Heptadecagon

Last updated

Regular heptadecagon
Regular polygon 17 annotated.svg
A regular heptadecagon
Type Regular polygon
Edges and vertices 17
Schläfli symbol {17}
Coxeter–Dynkin diagrams CDel node 1.pngCDel 17.pngCDel node.png
Symmetry group Dihedral (D17), order 2×17
Internal angle (degrees)≈158.82°
Properties Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon Self

In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon.

Contents

Regular heptadecagon

A regular heptadecagon is represented by the Schläfli symbol {17}.

Construction

Publication by C. F. Gauss in Intelligenzblatt der allgemeinen Literatur-Zeitung Gauss 17-Eck.gif
Publication by C. F. Gauss in Intelligenzblatt der allgemeinen Literatur-Zeitung

As 17 is a Fermat prime, the regular heptadecagon is a constructible polygon (that is, one that can be constructed using a compass and unmarked straightedge): this was shown by Carl Friedrich Gauss in 1796 at the age of 19. [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 the trigonometric functions of the common angle in terms of arithmetic operations and square 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 book Disquisitiones Arithmeticae [2] gives this (in modern notation) as [3]

Gaussian construction of the regular heptadecagon. 01-Siebzehneck-Formel Gauss-2.svg
Gaussian construction of the regular heptadecagon.

Constructions for the regular triangle, pentagon, pentadecagon, and polygons with 2h 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 Herbert William Richmond in 1893. The following method of construction uses Carlyle circles, 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 2h times as many sides.

Regular Heptadecagon Using Carlyle Circle.gif
Construction according to Duane W. DeTemple with Carlyle circles, animation 1 min 57 s 01-Heptadecagon-Carlyle circle.gif
Construction according to Duane W. DeTemple with Carlyle circles, animation 1 min 57 s

Another construction of the regular heptadecagon using straightedge and compass is the following:

Regular Heptadecagon Inscribed in a Circle.gif

T. P. Stowell of Rochester, N. Y., responded to Query, by W.E. Heal, Wheeling, Indiana in The 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 a mean proportional between OH and OQ" 01 Siebzehneck-1806.svg
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"
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 01 Siebzehneck-1818-Animation.gif
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."
Construction according to H. W. Richmond 01-Siebzehneck-Richmond.svg
Construction according to H. W. Richmond
Construction according to H. W. Richmond as animation 01.Siebzehneck-Animation-Richmond.gif
Construction 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 01-Siebzehneck-Variation.svg
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

Combine nested double-angle formula with supplementary-angle formula to get the nested quadratic polynomial below.

, AND

Therefore,

On simplifying and solving for X,

Exact value of sin and cos of mπ/(17 × 2n)

If , and then, depending on any integer m

For example, if m = 1

Here are the expressions simplified into the following table.

Cos and Sin (m π / 17) in first quadrant, from which other quadrants are computable.
m16 cos (m π / 17)8 sin (m π / 17)
1
2
3
4
5
6
7
8

Therefore, applying induction with m=1 and starting with n=0:

and

Symmetry

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. Symmetries of heptadecagon.png
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.

The regular heptadecagon has Dih17 symmetry, order 34. Since 17 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic 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 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 be seen as directed edges.

Heptadecagrams

A heptadecagram is a 17-sided star polygon. There are seven regular forms given by Schläfli 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.

Picture Regular star polygon 17-2.svg
{17/2}
Regular star polygon 17-3.svg
{17/3}
Regular star polygon 17-4.svg
{17/4}
Regular star polygon 17-5.svg
{17/5}
Regular star polygon 17-6.svg
{17/6}
Regular star polygon 17-7.svg
{17/7}
Regular star polygon 17-8.svg
{17/8}
Interior angle≈137.647°≈116.471°≈95.2941°≈74.1176°≈52.9412°≈31.7647°≈10.5882°

Petrie polygons

The regular heptadecagon is the Petrie polygon for one higher-dimensional regular convex polytope, projected in a skew orthogonal projection:

16-simplex t0.svg
16-simplex (16D)

Related Research Articles

<span class="mw-page-title-main">Antiprism</span> Polyhedron with parallel bases connected by triangles

In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An.

<span class="mw-page-title-main">Constructible number</span> Number constructible via compass and straightedge

In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length can be constructed with compass and straightedge in a finite number of steps. Equivalently, is constructible if and only if there is a closed-form expression for using only integers and the operations for addition, subtraction, multiplication, division, and square roots.

<span class="mw-page-title-main">Straightedge and compass construction</span> Method of drawing geometric objects

In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.

<span class="mw-page-title-main">Root of unity</span> Number that has an integer power equal to 1

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

<span class="mw-page-title-main">Octagon</span> Polygon shape with eight sides

In geometry, an octagon is an eight-sided polygon or 8-gon.

<span class="mw-page-title-main">Decagon</span> Shape with ten sides

In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.

In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.

<span class="mw-page-title-main">Heptagon</span> Shape with seven sides

In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon.

<span class="mw-page-title-main">Dodecagon</span> Polygon with 12 edges

In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon.

<span class="mw-page-title-main">Constructible polygon</span> Regular polygon that can be constructed with compass and straightedge

In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known.

<span class="mw-page-title-main">Cupola (geometry)</span> Solid made by joining an n- and 2n-gon with triangles and squares

In geometry, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares, while the base and its opposite face are regular polygons, the triangular, square, and pentagonal cupolae all count among the Johnson solids, and can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, and rhombicosidodecahedron, respectively.

<span class="mw-page-title-main">Disdyakis triacontahedron</span> Catalan solid with 120 faces

In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.

<span class="mw-page-title-main">Viète's formula</span> Infinite product converging to 2/π

In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant π: It can also be represented as

In geometry, the area enclosed by a circle of radius r is πr2. Here, the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

<span class="mw-page-title-main">257-gon</span> Polygon with 257 sides

In geometry, a 257-gon is a polygon with 257 sides. The sum of the interior angles of any non-self-intersecting 257-gon is 45,900°.

<span class="mw-page-title-main">65537-gon</span> Shape with 65537 sides

In geometry, a 65537-gon is a polygon with 65,537 (216 + 1) sides. The sum of the interior angles of any non–self-intersecting 65537-gon is 11796300°.

<span class="mw-page-title-main">Bicentric polygon</span>

In geometry, a bicentric polygon is a tangential polygon which is also cyclic — that is, inscribed in an outer circle that passes through each vertex of the polygon. All triangles and all regular polygons are bicentric. On the other hand, a rectangle with unequal sides is not bicentric, because no circle can be tangent to all four sides.

<span class="mw-page-title-main">Exact trigonometric values</span> Trigonometric values in terms of square roots and fractions

In mathematics, the values of the trigonometric functions can be expressed approximately, as in , or exactly, as in . While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The angles with trigonometric values that are expressible in this way are exactly those that can be constructed with a compass and straight edge, and the values are called constructible numbers.

<span class="mw-page-title-main">Pentagon</span> Shape with five sides

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

In geometry of 4 dimensions or higher, a double pyramid, duopyramid, or fusil is a polytope constructed by 2 orthogonal polytopes with edges connecting all pairs of vertices between the two. The term fusil is used by Norman Johnson as a rhombic-shape. The term duopyramid was used by George Olshevsky, as the dual of a duoprism.

References

  1. 1 2 Arthur Jones, Sidney A. Morris, Kenneth R. Pearson, Abstract Algebra and Famous Impossibilities, Springer, 1991, ISBN   0387976612, p. 178.
  2. Carl Friedrich Gauss "Disquisitiones Arithmeticae" eod books2ebooks, p. 662 item 365.
  3. 1 2 Callagy, James J. "The central angle of the regular 17-gon", Mathematical Gazette 67, December 1983, 290–292.
  4. Duane W. DeTemple "Carlyle Circles and the Lemoine Simplicity of Polygon Constructions" in The 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
  5. Hendricks, J. E. (1877). "Answer to Mr. Heal's Query; T. P. Stowell of Rochester, N. Y." The Analyst: A Monthly Journal of Pure and Applied Mathematicus Vol.1: 94–95. Query, by W. E. Heal, Wheeling, Indiana p. 64; accessdate 30 April 2017
  6. Herbert W. Richmond, description "A Construction for a regular polygon of seventeen side" illustration (Fig. 6), The Quarterly Journal of Pure and Applied Mathematics 26: pp. 206–207. Retrieved 4 December 2015
  7. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN   978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)

Further reading