalent, resulting in the constructability.
In 1825, John Erchinger published for the first time a construction manual for the regular seventeen-sided in 64 steps. An animated representation of this construction is given below.
The discovery of Gauss is based on the solutions, a resolution of the cyclotomic equation - form in the Gaussian plane of complex numbers a regular seventeen- radius 1 - is the 17- th roots of unity. This equation can be solved only by the use of nested square roots (see above for the real part of the "first " of 1 different solution). Gauss recognized in 1796 as a 18 -year-old this way " through strenuous thinking ... in the morning ... ( before I got up out of bed ) " due to general number-theoretic properties of prime numbers, in this case specifically the prime number 17: The modulo a prime formed from 0 different residue classes can in fact be enumerated with a so-called primitive root in the form, which can be chosen in the case of concrete:
Sorted now one of the 1 different 17- th roots of unity corresponding to, ie in the order
We obtain by partial summation of every second, every fourth and every eighth root of unity from this collection, the so-called Gaussian periods: two 8-membered cycles with 8 summands, four 4-membered cycles with 4 addends and eight 2 -unit periods, each with 2 summands. Due to fundamental characteristics, or by explicit calculation can be show for it:
- The two 8-membered periods are solutions of a quadratic equation with integer coefficients.
- 4, the four -membered periods are solutions of two quadratic equations, the coefficients of the 8-membered periods are calculated.
- 2, the eight -membered periods are solutions of four quadratic equations, the coefficients of the 4 -membered periods are calculated.
This applies to the two-tier period to the "first" unit root.
The approach described can be analogously carried out for every prime of the form. Five such primes called Fermat primes are known: 3, 5, 17, 257, 65537 therefore also include regular corner 257 and the corner of the regular 65537 constructible polygons..
The points of intersection of this tangent with the output circuit k1 are the points P3 and P14 of the regular Siebzehnecks. With A = P0 can be found by seven times removal of the distance d1 in each direction on the circle all the other points of the Siebzehnecks.
1 to 5 as previously
6 Construction of the bisector of the angle w1 OFA intersection with AB is the point Q.
7 Construction of the bisector of the angle between m and w2 w1, intersection with AB is point G.
8 Construction of a circle k4 to Q which passes through F, the intersection of k4 with AB is H.
Go to Step 10 of the previous construction.