查看“︁大三角化八面體”︁的源代码

{{NoteTA|G1=Math}}
{{Infobox polyhedron
| name = 大三角化八面體
| polyhedron = 大三角化八面體
| imagename = DU19_great_triakisoctahedron.png
| Type = [[均勻多面體對偶]]<br/>星形多面體
| Face = 24
| Edge = 36
| Vertice = 14
| Face_type = 24個[[等腰三角形]]<br/>[[File:DU19_facets.png|80px]]
| Vertice_type = 
| Schläfli = 
| Wythoff = 
| Symmetry_group = O<sub>h</sub>, [4,3], *432
| Index_references = DU<sub>19</sub>
| Rotation_group = 
| Properties = 等面、非凸
| dual_image = Stellated_truncated_hexahedron.png
}}
在[[幾何學]]中，'''大三角化八面體'''是一種星形多面體，由24個[[全等]]且互相[[相交]]的[[等腰三角形]]組成，其索引為DU<sub>19</sub>。溫尼爾在他的書中列出將大三角化八面體編為W<sub>92</sub><ref>{{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 = GreatTriakisOctahedron | title = Great Triakis Octahedron}}</ref>。

大三角化八面體的[[對偶多面體]]是[[星形截角立方體]]<ref>{{Cite web | url = http://archive.lib.msu.edu/crcmath/math/math/g/g309.htm | author = Eric W. Weisstein | title = Great Triakis Octahedron, The Dual of the Stellated Truncated Hexahedron. | publisher = [[密西根州立大學]][[圖書館]] | archiveurl = https://web.archive.org/web/20160319132133/http://archive.lib.msu.edu/crcmath/math/math/g/g309.htm | archivedate = 2016-03-19 | access-date = 2016-09-02 | dead-url = no }}</ref>。

== 性質 ==
大三角化八面體由24個[[全等]]且互相相交的[[等腰三角形|等腰銳角三角形]]組成，其共有24個面、36條邊和14個[[頂點 (幾何)|頂點]]，是一種[[二十四面體]]。

=== 構造 ===
[[File:Kleetope of octahedron.gif|thumb|right|八面體加入角錐的幾種變化]]
大三角化八面體的構成是在[[正八面體]]的每個面中加入穿過對面的面的倒角錐而成的<ref>{{cite mathworld| urlname = GreatTriakisOctahedron| title = 大三角化八面體}}</ref>，這種在面上加入倒角錐的做法使其與三角化八面體有一樣的[[拓樸]]結構，幾何上的差異在於，大三角化八面體和三角化八面體一個是向外[[Kleetope|加入角錐]]<ref>{{cite mathworld| urlname = SmallTriakisOctahedron | title = Triakis octahedron }}</ref>、一個是向內加入角錐。

=== 面的組成 ===
大三角化八面體由24個[[全等]]的[[等腰三角形|等腰銳角三角形]]組成，其36條邊中，有12條來自等腰三角形的[[底]]邊、24條來自等腰銳角三角形的腰<ref name="dmccooey.com"/>。
:[[File:DU19_facets.png|200px]]
此等腰三角形在立體中共露出了5個部分，有1個凹八邊形和4個三角形，如圖，以藍色表示。若這等腰三角形長邊為2個單位長，則底邊為<math>2-\sqrt{2}</math><ref name="dmccooey.com">{{cite web |url=http://dmccooey.com/polyhedra/GreatTriakisOctahedron.html |title=Self-Intersecting Truncated Regular Duals: Great Triakis Octahedron |publisher=dmccooey.com |archiveurl=https://web.archive.org/web/20160324081105/http://dmccooey.com/polyhedra/GreatTriakisOctahedron.html |archivedate=2016-03-24 |access-date=2016-09-02 |dead-url=no }}</ref>。

=== 二面角 ===
大三角化八面體的[[二面角]]為<ref name="dmccooey">{{cite web| url = http://dmccooey.com/polyhedra/GreatTriakisOctahedron.html| title = Self-Intersecting Truncated Regular Polyhedra: Great Triakis Octahedron| publisher = dmccooey.com| archiveurl = https://web.archive.org/web/20160312082758/http://dmccooey.com/polyhedra/GreatTriakisOctahedron.html| archivedate = 2016-03-12| access-date = 2016-09-02| dead-url = no}}</ref>：
:<math>\cos^{-1} (\frac{8\sqrt{2}-3}{17}) \approx 1.0598 \approx 60.722 ^{\circ}</math>

=== 頂點坐標 ===
若其[[對偶多面體]][[星形截角立方體]]的邊長為1單位，則大三角化八面體的[[頂點 (幾何)|頂點]][[坐標]]為<ref name="dmccooeydata">{{Cite web|url=http://dmccooey.com/polyhedra/GreatTriakisOctahedron.txt|title=Data of Great Triakis Octahedron|publisher=dmccooey.com|archiveurl=https://web.archive.org/web/20160902152838/http://dmccooey.com/polyhedra/GreatTriakisOctahedron.txt|archivedate=2016-09-02|access-date=2016-09-02|dead-url=no}}</ref>：
:<math>( 0,  0, \pm (\sqrt{2}-1))</math>、
:<math>( \pm (\sqrt{2}-1),  0,  0)</math>、
:<math>( 0, \pm (\sqrt{2}-1),  0)</math>、
:<math>( \pm 1, \pm 1, \pm 1)</math>。

== 相關多面體 ==
=== 對偶複合體 ===
星形截角立方體與其對偶的複合體為'''複合星形截角立方體大三角化八面體'''。其共有38個面、72條邊和38個[[頂點 (幾何)|頂點]]，其[[歐拉示性數]]為4，虧格為-1，有6個非凸面<ref>{{Cite web|url=http://bulatov.org/polyhedra/uniform_compounds/uc24.html|title=compound of stellated truncated hexahedron and great triakisoctahedron|publisher=bulatov.org|archiveurl=https://web.archive.org/web/20150906140734/http://bulatov.org/polyhedra/uniform_compounds/uc24.html|archivedate=2015-09-06|access-date=2016-09-02|dead-url=no}}</ref>。

== 參見 ==
* [[三角化八面體]]

== 參考文獻 ==
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== 外部連結 ==
* {{mathworld | urlname = GreatTriakisOctahedron| title = 大三角化八面體}}
* [http://mathworld.wolfram.com/GreatTriakisOctahedron.html Mathworld - Great Triakis Octahedron] {{Wayback|url=http://mathworld.wolfram.com/GreatTriakisOctahedron.html |date=20201107205627 }}
* [http://mathworld.wolfram.com/SmallTriakisOctahedron.html Mathworld - Small Triakis Octahedron] {{Wayback|url=http://mathworld.wolfram.com/SmallTriakisOctahedron.html |date=20201108055823 }}
* [https://web.archive.org/web/20171110075259/http://gratrix.net/polyhedra/uniform/summary/ Uniform polyhedra and duals]

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