查看“︁大鳶形二十四面體”︁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 =大鳶形二十四面體
| polyhedron = 大鳶形二十四面體
| imagename =DU17 great strombic icositetrahedron.png
| Type = [[均勻多面體對偶]]<br/>[[星形多面體]]
| Face = 24
| Edge = 48
| Vertice = 26
| Face_type = 24個凹鳶形<br/>[[File:DU17 facets.png|80px]]
| Vertice_type = 
| Vertice_configuration = 兩種[[頂點 (幾何)|頂點]]<br/>{{nowrap|3個凹鳶形的公共頂點}}<br/>{{nowrap|4個凹鳶形的公共頂點}}
| Schläfli = 
| Wythoff = 3/2 4 &#124; 2<ref>{{Cite web | author = Sam Gratrix | url = https://gratrix.net/polyhedra/uniform/w7_dual_17.html | title = Dual 17: great deltoidal icositetrahedron | publisher = gratrix.net | access-date = 2019-09-07 | archive-url = https://web.archive.org/web/20081205004528/http://gratrix.net/polyhedra/uniform/w7_dual_17.html | archive-date = 2008-12-05 | dead-url = no }}</ref><ref name="book weisstein2002crc">{{Cite book
 |title=CRC Concise Encyclopedia of Mathematics
 |author=Weisstein, E.W.
 |isbn=9781420035223
 |year=2002
 |page=p.3107
 |publisher=CRC Press}}</ref>
| Symmetry_group = [[八面體群對稱|O<sub>h</sub>]], [4,3], *432
| Index_references = DU<sub>17</sub>
| dual= [[非凸大斜方截半立方體]]
| Rotation_group = 
| Properties = 等面、非凸
| dual_image = Uniform_great_rhombicuboctahedron.png
}}
在[[幾何學]]中，'''大鳶形二十四面體'''是一種[[星形多面體|星形]][[二十四面體]]，由24個[[凹多邊形|凹]][[鳶形]]組成<ref name="bulatov.org">{{Cite web|author=Vladimir Bulatov|url=http://bulatov.org/polyhedra/dual/ud22.html|title=great deltoidal icositetrahedron|publisher=bulatov.org|access-date=2019-09-07|archive-url=https://web.archive.org/web/20171206235556/http://bulatov.org/polyhedra/dual/ud22.html|archive-date=2017-12-06|dead-url=no}}</ref>，其索引編號為DU<sub>17</sub><ref>{{Citation | last1=Wenninger | first1=Magnus | title=Dual Models | publisher=[[Cambridge University Press]] | isbn=978-0-521-54325-5 | mr= 730208| year=1983}}</ref>。大鳶形二十四面體的[[對偶多面體]]為[[非凸大斜方截半立方體]]<ref>{{Cite web | url = http://intranet-statique.enstimac.fr/crcmath/math/math/g/g256.htm | author = Eric W. Weisstein | title = Great Deltoidal Icositetrahedron | publisher = [[密西根州立大學]]圖書館 | archive-url = https://archive.today/20190908181215/http://intranet-statique.enstimac.fr/crcmath/math/math/g/g256.htm | archive-date = 2019-09-08 | access-date = 2019-09-07 | dead-url = no }}</ref>。

== 性質 ==
大鳶形二十四面體是一種[[星形多面體]]，由24個的凹鳶形組成，並具有面可遞（或稱等面）的特性，這意味著每個面皆[[全等]]，且這立體上的任意兩個面A和B，透過旋轉或鏡射這個立體，使A移動到B原來的位置時，其面仍然佔據了相同的空間區域<ref>{{citation|title=Dungeons, dragons, and dice|first=K. Robin|last=McLean|journal=The Mathematical Gazette|volume=74|issue=469|year=1990|pages=243–256|jstor=3619822}}.</ref>。這個立體共有24個面﹑48條稜和26個頂點。在其26個頂點中，有8個是3個凹鳶形的公共頂點、另外18個是4個凹鳶形的公共頂點<ref>{{Cite web | author = Chris (Kit) Wallace | url = http://kitwallace.co.uk/3d/solid-index.xq?mode=solid&id=GreatDeltoidalIcositetrahedron | title = Great Deltoidal Icositetrahedron | website = kitwallace.co.uk | archive-url = https://archive.today/20190908181617/http://kitwallace.co.uk/3d/solid-index.xq?mode=solid&id=GreatDeltoidalIcositetrahedron | archive-date = 2019-09-08 | dead-url = no | access-date = 2019-09-07 }}</ref>。

=== 面的組成 ===
大鳶形二十四面體由24個[[全等]]的凹鳶形（亦稱為鏢形或箭頭形）所組成：
{| class=wikitable
|[[File:DU17 facets.png|200px]]<br/>大鳶形二十四面體的其中一面。
|[[File:Great strombic icositetrahedron with difference color concave kite shape face.png|200px]]<br/>大鳶形二十四面體每面塗上不同顏色。
|}
凹鳶形與鳶形同樣皆有兩組邊等長。若對應的[[對偶多面體]]邊長為單位長，則在與之對應的大鳶形二十四面體中，凹鳶形較長的一組邊邊長為<ref name="dmccooey,GreatDeltoidalIcositetrahedron"/>：
:<math>\sqrt{2\left( 2+\sqrt{2}\right) } \approx 2.6131</math>單位
凹鳶形較短的一組邊邊長為<ref name="dmccooey,GreatDeltoidalIcositetrahedron"/>：
:<math>\frac{2\sqrt{10+\sqrt{2}}}{7} \approx 0.96528</math>單位

=== 二面角 ===
大鳶形二十四面體僅有一種[[二面角]]，其值為負的十七分之七減四根號二之[[反餘弦]]值<ref name="dmccooey,GreatDeltoidalIcositetrahedron">{{cite web | author = David I. McCooey | url = http://dmccooey.com/polyhedra/GreatDeltoidalIcositetrahedron.html | title = Self-Intersecting Quasi-Quasi-Regular Duals: Great Deltoidal Icositetrahedron | publisher = dmccooey.com | access-date = 2019-09-07 | archive-url = https://web.archive.org/web/20180423011743/http://dmccooey.com/polyhedra/GreatDeltoidalIcositetrahedron.html | archive-date = 2018-04-23 | dead-url = no }}</ref>：
:<math>\cos^{-1} \left( -\frac{7-4\sqrt{2}}{17}\right) \approx 94.5315^\circ</math>

=== 尺寸 ===
大鳶形二十四面體有兩種頂點，分別是8個3個凹鳶形的公共頂點和18個4個凹鳶形的公共頂點。這兩組頂點每組分別可以構成一個球。若對應的對偶多面體邊長為單位長，則在與之對應的大鳶形二十四面體中，由3個凹鳶形的公共頂點構成的球其半徑為<ref name="dmccooey,GreatDeltoidalIcositetrahedron"/>：
:<math>\frac{4\sqrt{3}-\sqrt{6}}{7} \approx 0.6398</math>單位
另一組頂點，由4個凹鳶形的公共頂點構成的球其半徑為[[2的算術平方根|二的平方根]]單位長<ref name="dmccooey,GreatDeltoidalIcositetrahedron"/>：
:<math>\sqrt{2} \approx 1.4142</math>單位

=== 頂點座標 ===
若一個大鳶形二十四面體其對應的[[對偶多面體]][[非凸大斜方截半立方體]]的邊長為單位長，且幾何中心位於原點，此時對應的大鳶形二十四面體最短邊長為<math>\tfrac{2\sqrt{10+\sqrt{2}}}{7}</math>單位長<ref name="dmccooey,GreatDeltoidalIcositetrahedron"/>，此時，大鳶形二十四面體的頂點座標為<ref>{{cite web |archiveurl=https://web.archive.org/web/20190921150416/http://dmccooey.com/polyhedra/GreatDeltoidalIcositetrahedron.txt |archivedate=2019-09-21 |url=http://dmccooey.com/polyhedra/GreatDeltoidalIcositetrahedron.txt |title=Data of Great Deltoidal Icositetrahedron |website=dmccooey.com |accessdate=2019-09-21 |dead-url=no }}</ref>：
:<math>\left(\pm \sqrt{2}\,,\quad 0\,,\quad 0\right)</math>、<math>\left( 0 \,,\quad\pm \sqrt{2}\,,\quad 0\right)</math>、<math>\left( 0 \,,\quad 0\,,\quad\pm \sqrt{2}\right)</math>
:<math>\left(0\,,\quad \pm 1\,,\quad\pm 1\right)</math>、<math>\left(\pm 1\,,\quad 0\,,\quad\pm 1\right)</math>、<math>\left(\pm 1\,,\quad\pm 1\,,\quad 0\right)</math>
:<math>\left(\pm\frac{4-\sqrt{2}}{7}\,,\quad\pm\frac{4-\sqrt{2}}{7}\,,\quad\pm\frac{4-\sqrt{2}}{7}\right)</math>

== 相關多面體與鑲嵌 ==
[[大六角二十四面體]]與[[反平行四邊形二十四面體]]的[[幾何中心]]重合可以組成一個[[大鳶形二十四面體]]<ref>{{cite web| author=Robert Webb| 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=2019-09-07| dead-url=no}}</ref>。
{| class="wikitable" width=300
|- align=center
| [[Image:DU14_great_hexacronic_icositetrahedron.png|240px]]<br/>[[大六角二十四面體]]
| [[Image:DU21_great_rhombihexacron.png|240px]]<br/>[[反平行四邊形二十四面體]]
| [[Image:DU17_great_strombic_icositetrahedron.png|240px]]<br/>[[大鳶形二十四面體]]
|}

== 參見 ==
*[[鳶形二十四面體]]
*[[反平行四邊形二十四面體]]
*[[二十四面體]]

== 參考文獻 ==
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==外部連結==
*{{mathworld | title=大鳶形二十四面體 | urlname=GreatDeltoidalIcositetrahedron }}

[[Category:星形多面体]]