查看“︁十角星”︁的源代码

{{NoteTA|G1=Math}}
{{Infobox polygon
| name        = 正十角星
| image       = Regular_star_polygon_10-3.svg
| caption    = 正十角星形
| edges       = 10
| schläfli    = 10/3<br>5/3
| coxeter     = {{CDD|node_1|10|rat|d3|node}}<br>{{CDD|node_1|5-3|node_1}}
| symmetry    = [[二面體群]] (D<sub>10</sub>)
| angle       =72
| dual       = 自身對偶
| properties  = [[星形多边形|星形]]、[[外接圓]]、[[等边多边形|等边]]、[[等角圖形|等角]]、[[isotoxal figure|isotoxal]]
}}

'''十角星'''，又稱十芒星，是指一種有十隻尖角，並以十條直線畫成的星星圖形。

==幾何學==
在幾何學中，十角星是邊自我相交的[[十邊形]]。

正十角星只有一種，其[[施萊夫利符號]]為{10/3}，與所述第二數字差別在繪製十角星時頂點間隔數。<ref>{{citation|title=Gems of Geometry|first=John|last=Barnes|publisher=Springer|year=2012|isbn=9783642309649|pages=28–29|url=https://books.google.com/books?id=7YCUBUd-4BQC&pg=PA28|accessdate=2015-08-16|archive-date=2019-06-08|archive-url=https://web.archive.org/web/20190608114208/https://books.google.com/books?id=7YCUBUd-4BQC&pg=PA28|dead-url=no}}.</ref>

正十角星每邊為<math>\sqrt 5 - 2</math>，正十角星各邊的長度比例，以及在每個邊的交叉點比例在以下圖形所示。
[[File:Decagram lengths.svg|320px]]

在幾何學上，只要擁有10個邊、10個角，並可用[[10邊形]]容納的圖形即可稱為十角星，其符號以{10/n}表示。只有{10/3}的十角星為正十角星，但還有三種十角星也可被解釋為正十角星。

*{10/5} 由5個5{2}[[退化多邊形]]組成。
*{10/4} 由兩個2{5/2}[[五角星]]組成。
*{10/2} 由兩個2{5}[[五邊形]]組成。<ref>Regular polytopes, p 93-95, regular star polygons, regular star compounds</ref><ref>Coxeter, Introduction to Geometry, second edition, 2.8 ''Star polygons'' p.36-38</ref>

{| class=wikitable width=360
!形式
!多邊形
!複合多邊形
!星形多邊形
!colspan=2|複合多邊形
|- align=center valign=top
!valign=center|圖形
|BGCOLOR="#ffe0e0"|[[File:Regular polygon 10.svg|120px]]
|BGCOLOR="#e0e0ff"|[[File:Regular star figure 2(5,1).svg|120px]]
|BGCOLOR="#ffe0e0"|[[File:Regular star polygon 10-3.svg|120px]]
|BGCOLOR="#e0e0ff"|[[File:Regular star figure 2(5,2).svg|120px]]
|[[File:Regular star figure 5(2,1).svg|120px]]
|- align=center
!符號
!{10/1} = {10}
|{10/2} = 2{5}
|{10/3}
|{10/4} = 2{5/2}
|{10/5} = 5{2}
|}

==與五角星及五邊形相關性==
十角星與五角星及五邊形有一定的關連性，當五角星或五邊形截斷邊角時，也可創造出十角星。<ref>The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'', {{le|布蘭科·格倫鮑姆|Branko Grünbaum|Grünbaum, B.}}.</ref><ref>*{{Cite journal | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=哈羅德·斯科特·麥克唐納·考克斯特 | last2=Longuet-Higgins | first2=M. S. | last3=Miller | first3=J. C. P. | title=Uniform polyhedra | jstor=91532 | mr=0062446  | year=1954 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=246 |issue=916 | pages=411 | publisher=The Royal Society | doi=10.1098/rsta.1954.0003 | ref=harv}}</ref><ref>Coxeter, The Densities of the Regular polytopes I, p.43 If d is odd, the truncation of the polygon {p/q} is naturally {2n/d}. But if not, it consists of two coincident {n/(d/2)}'s; two, because each side arises from an original side and once from an original vertex. Thus the density of a polygon is unaltered by truncation.</ref>

以下列表列出十角星與五角星及五邊形的關連性。
{| class=wikitable
|+ 十角星擬正多面體與五角星、五邊形的相關性
!擬正多面體
!colspan=2|等角多邊形
!擬正多面體<BR>雙層覆蓋形式
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular polygon truncation 5 1.svg|120px]]<BR>t{5} = {10}
|[[File:regular polygon truncation 5 2.svg|120px]]
|[[File:regular polygon truncation 5 3.svg|120px]]
|BGCOLOR="#e0e0ff"|[[File:Regular star polygon 5-2.svg|120px]]<BR>t{5/4} = {10/4} = 2{5/2}
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular star truncation 5-3 1.svg|120px]]<BR>t{5/3} = {10/3}
|[[File:regular star truncation 5-3 2.svg|120px]]
|[[File:regular star truncation 5-3 3.svg|120px]]
|BGCOLOR="#e0e0ff"|[[File:Regular polygon 5.svg|120px]]<BR>t{5/2} = {10/2} = 2{5}
|}

==應用==
十角星常出現在伊斯蘭教使用的{{le|綺理花磚|Girih tiles}}上。<ref>{{citation|contribution=Polyhedral Modularity in a Special Class of Decagram Based Interlocking Star Polygons|title=Bridges 2012: Mathematics, Music, Art, Architecture, Culture|first=Reza|last=Sarhangi|year=2012|url=http://archive.bridgesmathart.org/2012/bridges2012-165.pdf|pages=165–174|accessdate=2015-08-16|archive-date=2015-02-05|archive-url=https://web.archive.org/web/20150205034431/http://archive.bridgesmathart.org/2012/bridges2012-165.pdf|dead-url=yes}}.</ref>

[[File:Girih tiles.svg|160px]]
==參見==
*[[正圖形列表]]

==參考文獻==
{{Commons category|Ten pointed stars}}
{{reflist}}

{{多邊形}}

[[Category:多邊形]]
[[Category:星形符號]]
[[Category:10]]