查看“︁大星形截角十二面体”︁的源代码

{{NoteTA|G1=Math}}
{{Infobox polyhedron
| name = 大星形截角十二面体
| polyhedron = 大星形截角十二面体
| imagename = Great stellated truncated dodecahedron.png
| Type = [[星形均勻多面體]]
| Face = 32
| Edge = 90
| Vertice = 60
| Genu = 
| Face_type = 
| Vertice_type = <sup>10</sup>/<sub>3</sub>.<sup>10</sup>/<sub>3</sub>.3
| Vertice_configuration = 20{3}+12{<sup>10</sup>/<sub>3</sub>}
| Schläfli = 
| Wythoff = 2 3 &#124; <sup>5</sup>/<sub>3</sub> 
| Symmetry_group = I<sub>h</sub>, [5,3], *532
| Index_references = [[均勻多面體|U]]<sub>66</sub>, [[哈羅德·斯科特·麥克唐納·考克斯特|C]]<sub>83</sub>, [[溫尼爾多面體模型列表|W]]<sub>104</sub>
| Rotation_group = 
| Dihedral_angle = 
| Properties = 
| 3d_image = 
| Bowers acronym = Quit Gissid
| vfigimage = Great stellated truncated dodecahedron vertfig.png
| dual_image = DU66_great_triakisicosahedron.png
| net_image = 
}}

在[[幾何學]]中，'''大星形截角十二面體'''又稱為星形截角大十二面體是一種由十角星和三角形組成星形多面體，索引為U<sub>66</sub>，[[對偶多面體]]是{{link-en|大三角化二十面體|Great_triakis_icosahedron}}。

== 性質 ==
大星形截角十二面體共有32個面、90條邊和60個[[頂點 (幾何)|頂點]]<ref>{{cite web | url = http://bulatov.org/polyhedra/uniform/u71.html | title = great stellated truncated dodecahedron | publisher = bulatov.org | archiveurl = https://web.archive.org/web/20160326072416/http://bulatov.org/polyhedra/uniform/u71.html | archivedate = 2016-03-26 | access-date = 2016-09-03 | dead-url = no }}</ref>，在其32個面中，有20個正三角形和12個十角星，且每個頂點都是2個[[十角星]]和1個三角形的公共頂點，頂點圖可以用<sup>10</sup>/<sub>3</sub>.<sup>10</sup>/<sub>3</sub>.3表示<ref>{{Cite web | url = https://www.mathconsult.ch/static/unipoly/66.html | title = Uniform Polyhedra 66: great stellated truncated dodecahedron | publisher = mathconsult | archiveurl = https://web.archive.org/web/20160327004949/http://www.mathconsult.ch/static/unipoly/66.html | archivedate = 2016-03-27 | access-date = 2016-09-03 | dead-url = no }}</ref>。在[[施萊夫利符號]]中計為t<sub>0,1</sub>{<sup>5</sup>/<sub>3</sub>,3}、考克斯特記號中可以用{{CDD|node_1|5|rat|d3|node_1|3|node}}表示。

其可以視為在[[正二十面體]]每個面的角落放上[[三角錐]]，類似於將正二十面體進行[[截角 (幾何)|星形截角變換]]，但又與實際上真的進行星形截角變換的[[正二十面體]]不太相同。
=== 頂點座標 ===
邊長為1單位且[[幾何中心]]位於[[原點]]的大星形截角十二面體，其頂點[[座標]]為<ref name="dmccooeydata">{{Cite web | url = http://dmccooey.com/polyhedra/GreatStellatedTruncatedDodecahedron.txt | title = Data of Great Stellated Truncated Dodecahedron | publisher = dmccooey.com | archiveurl = https://web.archive.org/web/20160903160921/http://dmccooey.com/polyhedra/GreatStellatedTruncatedDodecahedron.txt | archivedate = 2016-09-03 | access-date = 2016-09-03 | dead-url = no }}</ref>：
: (0, ±τ, ±(2−1/τ))、
: (±τ, ±1/τ, ±2/τ)、
: (±1/τ<sup>2</sup>, ±1/τ, ±2)
其中，τ為[[黃金比例]]，其值為<math>\frac{1+\sqrt{5}}{2} \approx 1.68</math>，有時會以符號φ表示。

=== 二面角 ===
大星形截角十二面體有2種[[二面角]]，分別為十角星-十角星二面角以及十角星-三角形二面角。十角星-十角星二面角的角度為為[[5的算術平方根|五平方根]][[倒數]]的[[反餘弦]]值<ref>Johann Pitsch, Über Halbreguläre Sternpolyeder,
Zeitschrift für das Realschulwesen 6 (1881), 9-24, 64-65, 72-89, 216.</ref><ref name = "dmccooey GreatStellatedTruncatedDodecahedron">{{cite web | url=http://dmccooey.com/polyhedra/GreatStellatedTruncatedDodecahedron.html | title=Versi-Regular Polyhedra: Great Stellated Truncated Dodecahedron | publisher=dmccooey.com | archiveurl=https://web.archive.org/web/20160324082445/http://dmccooey.com/polyhedra/GreatStellatedTruncatedDodecahedron.html | archivedate=2016-03-24 | access-date=2016-09-03 | dead-url=no }}</ref>：
:<math>\cos^{-1} (\frac{\sqrt{5}}{5}) \approx 1.1071 \approx 63.4349 ^{\circ}</math>
而十角星-三角形二面角的角度為<ref name = "dmccooey GreatStellatedTruncatedDodecahedron"/>：
:<math>\cos^{-1} (\frac{\sqrt{15(5-2\sqrt{5})}}{15}) \approx 1.382 \approx 79.18768 ^{\circ}</math>

== 使用 ==
有一種[[魔術方塊]]使用大星形截角十二面體作為其外型<ref>{{Cite web | url = http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=3516 | title = Quasi Truncated Small Stellated Dodecahedron Rubik's Cube | publisher = twisty puzzles | archiveurl = https://web.archive.org/web/20150906234738/http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=3516 | archivedate = 2015-09-06 | access-date = 2016-09-03 | dead-url = no }}</ref>。

== 相關多面體 ==
{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
| [[Image:Great stellated truncated dodecahedron.png|100px]]<br/>大星形截角十二面體
| [[Image:Small icosicosidodecahedron.png|100px]]<br/>[[小二十面化截半二十面體]]
| [[Image:Small ditrigonal dodecicosidodecahedron.png|100px]]<br/>{{link-en|小二重三角十二面截半二十面體|Small ditrigonal dodecicosidodecahedron}}
| [[Image:Small dodecicosahedron.png|100px]]<br/>[[小十二面二十面體]]
|}

=== 對偶複合體 ===
星形截角立方體與其對偶的複合體為'''複合大星形截角十二面體大三角化二十面體'''。其共有92個面、180條邊和92個頂點，其[[歐拉示性數]]為4，虧格為-1，有12個非凸面<ref>{{Cite web|url=http://bulatov.org/polyhedra/uniform_compounds/uc71.html|title=compound of great stellated truncated dodecahedron and great triakisicosahedron|publisher=bulatov.org|archiveurl=https://web.archive.org/web/20150906135946/http://bulatov.org/polyhedra/uniform_compounds/uc71.html|archivedate=2015-09-06|access-date=2016-09-03|dead-url=no}}</ref>。

== 參見 ==
* [[截角_(幾何)#廣義截角|星形截角]]
* [[正十二面體]]

== 參考文獻 ==
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== 外部連結 ==
* {{mathworld | urlname = GreatStellatedTruncatedDodecahedron| title = 大星形截角十二面体}}
{{從星形正多面體變換的星形均勻多面體}}
{{均勻多面體導航}}
[[Category:星形多面体]]
[[Category:均勻多面體]]
[[Category:非凸多面體]]