[A couple of citations to the literature regarding constructions. ]
[First two are © Copyright American Mathematical Society 1997 -- djr.]
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89d:51028 51M15
Gleason, Andrew M.(1-HRV)
Angle trisection, the heptagon, and the triskaidecagon. (English)
Amer. Math. Monthly 95 (1988), no. 3, 185--194.
As an introduction, the author describes a simple construction of the
regular 7-gon (heptagon) which besides ruler and compass only requires
the trisection of an angle and which is related to an older
construction of J. Plemelj [Monatsh. Math. Phys. 23 (1912), 309--311;
Jbuch 43, 585]. Arbitrary $n$-section (particularly trisection) of an
angle can be done, given an Archimedean spiral.
The main goal of the paper is the answer to the question: which
regular polygons can be constructed with the aid of ruler, compass and
angle trisector? The following theorem is proved: A regular polygon of
$n$ sides can be constructed by ruler, compass and angle trisector if
and only if the prime factorization of $n$ is $2\sp r3\sp sp\sb 1p\sb
2\cdots p\sb k$, where $p\sb 1,p\sb 2,\cdots, p\sb k$ are distinct
primes $(>3)$ each of the form $2\sp t3\sp u+1$ ($k=0$, i.e. $n=2\sp
r3\sp s$, is included).
There exist 41 such primes $n<1 000\,000$.
An application of this result is the construction of the regular
13-gon (triskaidecagon), which requires one angle trisection. For the
19-gon one needs two angle trisections.
Reviewed by K. Strubecker
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97d:51028 51M15
Strommer, J.
Konstruktion des regularen $257$-Ecks mit Lineal und
Streckenubertrager. (German)
[Construction of the regular $257$-gon with straightedge and segment
transferer]
Acta Math. Hungar. 70 (1996), no. 4, 259--292.
As the author informs us in a short historical opening section, a
construction of the regular $257$-gon was already given in 1873,
though with straightedge and compass. Hilbert has shown that the
compass can be replaced by a segment transferer, so, having published
a construction of the regular $17$-gon with straightedge and segment
transferer in a preceding paper, the author now proceeds with a
corresponding solution for the regular $257$-gon.
To "construct" a $257$-gon by straightedge and segment transferer
means to compute coordinates of the $257$-gon's vertices such that
they only contain algebraic operations that can be carried out by
these instruments geometrically. The author succeeds by working
through long lists of equations, which in such a context are certainly
unavoidable. However, the reading would have been easier if the course
of argumentation had been pointed out more clearly. Readers who are
not familiar with the subject might therefore start with the final
section, where, rather unexpectedly, the idea of the paper is
practised once more on the regular $17$-gon.
Reviewed by Guido M. Pinkernell
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From Coxeter's "Introduction to Geometry" 1980:
"Euclid's postulates imply a restriction on the instruments that he
allowed for making constructions, namely the restriction to ruler (or
straightedge) and compasses. He constructed an equilateral triangle
(I.1 [Book I, sect. 1 of the Elements]), a square (IV.6), a regular
pentagon (IV.11), a regular hexagon (IV.15), and a regular 15-gon
(IV.16). THe number of sides may be doubled again and again by
repeated angle bisections. It is natural to ask which other regular
polygons can be constructed with Euclid's instrments. This question
was completely answered by Gauss (1777-1855) at the age of
nineteen. Gauss found that a regular n-gon... can be so constructed if
the odd prime factors of n are distinct "Fermat primes"
F_k = 2^(2^k)+1.
The only known primes of this kind are [3, 5, 17, 257, 65537].
"To inscribe a regular pentagon in a given circle, simpler constructions
than Euclid's were given by Ptolemy and Richmond... [latter given]
"Richmond also gave a simple construction for the [17-gon]... [given].
Richelot and Schwendenwein constructed the regular 257-gon in 1832.
J. Hermes spent ten years on the regular 65537-gon and deposited the
manuscript in a large box in the University of Goettingen, where it
may still be found."