Publication by C. F. Gauss inIntelligenzblatt der allgemeinen Literatur-Zeitung
As 17 is aFermat prime, the regular heptadecagon is aconstructible polygon (that is, one that can be constructed using acompass and unmarked straightedge): this was shown byCarl Friedrich Gauss in 1796.[1] This proof represented the first progress in regular polygon construction in over 2000 years.[1] Gauss's proof relies firstly on the fact that constructibility is equivalent to expressibility of thetrigonometric functions of the common angle in terms ofarithmetic operations andsquare root extractions, and secondly on his proof that this can be done if the odd prime factors of, the number of sides of the regular polygon, are distinct Fermat primes, which are of the form for some nonnegative integer. Constructing a regular heptadecagon thus involves finding the cosine of in terms of square roots. Gauss's bookDisquisitiones Arithmeticae[2] gives this (in modern notation) as[3]
Gaussian construction of the regular heptadecagon.
Constructions for theregular triangle,pentagon,pentadecagon, and polygons with2h 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 areFn forn = 0, 1, 2, 3, 4. They are 3, 5, 17, 257, and 65537.)
The explicit construction of a heptadecagon was given byHerbert William Richmond in 1893. The following method of construction usesCarlyle circles, as shown below. Based on the construction of the regular 17-gon, one can readily constructn-gons withn 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 regularn-gon with2h times as many sides.
Construction according to Duane W. DeTemple with Carlyle circles,[4] animation 1 min 57 s
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 inThe Analyst in the year 1877:[5]
"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."
Construction according to "sent by T. P. Stowell, credited to Leybourn's Math. Repository, 1818". Added:"take OK amean proportional between OH and OQ"Construction according to "sent by T. P. Stowell, credited to Leybourn's Math. Repository, 1818". Added:"take OK a mean proportional between OH and OQ", animation
The following simple design comes from Herbert William Richmond from the year 1893:[6]
"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 ofThales' theorem over AF.
Construction according to H. W. RichmondConstruction according to H. W. Richmond as animation
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.
Heptadecagon in principle according to H.W. Richmond, a variation of the design regarding to point N
Another more recent construction is given by Callagy.[3]
Trigonometric derivation using nested quadratic equations
Symmetries of a regular heptadecagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.
Theregular heptadecagon hasDih17 symmetry, order 34. Since 17 is aprime number there is one subgroup with dihedral symmetry: Dih1, and 2cyclic group 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.[7] Full symmetry of the regular form isr34 and no symmetry is labeleda1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), andi when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled asg for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only theg17 subgroup has no degrees of freedom but can be seen asdirected edges.
A heptadecagram is a 17-sidedstar polygon. There are seven regular forms given bySchläfli symbols: {17/2}, {17/3}, {17/4}, {17/5}, {17/6}, {17/7}, and {17/8}. Since 17 is a prime number, all of these are regular stars and not compound figures.
^Duane W. DeTemple "Carlyle Circles and the Lemoine Simplicity of Polygon Constructions" inThe American Mathematical Monthly, Volume 98, Issuc 1 (Feb. 1991), 97–108."4. Construction of the Regular Heptadecagon (17-gon)" pp. 101–104, p.103, web.archive document, selected on 28 January 2017
^John H. Conway, Heidi Burgiel,Chaim Goodman-Strauss, (2008) The Symmetries of Things,ISBN978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)
