十角星

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正十角星
Regular star polygon 10-3.svg
正十角星形
10
頂點 10
施萊夫利符號 10/3
5/3
考克斯特圖 CDel node 1.pngCDel 10.pngCDel rat.pngCDel d3.pngCDel node.png
CDel node 1.pngCDel 5-3.pngCDel node 1.png
對稱群 二面體群 (D10)
內角 72
對偶 自身對偶
特性 星形英语Star polygon外接圓等边英语Equilateral polygon等角英语Isogonal figureisotoxal

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[编辑]

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。[1]

正十角星每邊為,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。 Decagram lengths.svg

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。

形式 多邊形 複合多邊形 星形多邊形 複合多邊形
圖形 Regular polygon 10.svg Regular star figure 2(5,1).svg Regular star polygon 10-3.svg Regular star figure 2(5,2).svg Regular star figure 5(2,1).svg
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[编辑]

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。[4][5][6]

以下列表列出十角星與五角星及五邊形的關連性。

十角星擬正多面體與五角星、五邊形的相關性
擬正多面體 等角多邊形 擬正多面體
雙層覆蓋形式
Regular polygon truncation 5 1.svg
t{5} = {10}
Regular polygon truncation 5 2.svg Regular polygon truncation 5 3.svg Regular star polygon 5-2.svg
t{5/4} = {10/4} = 2{5/2}
Regular star truncation 5-3 1.svg
t{5/3} = {10/3}
Regular star truncation 5-3 2.svg Regular star truncation 5-3 3.svg Regular polygon 5.svg
t{5/2} = {10/2} = 2{5}

應用[编辑]

十角星常出現在伊斯蘭教使用的綺理花磚英语Girih tiles上。[7]

Girih tiles.svg

參見[编辑]

參考文獻[编辑]

  1. ^ Barnes, John, Gems of Geometry, Springer: 28–29, 2012, ISBN 9783642309649 .
  2. ^ Regular polytopes, p 93-95, regular star polygons, regular star compounds
  3. ^ Coxeter, Introduction to Geometry, second edition, 2.8 Star polygons p.36-38
  4. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Grünbaum, B.英语Branko Grünbaum.
  5. ^ *Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. Uniform polyhedra. Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences (The Royal Society). 1954, 246 (916): 411. ISSN 0080-4614. JSTOR 91532. MR 0062446. doi:10.1098/rsta.1954.0003. 
  6. ^ 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.
  7. ^ Sarhangi, Reza, Polyhedral Modularity in a Special Class of Decagram Based Interlocking Star Polygons, Bridges 2012: Mathematics, Music, Art, Architecture, Culture (PDF): 165–174, 2012 .