|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {| class="wikitable" align=right
| | Greetings! I am Dalton. Acting has been a thing that I appreciate totally addicted to. Vermont is just where my home could be described as. Managing professionals has been my big day job for a in addition to but I plan on changing it. I've been working out on my website for the some time now. Find it out here: http://prometeu.net<br><br>Here is my webpage: [http://prometeu.net clash of clans cheats android] |
| |+ '''Set of regular star polygons'''
| |
| |-
| |
| | colspan=2|
| |
| {| class="wikitable"
| |
| |- align=center
| |
| |[[File:Star polygon 5-2.svg|80px]]<br>[[Pentagram|{5/2}]]
| |
| |[[File:Star polygon 7-2.svg|80px]]<br>[[Heptagram|{7/2}]]
| |
| |[[File:Star polygon 7-3.svg|80px]]<br>[[Heptagram|{7/3}]]
| |
| |[[File:Star polygon 8-3.svg|80px]]<br>[[Octagram|{8/3}]]
| |
| |- align=center
| |
| |[[File:Star polygon 9-2.svg|80px]]<br>[[Enneagram (geometry)|{9/2}]]
| |
| |[[File:Star polygon 9-4.svg|80px]]<br>[[Enneagram (geometry)|{9/4}]]
| |
| |[[File:Star polygon 10-3.svg|80px]]<br>[[Decagram (geometry)|{10/3}]]
| |
| |...
| |
| |}
| |
| |-
| |
| ! [[Schläfli symbol]]<br>2<2q<p<br>[[Greatest common divisor|gcd]](p,q)=1
| |
| |{p/q}
| |
| |-
| |
| ! [[Vertex (geometry)|Vertices]] and [[Edge (geometry)|Edges]]
| |
| |p
| |
| |-
| |
| ! [[Density (polygon)|Density]] | |
| |q
| |
| |-
| |
| ! [[Coxeter–Dynkin diagram]]
| |
| |{{CDD|node_1|p|rat|dq|node}}
| |
| |-
| |
| ! [[Symmetry group]]
| |
| |[[Dihedral symmetry|Dihedral]] (D<sub>p</sub>)
| |
| |-
| |
| ! [[Dual polygon]]
| |
| |Self-dual
| |
| |-
| |
| ! [[Internal angle]]<br>([[degree (angle)|degree]]s)
| |
| |<math>\frac{180(p-2q)}{p}</math><ref>{{cite book |last=Kappraff |first=Jay |title=Beyond measure: a guided tour through nature, myth, and number |publisher=World Scientific |year=2002 |page=258 |isbn= 978-981-02-4702-7 |url=http://books.google.com/books?id=vAfBrK678_kC&pg=PA256&dq=star+polygon}}</ref>
| |
| |}
| |
| A regular '''star polygon''' (not to be confused with [[star-shaped polygon]]) is a regular non-convex polygon. Only the regular ones have been studied in any depth; star polygons in general appear not to have been formally defined. They should not be confused with [[star domain]]s.
| |
| | |
| ==Etymology==
| |
| Modern star polygon names are created by combining a [[numeral prefix]], such as ''[[wikt:penta-|penta-]]'', with the Greek suffix ''[[wikt:-gram|-gram]]'' (in this case creating ''[[pentagram]]''). The prefix is normally a Greek [[Cardinal number (linguistics)|cardinal]], but synonyms using other prefixes exist. For example, a nine-pointed polygon is called an ''[[Enneagram (geometry)|enneagram]]'', but is also known as a ''nonagram'', using the [[Ordinal number (linguistics)|ordinal]] ''nona'' from Latin.
| |
| | |
| Although this prefix+suffix formula can be used to create or find star polygon names, it does not necessarily reflect the word's history. For example, ''pentagram'' derives from ''pentagrammos'' / ''pentegrammos'' ("five lines") whose ''-grammos'' derives from ''grammē'' meaning "line". The ''-gram'' suffix, however, derives from ''gramma'' meaning "to write". ''Gramma'' and ''grammē'' are however very similar in sound, writing (γράμμα, γραμμή) and meaning ("written character, letter, that which is drawn", "stroke or line of a pen<ref>[http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dgrammh%2F γραμμή], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref>"), and are possibly [[cognate]]s.
| |
| | |
| ==Regular star polygons==
| |
| In [[geometry]], a "regular star polygon" is a self-intersecting, equilateral equiangular [[polygon]], created by connecting one [[vertex (geometry)|vertex]] of a simple, regular, ''p''-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again.<ref>{{cite book |last=Coxeter |first=Harold Scott Macdonald |title=Regular polytopes |publisher=Courier Dover Publications |year=1973 |isbn=978-0-486-61480-9}}</ref> Alternatively for integers ''p'' and ''q'', it can be considered as being constructed by connecting every ''q''th point out of ''p'' points regularly spaced in a circular placement.<ref>{{MathWorld |urlname=StarPolygon |title=Star Polygon}}</ref> For instance, in a regular pentagon, a five-pointed star can be obtained by drawing a line from the first to the third vertex, from the third vertex to the fifth vertex, from the fifth vertex to the second vertex, from the second vertex to the fourth vertex, and from the fourth vertex to the first vertex. The notation for such a polygon is {''p''/''q''} (''see [[Schläfli symbol]]''), which is equal to {''p''/''p-q''}. Regular star polygons will be produced when ''p'' and ''q'' are [[coprime|relatively prime]] (they share no factors). A regular star polygon can also be represented as a sequence of [[stellation]]s of a convex regular ''core'' polygon. Regular star polygons were first studied systematically by [[Thomas Bradwardine]].
| |
| | |
| ===Examples===
| |
| {| class="wikitable"
| |
| |align=center colspan=7|[[File:Regular Star Polygons.jpg|640px]]
| |
| |}
| |
| | |
| ===Star figures===<!--This section is linked from [[Polyhedral compound]]-->
| |
| [[File:Star polygon 6-2.svg|100px|thumb|Star figure<br>[[hexagram]]<br>2{3} or {6/2}]]
| |
| [[File:Star polygon 9-3.svg|100px|thumb|Star figure<br>[[enneagram (geometry)|enneagram]]<br>3{3} or {9/3}]]
| |
| If the number of sides ''n'' is divisible by ''m'', the star polygon obtained will be a regular polygon with ''n''/''m'' sides. A new figure is obtained by rotating these regular ''n''/''m''-gons one vertex to the left on the original polygon until the number of vertices rotated equals ''n''/''m'' minus one, and combining these figures. An extreme case of this is where ''n''/''m'' is 2, producing a figure consisting of ''n''/2 straight line segments; this is called a "[[Degeneracy (mathematics)|degenerate]] star polygon".
| |
| | |
| In other cases where ''n'' and ''m'' have a common factor, a star polygon for a lower ''n'' is obtained, and rotated versions can be combined. These figures are called "star figures" or "improper star polygons" or "compound polygons". The same notation {''n''/''m''} is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form ''k''{''n''} as being more correct, where usually ''k'' = ''m''.
| |
| | |
| A further complication comes when we compound two or more star polygons, as for example two pentagrams, differing by a rotation of 36°, inscribed in a decagon. This is correctly written in the form ''k''{''n''/''m''}, as 2{5/2}, rather than the commonly used {10/4}.
| |
| | |
| A six-pointed star, like a hexagon, can be created using a compass and a straight edge:
| |
| *Make a circle of any size with the compass.
| |
| *Without changing the radius of the compass, set its pivot on the circle's circumference, and find one of the two points where a new circle would intersect the first circle.
| |
| *With the pivot on the last point found, similarly find a third point on the circumference, and repeat until six such points have been marked.
| |
| *With a straight edge, join alternate points on the circumference to form two overlapping equilateral triangles.
| |
| | |
| ===Symmetry===
| |
| Regular star polygons and star figures can be thought of as diagramming [[coset]]s of the [[subgroup]]s <math>x\mathbb{Z}_n</math> of the [[finite group]] <math>\mathbb{Z}_n</math>.
| |
| | |
| The [[symmetry group]] of {''n''/''k''} is [[dihedral group]] ''D''<sub>n</sub> of order 2''n'', independent of ''k''.
| |
| | |
| ==Irregular star polygons==
| |
| [[File:Great retrosnub icosidodecahedron vertfig.png|280px|thumb|The white line in this graph is an irregular pentagonal cyclic polygon, defining the a [[vertex figure]] for the [[great retrosnub icosidodecahedron]]. The edge lengths are defined by the distance between alternate vertices in the faces of the [[uniform polyhedron]].]]
| |
| A star polygon need not be regular. Irregular [[Cyclic polygon|cyclic]] star polygons occur as [[Vertex (geometry)|vertex]] figures for the [[uniform polyhedra]], defined by the sequence of regular polygon faces around each vertex, allowing for both multiple turns, and retrograde directions. (See vertex figures at [[List of uniform polyhedra]])<ref>[[H. S. M. Coxeter]], [[M. S. Longuet-Higgins]], [[J. C. P. Miller]], ''Uniform polyhedra'', Phil. Trans. 1954 (Tables 6-8)</ref>
| |
| | |
| The [[Final_stellation_of_icosahedron#As_a_star_polyhedron|Final stellation of icosahedron]] can be seen as a polyhedron with irregular {9/4} star polygon faces with Dih<sub>3</sub> [[dihedral symmetry]].
| |
| :[[File:Enneagram 9-4 icosahedral.svg|120px]] | |
| | |
| The [[unicursal hexagram]] is another example of a cyclic irregular star polygon, containing Dih<sub>2</sub> [[dihedral symmetry]].
| |
| : [[File:Solid unicursal hexagram.svg|120px]]
| |
| | |
| ==Interiors of star polygons==
| |
| Star polygons leave an ambiguity of interpretation for interiors. This diagram demonstrates three ''interpretations'' of a pentagram.
| |
| | |
| [[File:Pentagram interpretations.svg|500px]]
| |
| *The left-hand interpretation has the 5 vertices of a regular pentagon connected alternately on a cyclic path, skipping alternate vertices. The interior is everything immediately left (or right) from each edge (until the next intersection). This makes the core convex pentagonal region actually "outside", and in general you can determine inside by a binary [[even-odd rule]] of counting how many edges are intersected from a point along a ray to infinity.
| |
| *The middle interpretation also has the 5 vertices of a regular pentagon connected alternately on a cyclic path. The interior may be treated either:
| |
| **as the inside of a simple 10-sided polygon perimeter boundary, as below.
| |
| **with the central convex pentagonal region surrounded twice, because the starry perimeter winds around it twice.
| |
| *The right-hand interpretation creates new vertices at the intersections of the edges (5 in this case) and defines a new concave decagon (10-pointed polygon) formed by perimeter path of the middle interpretation; it is in fact no longer a pentagram.
| |
| | |
| What is the area inside the pentagram? Each interpretation leads to a [[Polygon#Area and centroid|different answer]].
| |
| | |
| ===Example interpretations of a star prism===
| |
| '''[[Heptagrammic prism (7/2)|{7/2} heptagrammic prism]]:'''
| |
| {| class="wikitable"
| |
| |[[File:Septagram prism-2-7.png|150px]]<br>Heptagrams with<br>''2-sided'' interior
| |
| |[[File:Heptagrammic prism 7-2.png|150px]]<br>Heptagrams with<br>a simple perimeter interior
| |
| |}
| |
| | |
| The heptagrammic prism above shows different interpretations can create very different appearances.
| |
| | |
| Builders of [[polyhedron model]]s, like [[List of Wenninger polyhedron models|Magnus Wenninger]], usually represent ''star polygon'' faces in the concave form, without ''internal edges'' shown.
| |
| | |
| ==Star polygons in art and culture==
| |
| Star polygons feature prominently in art and culture. Such polygons may or may not be [[regular polygon|regular]] but they are always highly [[symmetrical]]. Examples include:
| |
| *The {5/2} star pentagon is also known as a [[pentagram]], pentalpha or pentangle, and historically has been considered by many [[Magic (paranormal)|magic]]al and [[religious]] cults to have [[occult]] significance.
| |
| *The simplest compound star polygon is two opposed triangles, sometimes written as {6/2} and known variously as the [[hexagram]], ([[Star of David]] or [[Seal of Solomon]]).
| |
| *The {7/3} and {7/2} star polygons which are known as [[heptagram]]s and also have occult significance, particularly in the [[Kabbalah]] and in [[Wicca]].
| |
| *The compound of two squares, sometimes written as {8/2}, is known in [[Hinduism]] as the [[Star of Lakshmi]] and in [[Islam]] as the [[Rub el Hizb]].
| |
| *The {8/3} star polygon ([[octagram]]), and the compound {16/6} are frequent geometrical motifs in [[Mughal Empire|Mughal]] [[Islamic art history|Islamic art]] and [[Islamic architecture|architecture]]; the first is on the [[emblem of Azerbaijan]].
| |
| *An eleven pointed star called the [[hendecagram]] was used on the tomb of Shah Nemat Ollah Vali.
| |
| | |
| {|
| |
| |- valign=top
| |
| |[[File:Octagram.svg|thumb|125px|right|An {8/3} star polygon (octagram) constructed in an octagon]]
| |
| |[[File:Seal of Solomon (Simple Version).svg|thumb|125px|Seal of Solomon ([[interlaced]] hexagram, with circle and dots)]]
| |
| |}
| |
| | |
| ==See also==
| |
| *[[Complex polygon]]
| |
| *[[List of regular polytopes#Two Dimensions 2|List of regular polytopes – Nonconvex forms (2D)]]
| |
| *[[Magic star]]
| |
| *[[Star polyhedron]]
| |
| *[[Star polychoron]] (4-polytopes)
| |
| *[[Star-shaped polygon]]
| |
| *[[Stellation#Stellated polygons]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| *Cromwell, P.; ''Polyhedra'', CUP, Hbk. 1997, ISBN 0-521-66432-2. Pbk. (1999), ISBN 0-521-66405-5. p.175
| |
| *[[Branko Grünbaum|Grünbaum, B.]] and G.C. Shephard; ''Tilings and Patterns'', New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1.
| |
| *Grünbaum, B.; Polyhedra with Hollow Faces, ''Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993)'', ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43–70.
| |
| *[[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)
| |
| | |
| ==External links==
| |
| *{{Mathworld |urlname=Polygram |title=Polygram}}
| |
| *[http://public.beuth-hochschule.de/~meiko/applets/star1.html Star Polygons – java applet]
| |
| | |
| {{Polygons}}
| |
| | |
| {{DEFAULTSORT:Star Polygon}}
| |
| [[Category:Polygons]]
| |
| [[Category:Star symbols]]
| |