查看“︁大星形五角化十二面體”︁的源代码

{{NoteTA|G1=Math}}
{{Infobox polyhedron
| name = 大星形五角化十二面體
| polyhedron = 大星形五角化十二面體
| imagename = DU55 great stellapentakisdodecahedron.png
| Type = [[均勻多面體對偶]]<br/>星形多面體
| Face = 60
| Edge = 90
| Vertice = 32
| Face_type = 60個鈍角[[等腰三角形]]<br/>[[File:DU55_facets.png|90px]]
| Vertice_type = 
| Schläfli = 
| Wythoff = 
| Symmetry_group = I<sub>h</sub>, [5,3], *532
| Index_references = DU<sub>55</sub>
| Rotation_group = 
| Properties = 等面、非凸
| dual_image = Great_truncated_icosahedron.png
}}
在[[幾何學]]中，'''大星形五角化十二面體'''是一種[[星形多面體]]，由60個互相相交的鈍角等腰三角形組成，在[[均勻多面體]]中，其索引編號為DU<sub>55</sub><ref name="Magnus J. Wenninger">{{cite book | author = {{link-en|馬格努斯·J·溫尼爾|Magnus J. Wenninger|Wenninger, Magnus}} | title = Polyhedron Models | publisher = Cambridge University Press | year = 1974 | isbn = 0-521-09859-9 }}</ref><ref name="math world">{{cite mathworld| urlname =  GreatStellapentakisDodecahedron | title = Great Stellapentakis Dodecahedron}}</ref>，[[對偶多面體]]為[[截角大二十面體]]<ref>{{Cite web |url=http://archive.lib.msu.edu/crcmath/math/math/g/g304.htm |author=Eric W. Weisstein |title=Great Stellapentakis Dodecahedron is The Dual Polyhedron of the Great Truncated Icosahedron. |publisher=[[密西根州立大學]][[圖書館]] |archiveurl=https://web.archive.org/web/20140713032802/http://archive.lib.msu.edu/crcmath/math/math/g/g304.htm |date=1999-05-25 |archivedate=2014-07-13 |access-date=2016-09-03 |dead-url=no }}</ref>。

== 性質 ==
大星形五角化十二面體由60個面、90條邊和32個[[頂點 (幾何)|頂點]]組成<ref name = bulatov.org>{{Cite web | url = http://bulatov.org/polyhedra/dual/ud60.html | title = great stellapentakisdodecahedron | publisher = bulatov.org | archiveurl = https://web.archive.org/web/20150906122726/http://bulatov.org/polyhedra/dual/ud60.html | archivedate = 2015-09-06 | access-date = 2016-09-03 | dead-url = no }}</ref>，是一種[[六十面體]]。其具有互相相交的面，是一種[[複雜多面體]]，但其僅有面互相相交，其所有面都是凸多邊形<ref name=bulatov.org/>。

=== 面的組成 ===
大星形五角化十二面體的面由60個[[全等]]的[[等腰三角形|等腰鈍角三角形]]組成，每個等腰頓角三角形彼此互相[[相交]]，每個等腰三角形皆露出兩個頓角三角形部分，其餘皆隱藏於該立體的內部。露在該立體外部的部分如下圖，以藍色表示，其中黑線代表等腰頓角三角形彼此互相[[相交]]的位置：
:[[Image:DU55_facets.png|150px]]

大星形五角化十二面體一共有兩種邊長，分別為[[等腰三角形]]的[[底邊]]，和等腰三角形的腰長。
若其[[對偶多面體]]截角大二十面體的邊長為單位長，則[[等腰三角形]]的腰長為<ref name="dmccooey GreatStellapentakisDodecahedron"/>：
:<math>\frac{9(1+2\sqrt{5})}{19} \approx 2.592 </math>

在同樣情況下，等腰三角形的[[底|底邊]]長為<ref name="dmccooey GreatStellapentakisDodecahedron"/>：
:<math>\frac{3(1+\sqrt{5})}{2} \approx 4.8541 </math>

=== 二面角 ===
大星形五角化十二面體的[[二面角]]僅有一種，其值為<ref name="dmccooey GreatStellapentakisDodecahedron">{{cite web| url = http://dmccooey.com/polyhedra/GreatStellapentakisDodecahedron.html| title = Self-Intersecting Truncated Regular Duals: Great Stellapentakis Dodecahedron| publisher = dmccooey.com| archiveurl = https://web.archive.org/web/20160324083933/http://dmccooey.com/polyhedra/GreatStellapentakisDodecahedron.html| archivedate = 2016-03-24| access-date = 2016-09-03| dead-url = no}}</ref>：
:<math>\cos^{-1} (-\frac{80-9\sqrt{5}}{109}) \approx 2.15234 \approx 123.32 ^{\circ}</math>

=== 頂點座標 ===
若一個大星形五角化十二面體[[幾何中心]]位於原點，且其經對偶變換後的立體邊長為1單位長，則其頂點[[座標]]為<ref name="dmccooeydata">{{Cite web|url=http://dmccooey.com/polyhedra/GreatStellapentakisDodecahedron.txt|title=Data of Great Stellapentakis Dodecahedron|publisher=dmccooey.com|archiveurl=https://web.archive.org/web/20161001050400/http://dmccooey.com/polyhedra/GreatStellapentakisDodecahedron.txt|archivedate=2016-10-01|access-date=2016-10-01|dead-url=no}}</ref>：
:<math>( \pm \frac{3(1+\sqrt{5})}{4},  0,  \pm \frac{3(\sqrt{5}-1)}{4})</math>、
:<math>( 0,  \pm \frac{3(\sqrt{5}-1)}{4},  \pm \frac{3(1+\sqrt{5})}{4})</math>、
:<math>( \pm \frac{3(\sqrt{5}-1)}{4},  \pm \frac{3(1+\sqrt{5})}{4},  0)</math>、
:<math>( 0,  \pm \frac{9(9-\sqrt{5})}{76},  \pm \frac{9(5\sqrt{5}-7)}{76})</math>、
:<math>( \pm \frac{9(9-\sqrt{5})}{76},  \pm \frac{9(5\sqrt{5}-7)}{76},  0)</math>、
:<math>( \pm \frac{9(5\sqrt{5}-7)}{76},  0,  \pm \frac{9(9-\sqrt{5})}{76})</math>、
:<math>(\pm \frac{3}{2}, \pm \frac{3}{2}, \pm \frac{3}{2})</math>。

== 相關多面體 ==
=== 對偶複合體 ===
對偶複合體，即一個多面體與其[[對偶多面體]]組合成的複合圖形。大星形五角化十二面體與其對偶的複合體為'''複合截角大二十面體大星形五角化十二面體'''。其共有92個面、180條邊和92個頂點，其尤拉示性數為4，虧格為-1，有12個非凸面<ref>{{Cite web|url=http://bulatov.org/polyhedra/uniform_compounds/uc60.html|title=compound of great truncated icosahedron and great stellapentakisdodecahedron|publisher=bulatov.org|archiveurl=https://web.archive.org/web/20150906143731/http://bulatov.org/polyhedra/uniform_compounds/uc60.html|archivedate=2015-09-06|access-date=2016-09-03|dead-url=no}}</ref>。

== 參見 ==
*[[大星形十二面體]]

== 參考文獻 ==
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== 外部連結 ==
*{{mathworld | urlname = GreatStellapentakisDodecahedron | title = 大星形五角化十二面體}}

[[Category:多面體]]
[[Category:星形多面体]]
[[Category:均勻多面體對偶]]