|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{no footnotes|date=December 2011}}
| | ahead of, relies upon sony vegas master 12 break.<br><br>My blog post - Sony Vegas Pro 12 - [http://sonyvegaspro12.info/ http://sonyvegaspro12.info], |
| {{Odd polygon db|Odd polygon stat table|p7}}
| |
| In [[geometry]], a '''heptagon''' is a [[polygon]] with seven sides and seven [[angle]]s. In a [[regular polygon|regular]] heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5π/7 [[radian]]s, 128.5714286 [[degree (angle)|degree]]s. Its [[Schläfli symbol]] is {7}. The area (A) of a regular heptagon of side length ''a'' is given by
| |
| :<math>A = \frac{7}{4}a^2 \cot \frac{\pi}{7} \simeq 3.633912444 a^2.</math>
| |
| | |
| The heptagon is also occasionally referred to as the '''septagon''', using "sept-" (an elision of ''[[Wikt:septua-|septua-]]'', a [[Latin]]-derived [[numerical prefix]], rather than ''[[Wikt:hepta-|hepta-]]'', a [[Greek language|Greek]]-derived numerical prefix) together with the Greek suffix "-agon" meaning angle).
| |
| | |
| ==Construction==
| |
| A regular heptagon is not [[Constructible polygon|constructible]] with [[compass and straightedge]] but is constructible with a marked [[ruler]] and compass. This type of construction is called a [[neusis construction]]. It is also constructible with compass, straightedge and angle trisector. The impossibility of straightedge and compass construction follows from the observation that <math>\scriptstyle {2\cos{\tfrac{2\pi}{7}} \approx 1.247}</math> is a zero of the [[irreducible polynomial|irreducible]] [[cubic function|cubic]] {{nowrap|''x''<sup>3</sup> + ''x''<sup>2</sup> − 2''x'' − 1}}. Consequently this polynomial is the [[minimal polynomial (field theory)|minimal polynomial]] of 2cos({{frac|2π|7}}), whereas the degree of the minimal polynomial for a [[constructible number]] must be a power of 2.
| |
| | |
| {| class=wikitable width=480
| |
| |[[Image:Neusis-heptagon.png|240px]]<br>A ''neusis construction'' of the interior angle in a regular heptagon.
| |
| |[[File:Heptagonneusis.gif|240px]]<br>A ''neusis construction'' of the interior angle in a regular heptagon. (method by [[John Horton Conway]]
| |
| |-
| |
| |colspan=2|[[File:Approximated Heptagon Inscribed in a Circle.gif]]<br>An animation of an approximate compass-and-straightedge construction of a regular heptagon.
| |
| |}
| |
| | |
| ===Approximation===
| |
| A decent approximation for practical use with an error of ≈0.2% is shown in the drawing. Let ''A'' lie on the circumference of the circumcircle. Draw arc ''BOC''. Then <math>\scriptstyle {BD = {1 \over 2}BC}</math> gives an approximation for the edge of the heptagon.
| |
| | |
| [[Image:7-gone_approx.png|240px]]
| |
| | |
| ===A more accurate approximation===
| |
| A regular heptagon with sides <math>\scriptstyle {S = 3\tfrac{2}{11}}</math> can be inscribed in a circle of the diameter <math>\scriptstyle {R = 3\tfrac{2}{3}}</math> with an error of less than 0.00013%.
| |
|
| |
| This follows from a rational approximation of <math>\scriptstyle {\tfrac{S}{R} =\ 2 \sin{\tfrac{\pi}{7}} \approx 1-(\tfrac{4}{11})^2}</math>.
| |
| | |
| =={{anchor|Star Heptagons}} Star heptagons==
| |
| Two kinds of [[heptagram|star heptagons]] can be constructed from regular heptagons, labeled by ''Schläfli symbols'' {7/2}, and {7/3}, with the [[divisor]] being the interval of connection.
| |
| | |
| [[Image:Heptagrams.svg|200px]]<br>Blue, {7/2} and green {7/3} star heptagons inside a red heptagon.
| |
| | |
| ==Uses==
| |
| The [[United Kingdom]] currently (2011) has two heptagonal [[coin]]s, the [[british coin Fifty Pence|50p]] and [[british coin Twenty Pence|20p]] pieces, and the [[Barbados]] Dollar is also heptagonal. The 20-eurocent coin has cavities placed similarly. Strictly, the shape of the coins is a [[Reuleaux heptagon]], a [[curvilinear]] heptagon to make them [[Curve of constant width|curves of constant width]]: the sides are curved outwards so that the coin will roll smoothly in [[vending machine]]s. [[Botswana pula]] coins in the denominations of 2 Pula, 1 Pula, 50 Thebe and 5 Thebe are also shaped as equilateral-curve heptagons. Coins in the shape of Reuleaux heptagons are in circulation in Mauritius, U.A.E., Tanzania, Samoa, Papua New Guinea, São Tomé and Príncipe, Haiti, Jamaica, Liberia, Ghana, the Gambia, Jordan, Jersey, Guernsey, Isle of Man, Gibraltar, Guyana, Solomon Islands, Falkland Islands and Saint Helena. The 1000 [[Zambian Kwacha|Kwacha]] coin of Zambia is a true heptagon.
| |
| | |
| The [[Brazil]]ian 25-cent coin has a heptagon inscribed in the coin's disk. Some old versions of the [[Coat of arms of Georgia (country)|coat of arms of Georgia]], including in [[Georgian Soviet Socialist Republic|Soviet days]], used a {7/2} heptagram as an element.
| |
| | |
| In architecture, heptagonal floor plans are very rare. A remarkable example is the [[Mausoleum of Prince Ernst]] in [[Stadthagen]], [[Germany]].
| |
| | |
| Apart from the [[heptagonal prism]] and [[heptagonal antiprism]], no convex polyhedron made entirely out of regular polygons contains a heptagon as a face.
| |
| | |
| Regular heptagons can tile the [[Hyperbolic geometry|hyperbolic plane]], as shown in this [[Poincaré disk model]] projection:
| |
| :[[File:Uniform tiling 73-t0.png|250px]]<br>[[Order-3 heptagonal tiling|heptagonal tiling]]
| |
| | |
| ==Graphs==
| |
| The K<sub>7</sub> [[complete graph]] is often drawn as a ''regular heptagon'' with all 21 edges connected. This graph also represents an [[orthographic projection]] of the 7 vertices and 21 edges of the [[6-simplex]]. The 21 and 35 vertices of the rectified and birectified 6-simplex also orthogonally project into regular heptagons.
| |
| | |
| {| class=wikitable
| |
| |- align=center
| |
| |[[File:6-simplex_t0.svg|150px]]<BR>[[6-simplex]] (6D)
| |
| |[[File:6-simplex_t1.svg|150px]]<BR>[[Rectified 6-simplex]] (6D)
| |
| |[[File:6-simplex_t2.svg|150px]]<BR>[[Birectified 6-simplex]] (6D)
| |
| |}
| |
| | |
| ==Heptagon in Natural Structures==
| |
| {| class=wikitable
| |
| |[[File:Get_the_point_(301733819).jpg|thumb|left|<center> C a c t u s </center>]]
| |
| |}
| |
| | |
| | |
| | |
| ==See also==
| |
| {{wiktionarypar|Heptagon}}
| |
| * [[Heptagram]]
| |
| | |
| ==External links==
| |
| *[http://www.mathopenref.com/heptagon.html Definition and properties of a heptagon] With interactive animation
| |
| *{{MathWorld|title=Heptagon|urlname=Heptagon}}
| |
| *[http://toon.macharis.be/heptagon/ Another approximate construction method]
| |
| *[http://www.coolmath.com/reference/polygons-07-heptagons.html Polygons – Heptagons]
| |
| *[http://precedings.nature.com/documents/2153/version/1 Recently discovered and highly accurate approximation for the construction of a regular heptagon.]
| |
| [[Category:Polygons]]
| |
| [[Category:Elementary shapes]]
| |
| | |
| {{Polygons}}
| |
ahead of, relies upon sony vegas master 12 break.
My blog post - Sony Vegas Pro 12 - http://sonyvegaspro12.info,