Regular seventeen

Seventeen
Seventeen
	; Regular seventeen
Type of	Regular polygon
Ribs	17
Shlefly Symbol	{17}
Coxeter - Dynkin diagram	;
Type of symmetry	Dihedral group (D 18 ) order 2 × 18
Inside corner	≈158.82 °
The properties
	convex , inscribed , equilateral , , isotoxic

The correct seventeenth angle is a geometric figure belonging to the group of regular polygons . It has seventeen sides and seventeen angles , all its angles and sides are equal to each other, all the vertices lie on the same circle . Among other regular polygons with a large (more than five ) simple number of sides, it is interesting in that it can be built with the help of a compass and a ruler (thus, it is impossible to build seven- , eleven- and thirteen- triangles with a compass and a ruler).

Properties

The central angle α is ${\frac {360^{\circ }}{17}}\approx 21{,}17647059^{\circ }$ ${\ displaystyle {\ frac {360 ^ {\ circ}} {17}} \ approx 21 {,} 17647059 ^ {\ circ}}$ ${\frac {360^{\circ }}{17}}\approx 21{,}17647059^{\circ }$ .

The ratio of the side length to the radius of the circumscribed circle is

s=2\cdot r_{u}\cdot \sin \left({\frac {\alpha }{2}}\right)\approx r_{u}\cdot 0{,}3675.

{\ displaystyle s = 2 \ cdot r_ {u} \ cdot \ sin \ left ({\ frac {\ alpha} {2}} \ right) \ approx r_ {u} \ cdot 0 {,} 3675.}

s=2\cdot r_{u}\cdot \sin \left({\frac {\alpha }{2}}\right)\approx r_{u}\cdot 0{,}3675.

A regular seventeenthagon can be constructed using a compass and a ruler , which was proved by Gauss in the monograph Arithmetic Studies (1796). He also found the value of the cosine of the central angle of the hexagon:

\cos {\frac {360^{\circ }}{17}}=

{\ displaystyle \ cos {\ frac {360 ^ {\ circ}} {17}} =}

\cos {\frac {360^{\circ }}{17}}=

={\frac {1}{16}}\left(-1+{\sqrt {17}}+{\sqrt {2\left(17-{\sqrt {17}}\right)}}+2{\sqrt {17+3{\sqrt {17}}-{\sqrt {2\left(17-{\sqrt {17}}\right)}}-2{\sqrt {2\left(17+{\sqrt {17}}\right)}}}}\right).

{\ displaystyle = {\ frac {1} {16}} \ left (-1 + {\ sqrt {17}} + {\ sqrt {2 \ left (17 - {\ sqrt {17}} \ right)}} +2 {\ sqrt {17 + 3 {\ sqrt {17}} - {\ sqrt {2 \ left (17 - {\ sqrt {17}} \ right)}} - 2 {\ sqrt {2 \ left (17 + {\ sqrt {17}} \ right)}}}} \ right).}

={\frac {1}{16}}\left(-1+{\sqrt {17}}+{\sqrt {2\left(17-{\sqrt {17}}\right)}}+2{\sqrt {17+3{\sqrt {17}}-{\sqrt {2\left(17-{\sqrt {17}}\right)}}-2{\sqrt {2\left(17+{\sqrt {17}}\right)}}}}\right).

In the same work, Gauss proved that if the odd prime divisors of n are different Fermat primes ( Fermat numbers ), that is, primes of the form $F_{m}=2^{2^{m}}+1,$ ${\ displaystyle F_ {m} = 2 ^ {2 ^ {m}} + 1,}$ $F_{m}=2^{2^{m}}+1,$ then a regular n-gon can be constructed using a compass and a ruler (see the Gauss – Wanzel theorem ).

Facts

Gauss was so inspired by his discovery that at the end of his life he bequeathed that a regular seventeenthagon be carved on his grave. The sculptor refused to do this, claiming that the construction would be so complicated that the result could not be distinguished from a circle.

In 1893, published an explicit description of constructing a regular seventeenthagon in 64 steps. The following is this construction.

Build

Exact construction

Draw a large circle k ₁ (the future circumscribed circle of the hexagon) with center O.
We draw its diameter AB .
We construct a perpendicular m to it that intersects k₁ at points C and D.
Mark the point E - the middle of DO .
In the middle of EO, mark point F and draw a segment FA .
We construct the bisector w₁ of the angle ∠OFA.
We construct w₂, the bisector of the angle between m and w₁, which intersects AB at the point G.
Draw s - perpendicular to w₂ from point F.
We construct w₃ - the bisector of the angle between s and w₂. She crosses AB at point H.
We construct the Thales circle ( k ₂) on the diameter HA . It intersects the CD at points J and K.
Draw a circle k₃ with center G through points J and K. It intersects with AB at points L and N. It is important not to confuse N with M , they are located very close.
We construct a tangent to k₃ through N.

The points of intersection of this tangent with the original circle k₁ are the points P₃ and P₁₄ of the desired seventagon. If we take the middle of the resulting arc for P₀ and lay the arc P₀P₁₄ three times in a circle, all the vertices of the hexagon will be built.

Approximate construction

The following construction, although approximately, is much more convenient.

We put a point M on the plane, build a circle k around it and draw its diameter AB ;
Divide the radius AM in half three times in succession towards the center (points C , D and E ).
Divide in half the segment EB (point F ).
draw a perpendicular to AB at point F.

In short: build the perpendicular to the diameter at a distance of 9/16 of the diameter from B.

The intersection points of the last perpendicular with the circle are a good approximation for the points P₃ and P₁₄.

With this construction, a relative error of 0.83% is obtained. The angles and sides are obtained in this way a little more than necessary. With a radius of 332.4 mm, the side is longer by 1 mm.

Erhinger's Animated Build

Building a seventeen-sided compass and ruler in 64 steps by Johannes Erhinger

Star Shapes

A regular seventeen has 7 regular star shapes.

{17/2}
{17/3}
{17/4}
{17/5}
{17/6}
{17/7}
{17/8}

Links

Karin Reich . Die Entdeckung und frühe Rezeption der Konstruierbarkeit des regelmäßigen 17-Ecks und dessen geometrische Konstruktion durch Johannes Erchinger (1825). // In: Mathesis, Festschrift zum siebzigsten Geburtstag von Matthias Schramm . Hrsg. von Rüdiger Thiele, Berlin, Diepholz 2000, pp. 101-118.
Weisstein, Eric W. Hexagon on the Wolfram MathWorld website.