In

p. 178.

/ref> This proof represented the first progress in regular polygon construction in over 2000 years. Gauss's proof relies firstly on the fact that constructibility is equivalent to expressibility of the^{''h''} times as many sides had been given by Euclid, but constructions based on the Fermat primes other than 3 and 5 were unknown to the ancients. (The only known Fermat primes are ''F_{n}'' for ''n'' = 0, 1, 2, 3, 4. They are 3, 5, 17, 257, and 65537.)
The explicit construction of a heptadecagon was given by in 1893. The following method of construction uses ^{''h''} times as many sides.
Another construction of the regular heptadecagon using straightedge and compass is the following:
T. P. Stowell of Rochester, N. Y., responded to Query, by W.E. Heal, Wheeling, Indiana in ''The Analyst'' in the year 1874:Query, by W. E. Heal, Wheeling, Indiana

p. 64; accessdate 30 April 2017 ''"To construct a regular polygon of seventeen sides in a circle.'' ''Draw the radius CO at right-angles to the diameter AB: On OC and OB, take OQ equal to the half, and OD equal to the eighth part of the radius: Make DE and DF each equal to DQ and EG and FH respectively equal to EQ and FQ; take OK a mean proportional between OH and OQ, and through K, draw KM parallel to AB, meeting the semicircle described on OG in M; draw MN parallel to OC, cutting the given circle in N – the arc AN is the seventeenth part of the whole circumference."'' The following simple design comes from Herbert William Richmond from the year 1893: ::''"LET OA, OB (fig. 6) be two perpendicular radii of a circle. Make OI one-fourth of OB, and the angle OIE one-fourth of OIA; also find in OA produced a point F such that EIF is 45°. Let the circle on AF as diameter cut OB in K, and let the circle whose centre is E and radius EK cut OA in N_{3} and N_{5}; then if ordinates N_{3}P_{3}, N_{5}P_{5} are drawn to the circle, the arcs AP_{3}, AP_{5} will be 3/17 and 5/17 of the circumference."''
*The point N_{3} is very close to the center point of _{2} determines the point H instead of the bisector w_{3}.
*The circle k_{4} around the point G' (reflection of the point G at m) yields the point N, which is no longer so close to M, for the construction of the tangent.
*Some names have been changed.
Another more recent construction is given by Callagy.

_{17} symmetry, order 34. Since 17 is a _{1}, and 2 _{17}, and Z_{1}.
These 4 symmetries can be seen in 4 distinct symmetries on the heptadecagon. John Conway labels these by a letter and group order.John H. Conway, Heidi Burgiel,

Heptadecagon trigonometric functions

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geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

.
Regular heptadecagon

A ''regular
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* Regular (Badfinger song), "Regular" (Badfinger song)
* Regular tunin ...

heptadecagon'' is represented by the Schläfli symbol
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

.
Construction

As 17 is aFermat prime
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, the regular heptadecagon is a constructible polygon
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

(that is, one that can be constructed using a compass and unmarked straightedge): this was shown by Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of m ...

in 1796 at the age of 19.Arthur Jones, Sidney A. Morris, Kenneth R. Pearson, ''Abstract Algebra and Famous Impossibilities'', Springer, 1991, p. 178.

/ref> This proof represented the first progress in regular polygon construction in over 2000 years. Gauss's proof relies firstly on the fact that constructibility is equivalent to expressibility of the

trigonometric function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s of the common angle in terms of arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...

operations and square root
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

extractions, and secondly on his proof that this can be done if the odd prime factors of $N$, the number of sides of the regular polygon, are distinct Fermat primes, which are of the form $F\_n\; =\; 2^\; +\; 1$ for some nonnegative integer $n$. Constructing a regular heptadecagon thus involves finding the cosine of $2\backslash pi/17$ in terms of square roots, which involves an equation of degree 17—a Fermat prime. Gauss's book ''Disquisitiones Arithmeticae
Title page of the first edition
The (Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through ...

'' gives this as (in modern notation):Callagy, James J. "The central angle of the regular 17-gon", ''Mathematical Gazette'' 67, December 1983, 290–292.
:$\backslash begin\; 16\backslash ,\backslash cos\backslash frac\; =\; \&\; -1+\backslash sqrt+\backslash sqrt+\; \backslash \backslash \; \&\; 2\backslash sqrt\backslash \backslash \; =\; \&\; -1+\backslash sqrt+\backslash sqrt+\; \backslash \backslash \; \&\; 2\backslash sqrt.\; \backslash end$
Constructions for the , pentagon
In geometry, a pentagon (from the Greek language, Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is ...

, pentadecagon
In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon. Good
Regular pentadecagon
A ''regular polygon, regular pentadecagon'' is represented by Schläfli symbol .
A Regular polygon, regular pentadecagon has interio ...

, and polygons with ''2''Carlyle circleIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s, as shown below. Based on the construction of the regular 17-gon, one can readily construct ''n''-gons with ''n'' being the product of 17 with 3 or 5 (or both) and any power of 2: a regular 51-gon, 85-gon or 255-gon and any regular ''n''-gon with ''2''p. 64; accessdate 30 April 2017 ''"To construct a regular polygon of seventeen sides in a circle.'' ''Draw the radius CO at right-angles to the diameter AB: On OC and OB, take OQ equal to the half, and OD equal to the eighth part of the radius: Make DE and DF each equal to DQ and EG and FH respectively equal to EQ and FQ; take OK a mean proportional between OH and OQ, and through K, draw KM parallel to AB, meeting the semicircle described on OG in M; draw MN parallel to OC, cutting the given circle in N – the arc AN is the seventeenth part of the whole circumference."'' The following simple design comes from Herbert William Richmond from the year 1893: ::''"LET OA, OB (fig. 6) be two perpendicular radii of a circle. Make OI one-fourth of OB, and the angle OIE one-fourth of OIA; also find in OA produced a point F such that EIF is 45°. Let the circle on AF as diameter cut OB in K, and let the circle whose centre is E and radius EK cut OA in N

Thales' theorem
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

over AF.
The following construction is a variation of H. W. Richmond's construction.
The differences to the original:
*The circle kSymmetry

The ''regular heptadecagon'' has Dihprime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

there is one subgroup with dihedral symmetry: Dihcyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

symmetries: ZChaim Goodman-Strauss
Chaim Goodman-Strauss (born June 22, 1967 in Austin TX) is an American mathematician who works in convex geometry, especially aperiodic tiling. He is on the faculty of the University of Arkansas and is a co-author with John H. Conway of ''The Sym ...

, (2008) The Symmetries of Things, (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278) Full symmetry of the regular form is r34 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g17 subgroup has no degrees of freedom but can seen as directed edge
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s.
Related polygons

Heptadecagrams

A heptadecagram is a 17-sidedstar polygon
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

. There are seven regular forms given by Schläfli symbol
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

s: , , , , , , and . Since 17 is a prime number, all of these are regular stars and not compound figures.
Petrie polygons

The regular heptadecagon is thePetrie polygon
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

for one higher-dimensional regular convex polytope, projected in a skew orthogonal projection
In linear algebra and functional analysis
Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common const ...

:
References

Further reading

* * Klein, Felix et al. ''Famous Problems and Other Monographs''. – Describes the algebraic aspect, by Gauss.External links

* Contains a description of the construction. *Heptadecagon trigonometric functions

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