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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 figures. A mat ...

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 ...

.

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 ...
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 a
Fermat 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 $2\pi/17$ 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. :$\begin 16\,\cos\frac = & -1+\sqrt+\sqrt+ \\ & 2\sqrt\\ = & -1+\sqrt+\sqrt+ \\ & 2\sqrt. \end$ 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''''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 ''Fn'' 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
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''''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 N3 and N5; then if ordinates N3P3, N5P5 are drawn to the circle, the arcs AP3, AP5 will be 3/17 and 5/17 of the circumference."'' *The point N3 is very close to the center point of
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 k2 determines the point H instead of the bisector w3. *The circle k4 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.

# Symmetry

The ''regular heptadecagon'' has Dih17 symmetry, order 34. Since 17 is a
prime 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: Dih1, and 2
cyclic 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: Z17, and Z1. 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,
Chaim 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

star 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

Petrie 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 ...

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