查看“︁反平行四邊形二十四面體”︁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 = 反平行四邊形二十四面體
| polyhedron = 反平行四邊形二十四面體
| imagename = DU21_great_rhombihexacron.png
| Type = [[均勻多面體對偶]]<br/>[[星形多面體]]
| Face = 24
| Edge = 48
| Vertice = 18
| Face_type = 24個[[反平行四邊形]]<br/>[[File:DU21 facets.png|80px]]
| Vertice_type = 
| Vertice_configuration = 兩種[[頂點 (幾何)|頂點]]<br/>{{nowrap|4個[[反平行四邊形]]的公共頂點}}<br/>{{nowrap|8個反平行四邊形的公共頂點}}<br/>(星形排佈)
| Schläfli = 
| dual=[[大斜方立方體]]
| Wythoff = 4/3 3/2 2 &#124;<ref>{{Cite web | url = https://gratrix.net/polyhedra/uniform/w7_dual_21.html | title = Dual 21: great rhombihexacron | publisher = gratrix.net | access-date = 2019-09-07 | archive-url = https://web.archive.org/web/20081204232300/http://gratrix.net/polyhedra/uniform/w7_dual_21.html | archive-date = 2008-12-04 | dead-url = no }}</ref>
| Symmetry_group = O<sub>h</sub>, [4,3], *432
| Index_references = DU<sub>21</sub>
| Rotation_group = 
| Properties = 等面、非凸
| dual_image = Great_rhombihexahedron.png
}}
{{Not|小反平行四邊形二十四面體}}
在[[幾何學]]中，'''反平行四邊形二十四面體'''是一種星形[[二十四面體]]，由24個的[[反平行四邊形]]組成，其索引編號為DU<sub>21</sub><ref>{{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=[[Cambridge University Press]] | isbn=978-0-521-54325-5 | mr= 730208| year=1983}}</ref>。反平行四邊形二十四面體的[[對偶多面體]]為[[大斜方立方體]]<ref>{{Cite web | url = https://archive.lib.msu.edu/crcmath/math/math/g/g299.htm | author = Eric W. Weisstein | title = Great Rhombihexacron | publisher = [[密西根州立大學]][[圖書館]] | access-date = 2019-09-07 | archive-url = https://web.archive.org/web/20140711083312/http://archive.lib.msu.edu/crcmath/math/math/g/g299.htm | archive-date = 2014-07-11 | dead-url = no }}</ref>。反平行四邊形二十四面體為數學家溫尼爾的著作《[[溫尼爾多面體模型列表|多面體模型]]》中之形狀W<sub>103</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>的[[對偶多面體]]<ref>{{Cite mathworld | urlname = GreatRhombihexacron | title = Great Rhombihexacron }}</ref>。

== 性質 ==
反平行四邊形二十四面體，由24個[[全等]]的[[反平行四邊形]]組成，其具有48條[[稜]]和18個頂點。在其18個頂點中，有12個是4個反平行四邊形的公共頂點、另外6個是8個反平行四邊形的公共頂點<ref>{{Cite web | url = http://kitwallace.co.uk/3d/solid-index.xq?mode=solid&id=GreatRhombihexacron | title = Great Rhombihexacron | website = kitwallace.co.uk | access-date = 2019-09-07 | archive-date = 2021-09-03 | archive-url = https://web.archive.org/web/20210903101006/http://kitwallace.co.uk/3d/solid-index.xq?mode=solid&id=GreatRhombihexacron | dead-url = no }}</ref>，然而這6個頂角並非一般的八面角，其對應的頂點圖為[[八角星]]，表示其排列方式同於八角星的稜之排佈。

=== 面的組成 ===
反平行四邊形二十四面體由24個[[全等]]的[[反平行四邊形]]（亦稱為領結形）所組成<ref name="dmccooey,GreatDeltoidalIcositetrahedron">{{cite web | author = David I. McCooey | url = http://dmccooey.com/polyhedra/GreatRhombihexacron.html | title = Versi-Quasi-Regular Duals: Great Rhombihexacron | publisher = dmccooey.com | access-date = 2019-09-07 | archive-url = https://web.archive.org/web/20180310213802/http://dmccooey.com/polyhedra/GreatRhombihexacron.html | archive-date = 2018-03-10 | dead-url = no }}</ref>：
{| class=wikitable
|[[File:DU21 facets.png|200px]]
|[[File:Great rhombihexacron with one blue antiparallelogram face.png|200px]]<br/>反平行四邊形在立體中的位置
|}
反平行四邊形具有兩對邊等長的特性<ref name="round">{{citation|title=How round is your circle? Where Engineering and Mathematics Meet|first1=John|last1=Bryant|first2=Christopher J.|last2=Sangwin|publisher=Princeton University Press|year=2008|isbn=978-0-691-13118-4|contribution=3.3 The Crossed Parallelogram|pages=54–56}}.</ref>，因此組成反平行四邊形二十四面體的反平行四邊形有兩種長度的邊。若反平行四邊形二十四面體對應的對偶多面體{{link-ja|大斜方立方體|Great rhombihexahedron}}其邊長為單位長，則對應的反平行四邊形二十四面體中反平行四邊形面上較短的邊長為<ref name="dmccooey,GreatDeltoidalIcositetrahedron"/>：
:<math>\sqrt{2\left( 2-\sqrt{2}\right) }</math>單位長
此時，較長的邊長為<ref name="dmccooey,GreatDeltoidalIcositetrahedron"/>：
:<math>2\sqrt{2-\sqrt{2}}</math>單位長
而其邊長比為<math>\left( 1:\sqrt{2}\right)</math>。

== 相關多面體與鑲嵌 ==
反平行四邊形二十四面體和星形四角化菱形十二面體皆可以視為將[[菱形十二面體]]每個面替換成一個頂點和四個三角形的結果<ref name="etc.usf.edu Hexakis Octahedron">{{cite web | url = https://etc.usf.edu/clipart/58400/58482/58482_hexakis-octa.htm | title = Hexakis Octahedron | publisher = Florida Center for Instructional Technology, College of Education, University of South Florida. | access-date = 2019-09-03 | archive-url = https://web.archive.org/web/20150121184454/http://etc.usf.edu/clipart/58400/58482/58482_hexakis-octa.htm | archive-date = 2015-01-21 | dead-url = no }}</ref>，換句話說即將菱形十二面體每個面替換成一個菱形錐，根據替換的角錐錐高的不同，可以產生不同的立體：
{| class="wikitable"
|-
| [[File:Rhombic dodecahedron with 3 colors.svg|120px]]
| [[File:Disdyakis_dodecahedron_3color.png|120px]]
| [[File:Three_flattened_octahedra_compound.png|120px]]
| [[File:Great Rhombihexacron with 3 Colors.svg|120px]]
|- align="center"
| [[菱形十二面體]] || [[四角化菱形十二面體]] || [[星形四角化菱形十二面體]] || 反平行四邊形二十四面體
|}
[[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|accessdate = 2019-09-07|dead-url = no}}</ref>。
{| class="wikitable" width=300px
|- 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]]
|}
反平行四邊形二十四面體由[[反平行四邊形]]（亦稱為領結形）所組成<ref name="dmccooey,GreatDeltoidalIcositetrahedron"/>，其他同樣由[[四邊形]]組成，且具有八面體群對稱性的二十四面體有：
{| class="wikitable" width="300"
|- align=center
! 圖像
| [[Image:Strombic_icositetrahedron.png|120px]]<br/>[[鳶形二十四面體]]
| [[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]]
| [[Image:DU13_small_hexacronic_icositetrahedron.png|120px]]<br/>{{link-en|小六角二十四面體|Small hexacronic icositetrahedron}}
| [[Image:DU14_great_hexacronic_icositetrahedron.png|120px]]<br/>[[大六角二十四面體]]
| [[Image:DU18_small_rhombihexacron.png|120px]]<br/>[[小反平行四邊形二十四面體]]
|-
!面
| [[File:DU10_facets.png|100px]]<br/>[[鳶形]]
| [[File:DU21_facets.png|100px]]<br/>[[反平行四邊形]]
| [[File:DU17_facets.png|100px]]<br/>凹鳶形
| [[File:DU13 facets.png|100px]]<br/>凹鳶形<br/>露出的部分為<br/>[[反平行四邊形]]
| [[File:DU14_facets.png|100px]]<br/>[[鳶形]]
| [[File:DU18_facets.png|100px]]<br/>反平行四邊形
|}

== 參見 ==
*[[鳶形二十四面體]]
*[[星形四角化菱形十二面體]]
*[[小反平行四邊形二十四面體]]

== 參考文獻 ==
{{reflist|2}}

==外部連結==
*{{mathworld|title=反平行四邊形二十四面體|urlname=GreatRhombihexacron}}

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