Gauss - Wantsel theorem

Building a regular pentagon

Gauss Theorem - Wantsel argues that the correct $n$ ${\ displaystyle n}$ $n$ -gon can be constructed using a compass and a ruler if and only if $n=2^{k}\cdot p_{1}\cdot \ldots \cdot p_{m}$ ${\ displaystyle n = 2 ^ {k} \ cdot p_ {1} \ cdot \ ldots \ cdot p_ {m}}$ ${\ displaystyle n = 2 ^ {k} \ cdot p_ {1} \ cdot \ ldots \ cdot p_ {m}}$ where $p_{i}$ ${\ displaystyle p_ {i}}$ $p_ {i}$ - various Fermat primes. This condition is also equivalent to the value of the Euler function $\varphi (n)$ ${\ displaystyle \ varphi (n)}$ $\ varphi (n)$ is a power of two.

History

Ancient geometers knew how to construct the right $n$ ${\ displaystyle n}$ $n$ -gons for $n=2^{k},3\cdot 2^{k},5\cdot 2^{k}$ ${\ displaystyle n = 2 ^ {k}, 3 \ cdot 2 ^ {k}, 5 \ cdot 2 ^ {k}}$ $n=2^{k},3\cdot 2^{k},5\cdot 2^{k}$ and $3\cdot 5\cdot 2^{k}$ ${\ displaystyle 3 \ cdot 5 \ cdot 2 ^ {k}}$ $3\cdot 5\cdot 2^{k}$ .

In 1796, Gauss showed the possibility of building the right $n$ ${\ displaystyle n}$ $n$ -gons at $n=2^{k}\cdot p_{1}\cdots p_{m}$ ${\ displaystyle n = 2 ^ {k} \ cdot p_ {1} \ cdots p_ {m}}$ $n=2^{k}\cdot p_{1}\cdots p_{m}$ where $p_{i}$ ${\ displaystyle p_ {i}}$ $p_{i}$ - various Fermat primes. (Here is the case $m=0$ ${\ displaystyle m = 0}$ $m=0$ corresponds to the number of parties $n=2^{k}$ ${\ displaystyle n = 2 ^ {k}}$ $n=2^{k}$ .)

In 1837, Wanzel proved that there are no other regular polygons that can be built with a pair of compasses and a ruler.

The specific implementation of the construction is very time-consuming:

The construction of a regular 17-gon was directly carried out by Gauss himself, but was first published by K.F. von Pfeiderer in 1802 .
A regular 257-gon was built by F. Yu. Richelo in 1832 ^[1] .
A manuscript is stored in the library of the University of Göttingen , which is the result of 10 years of work by I. G. Hermes , which contains a method for constructing a regular 65537-gon .

One too intrusive postgraduate student brought his supervisor to the point that he told him: "Go and work out the construction of a regular polygon with 65,537 sides." The graduate student retired to return after 20 years with the appropriate construction ^[2] .

J. Littlewood

Notes

Jacques Cesiano The history of constructing regular polygons using a compass and a ruler from Euclid to Gauss Seminar on the history of mathematics May 4, 2017 18:00, St. Petersburg, POMI , Fontanka 27, auditorium 106.

↑ Friedrich Julius Richelot. De resolutione algebraica aequationis x ²⁵⁷ = 1, sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata (neopr.) // Journal für die reine und angewandte Mathematik . - 1832. - T. 9 . - S. 1–26, 146–161, 209–230, 337–358 .
↑ J. Littlewood. The mathematical mixture . - M .: Nauka, 1990 .-- S. 43. - ISBN 5-02-014332-4 .

[1] Friedrich Julius Richelot. De resolutione algebraica aequationis x ²⁵⁷ = 1, sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata (neopr.) // Journal für die reine und angewandte Mathematik . - 1832. - T. 9 . - S. 1–26, 146–161, 209–230, 337–358 .

[2] J. Littlewood. The mathematical mixture . - M .: Nauka, 1990 .-- S. 43. - ISBN 5-02-014332-4 .

Gauss - Wantsel theorem

History

Notes

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