4294967295-corner

Regular 4294967295-Eck (4 billion 294 million 967-thousand-295-gon) is the regular polygon with the - biggest odd corners - if known number , which is theoretically construct ruler and compass can.

The work of Gauß and Wantzel

It has been known since ancient times that equilateral triangles, squares and pentagons can be constructed with rulers and compasses without the aid of other aids. In 1796, the then 19-year-old Carl Friedrich Gauß proved that this is also possible for the regular seventeen corner. A few years later, in his Disquisitiones Arithmeticae, he gave the more general proof that a regular polygon can be constructed precisely when its number of vertices can be represented as the product of a power of two with pairwise different Fermat's prime numbers , i.e. prime numbers of the form . The French mathematician Pierre Wantzel completed this proof in 1837 . ${\displaystyle 2^{2^{n}}+1}$

At the moment only Fermat's prime numbers are known. This 5-element set has exactly non-empty subsets , so with a compass and ruler exactly 31 regular polygons with an uneven number of corners can be constructed. The largest product of pairwise different numbers of this set is , so the largest possible number of corners is 4294967295 . It is unknown whether further regular polygons with an odd number of corners can be constructed and depends on the question of whether there are any other than the five known Fermat's prime numbers - a mathematical problem that has not yet been solved. ${\displaystyle \{3,5,17,257,65537\}}$ ${\displaystyle 2^{5}-1=31}$ ${\displaystyle 3\cdot 5\cdot 17\cdot 257\cdot 65537=4294967295=2^{32}-1}$

For regular polygons with an even number of corners, it is not possible to specify a maximum number of corners for constructability, because according to Gauss’s formula, the polygon with corners can also be constructed for every constructible polygon with corners . This means that there are infinitely many constructible regular polygons with an even number of corners. ${\displaystyle n}$ ${\displaystyle 2\cdot n}$

Mathematical relationships

The results for side length, incircle radius and area relate to a perimeter radius as a unit of length ( unit circle ). ${\displaystyle 1[\mathrm {LE} ]=R=1}$

The same applies to the area unit ${\displaystyle 1[\mathrm {FE} ]=R^{2}=1}$

Interior angle

The interior angle is enclosed by two adjacent sides of the length , is the number of sides or corners. ${\displaystyle \alpha }$ ${\displaystyle a}$ ${\displaystyle n}$

${\displaystyle \alpha ={\frac {n-2}{n}}\cdot 180^{\circ }={\frac {4294967293}{4294967295}}\cdot 180^{\circ }=\left(179{\frac {286331129}{286331153}}\right)^{\circ }=179{,}999999916180968\ldots ^{\circ }}$

Central angle

The central angle or center angle is enclosed by two adjacent circumferential radii of the length . ${\displaystyle \mu }$ ${\displaystyle R}$

${\displaystyle \mu ={\frac {360^{\circ }}{n}}={\frac {360^{\circ }}{4294967295}}=\left({\frac {24}{286331153}}\right)^{\circ }=0{,}0000000838190317349087055\ldots ^{\circ }=8{,}38190317349087055\ldots \cdot 10^{-8}\mathrm {^{\circ }} }$

Side length

The side length is the distance between two neighboring corner points. ${\displaystyle a}$

${\displaystyle a=R\cdot 2\cdot \sin \left({\frac {180^{\circ }}{n}}\right)=1\cdot 2\cdot \sin \left({\frac {180^{\circ }}{4294967295}}\right)=0{,}0000000014629180796077718390\ldots \;[\mathrm {R} ]=1{,}4629180796077718390\ldots \cdot 10^{-9}\;[\mathrm {R} ]}$

Example: In the case of a radius , the side length has the value . ${\displaystyle R=1000\;\mathrm {km} }$ ${\displaystyle a}$ ${\displaystyle 1{,}46\;\mathrm {mm} }$

scope

The scope differs from that of the perimeter by . ${\displaystyle 5{,}6028454255198359922307980967513\cdot 10^{-19}\;[\mathrm {LE} ]}$

Inscribed radius

The incircle radius is the height of an isosceles partial triangle with the circumferential radius as the length of the legs and that of the side as the base line : ${\displaystyle r}$ ${\displaystyle R}$ ${\displaystyle a}$

${\displaystyle r=R\cdot \cos \left({\frac {\mu }{2}}\right)=1\cdot \cos \left({\frac {180^{\circ }}{4294967295}}\right)=0{,}99999999999999999973248383654459\ldots \;[\mathrm {LE} ]}$

Area

The area of a triangle is generally calculated as . The following applies to the calculation of the 4294967295 corner ${\displaystyle A_{\Delta }={\tfrac {1}{2}}a\cdot h_{a}}$

${\displaystyle A=R^{2}\cdot {\frac {n}{2}}\cdot \sin \left({\frac {360^{\circ }}{n}}\right)}$

${\displaystyle A=R^{2}\cdot {\frac {4294967295}{2}}\cdot \sin \left({\frac {360^{\circ }}{4294967295}}\right)}$

${\displaystyle A=R^{2}\cdot 3{,}1415926535897932373420742981754}$

${\displaystyle A=3{,}1415926535897932373420742981754[\mathrm {FE} ]}$

The area deviates from the area of ​​the perimeter only by the factor

${\displaystyle 3{,}5668821794054759258295065324085\cdot 10^{-19}}$

from. In the case of the unit radius, these are

${\displaystyle \Delta A=A-\pi =-0{,}00000000000000000112\ldots \;[\mathrm {FE} ]=-1{,}12\ldots \cdot 10^{-18}\;[\mathrm {FE} ].}$

With a radius of 1000 km the deviation is only 1.12 mm².

Illustrations

If one were to place such a polygon in the lunar orbit ( orbit length ≈ 2,400,000 km), there would be a corner approximately every 56 cm.

If one were to place one around the earth's equator (length ≈ 40,000 km), a corner would be found approximately every 9.3 mm.

Web links

• Follow A045544 in OEIS

Individual evidence

1. ↑ Number dictionary. (PDF; 3.8 MB) Retrieved on April 10, 2020 : “4294967295. Exactly 31 polygons with an uneven number of corners can only be constructed with a compass and ruler. The largest has 4294967295 corners. "