查看“︁大六角二十四面體”︁的源代码

{{NoteTA
|G1=Math
|1=zh:鷂形;zh-cn:筝形;zh-hk:鳶形;zh-tw:鳶形;
}}
{{Infobox polyhedron
| name = 大六角二十四面體
| polyhedron = 大六角二十四面體
| imagename = DU14_great_hexacronic_icositetrahedron.png
| Type = [[均勻多面體對偶]]<br/>星形多面體
| Face = 24
| Edge = 48
| Vertice = 20
| Genu = 3
| Face_type = 24個[[鳶形]]<br/>[[File:DU14 facets.png|80px]]
| Vertice_type = 
| Vertice_configuration = 兩種[[頂點 (幾何)|頂點]]<br/>{{nowrap|3個[[鳶形]]的公共頂點}}<br/>{{nowrap|4個鳶形的公共頂點}}
| Schläfli = 
| Wythoff = 
| Symmetry_group = O<sub>h</sub>, [4,3], *432
| Index_references = DU<sub>14</sub>
| Rotation_group = 
| Properties = 等面、非凸
| dual_image = Great_cubicuboctahedron.png
}}
在[[幾何學]]中，'''大六角二十四面體'''是一種[[星形多面體]]，由24個互相相交的[[鷂形]]組成，其索引編號為DU<sub>14</sub>。大六角二十四面體的[[對偶多面體]]為[[大立方截半立方體]]<ref>{{Cite web | url = https://archive.lib.msu.edu/crcmath/math/math/g/g275.htm | author = Eric W. Weisstein | title = Great Hexacronic Icositetrahedron is The Dual of the Great Cubicuboctahedron | publisher = [[密西根州立大學]][[圖書館]] | archiveurl = https://web.archive.org/web/20140711085213/http://archive.lib.msu.edu/crcmath/math/math/g/g275.htm | archivedate = 2014-07-11 | access-date = 2016-09-01 | dead-url = no }}</ref>，與其對偶不同之處在於，其對偶有6個非凸多邊形面<ref>{{Cite web | url = http://bulatov.org/polyhedra/uniform/u19.html | title = great cubicuboctahedron, dual of great hexacronic icositetrahedron | publisher = bulatov.org | archiveurl = https://web.archive.org/web/20160326145638/http://bulatov.org/polyhedra/uniform/u19.html | archivedate = 2016-03-26 | access-date = 2016-09-01 | dead-url = no }}</ref>，而大六角二十四面體的面全部都是[[凸多邊形]]。

== 性質 ==
大六角二十四面體共有24個面、48條邊和20個[[頂點 (幾何)|頂點]]<ref name="bulatov data">{{Cite web | url = http://bulatov.org/polyhedra/dual/ud19.html | title = great hexacronic icositetrahedron | archiveurl = https://web.archive.org/web/20160328043120/http://bulatov.org/polyhedra/dual/ud19.html | archivedate = 2016-03-28 | publisher = bulatov.org | access-date = 2016-09-01 | dead-url = no }}</ref>，是一種[[二十四面體]]。

=== 面的組成 ===
大六角二十四面體由24個[[全等]]的[[鳶形]]組成。每個鳶形皆露出兩個頓角兩個部分，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：
:[[File:DU14_facets.png|100px]]

=== 頂角性質 ===
大六角二十四面體有三種頂角，一種是由4個鳶形面構成的四面角，為最外側的向外尖角，對應對偶多面體大立方截半立方體中的正方形面，共有6個；另一種是由3個面構成的三面角，為較內側、未隱沒於圖形內部的向外尖角，對應對偶多面體中的正三角形面，共有8個；還有一種是由8個面構成的八面角，其8個面互相相交，其[[頂點圖]]為施萊夫利符號計為{8/3}的八角星，且引麼於整個立體圖形的內側，對應對偶多面體中的八角星面，整個立體中共有6個這樣的頂點<ref name="bulatov data"/><ref name="dmccooey_info">{{Cite web | url = http://dmccooey.com/polyhedra/GreatCubicuboctahedron.html | title = Self-Intersecting Quasi-Quasi-Regular Polyhedra: Great Cubicuboctahedron | publisher = dmccooey.com | archiveurl = https://web.archive.org/web/20160324084419/http://dmccooey.com/polyhedra/GreatCubicuboctahedron.html | archivedate = 2016-03-24 | access-date = 2017-06-28 | dead-url = no }}</ref><ref>[[哈罗德·斯科特·麦克唐纳·考克斯特|H. S. M. Coxeter]], ''Regular Polytopes'', Hbk (1948), ppbk (1973).</ref>。

=== 頂點座標 ===
幾何中心位於原點且經過[[對偶|對偶變換]]後的像邊長為單位長的大六角二十四面體，其[[頂點 (幾何)|頂點]][[座標]]為<ref name="dmccooeydata">{{Cite web | url = http://dmccooey.com/polyhedra/GreatHexacronicIcositetrahedron.txt | title = Data of Great Hexacronic Icositetrahedron | publisher = dmccooey.com | archiveurl = https://web.archive.org/web/20160901012456/http://dmccooey.com/polyhedra/GreatHexacronicIcositetrahedron.txt | archivedate = 2016-09-01 | access-date = 2016-09-01 | dead-url = no }}</ref>：
:<math>(0, 0, \pm (2 - \sqrt{2}))</math>、
:<math>(\pm (2 - \sqrt{2}), 0, 0)</math>、
:<math>(0, \pm (2 - \sqrt{2}), 0)</math>、
:<math>(0, 0, \pm \sqrt{2})</math>、
:<math>( \pm \sqrt{2}, 0, 0)</math>、
:<math>(0,  \pm \sqrt{2}, 0)</math>、
:<math>(\pm \frac{4 - \sqrt{2}}{7}, \pm \frac{4 - \sqrt{2}}{7}, \pm \frac{4 - \sqrt{2}}{7})</math>。

=== 二面角 ===
大六角二十四面體的[[二面角]]為<ref>{{cite web | url = http://dmccooey.com/polyhedra/GreatHexacronicIcositetrahedron.html | title = Versi-Regular Polyhedra: Octahemioctahedron | publisher = dmccooey.com | archiveurl = https://web.archive.org/web/20160324084300/http://dmccooey.com/polyhedra/GreatHexacronicIcositetrahedron.html | archivedate = 2016-03-24 | access-date = 2016-09-01 | dead-url = no }}</ref>：
:<math>\cos^{-1} (-\frac{7-4\sqrt{2}}{17}) \approx 1.64988733 \approx 94.5315 ^{\circ}</math>

== 對偶多面體 ==
[[File:Great_cubicuboctahedron.png|thumb|大六角二十四面體的對偶多面體。]]
{{main|大立方截半立方體}}
大六角二十四面體的對偶多面體是[[大立方截半立方體]]<ref>{{cite mathworld| urlname = GreatHexacronicIcositetrahedron |title= Great Hexacronic Icositetrahedron}}</ref>，其在非凸均勻多面體被編號為U<sub>14</sub>。其在威佐夫符號中可以用3 4 | <sup>4</sup>/<sub>3</sub>或4 <sup>3</sup>/<sub>2</sub> | 4表示。

== 相關多面體 ==
大六角二十四面體與[[反平行四邊形二十四面體]]幾何中心重合可以組成一個[[大鳶形二十四面體]]<ref>{{cite web|url = http://www.software3d.com/GreatHexIcositetra.php|title = Great Hexacronic Icositetrahedron|publisher = software3d.com|archiveurl = https://web.archive.org/web/20151121153312/http://www.software3d.com/GreatHexIcositetra.php|archivedate = 2015-11-21|access-date = 2016-09-01|dead-url = no}}</ref>。
{| class="wikitable" width="300"
|- align=center
| [[Image:DU14_great_hexacronic_icositetrahedron.png|120px]]<br/>大六角二十四面體
| [[Image:DU21_great_rhombihexacron.png|120px]]<br/>[[反平行四邊形二十四面體]]
| [[Image:DU17_great_strombic_icositetrahedron.png|120px]]<br/>大鳶形二十四面體
|}

=== 對偶複合體 ===
大六角二十四面體與其對偶的複合體為'''複合大立方截半立方體大六角二十四面體'''。其共有44個面、96條邊和44個頂點，其[[歐拉特徵數|尤拉示性數]]為-8，虧格為5，具有6個非凸面<ref>{{Cite web|url=http://bulatov.org/polyhedra/uniform_compounds/uc19.html|title=compound of great cubicuboctahedron and great hexacronic icositetrahedron|publisher=bulatov.org|archiveurl=https://web.archive.org/web/20150906135651/http://bulatov.org/polyhedra/uniform_compounds/uc19.html|archivedate=2015-09-06|access-date=2016-09-01|dead-url=no}}</ref>。

== 參見 ==
* [[大立方截半立方體]]

== 參考文獻 ==
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== 外部連結 ==
* {{mathworld | urlname = GreatHexacronicIcositetrahedron| title = 大六角二十四面體}}

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