查看“︁小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 = 小反平行四邊形二十四面體
| polyhedron = 小反平行四邊形二十四面體
| imagename = DU18_small_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:DU18 facets.png|80px]]
| Vertice_type = 
| Vertice_configuration = {{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}}
| Schläfli = 
| Wythoff = 3/2 2 4&#124;<ref name="w7_dual_18">{{Cite web | url = https://gratrix.net/polyhedra/uniform/w7_dual_18.html | title = Dual 18: small rhombihexacron | publisher = gratrix.net | access-date = 2021-09-03 | archive-date = 2021-04-01 | archive-url = https://web.archive.org/web/20210401061212/http://gratrix.net/polyhedra/uniform/w7_dual_18.html | dead-url = no }}</ref>
| Symmetry_group = O<sub>h</sub>, [4,3], *432
| Index_references = DU<sub>18</sub>
| Rotation_group = 
| Properties = 等面、非凸
| dual_image = Small_rhombihexahedron.png
| 3d_image = Small_rhombihexacron.stl
}}
{{Not|反平行四邊形二十四面體}}
在[[幾何學]]中，'''小反平行四邊形二十四面體'''是一種星形[[二十四面體]]，由24個[[反平行四邊形]]組成，其索引編號為DU<sub>18</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>，[[對偶多面體]]為[[小斜方立方體]]<ref name="s/s426">{{Cite web | url = https://archive.lib.msu.edu/crcmath/math/math/s/s426.htm | author = Eric W. Weisstein | title = Small Rhombihexacron | publisher = [[密西根州立大學]][[圖書館]] | access-date = 2021-09-03 | archive-date = 2013-06-04 | archive-url = https://web.archive.org/web/20130604223958/http://archive.lib.msu.edu/crcmath/math/math/s/s426.htm | dead-url = no }}</ref>。其外觀與{{link-wd|Q7543070}}相同。

== 性質 ==
小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 (幾何)|頂點]]<ref>{{cite web|url=http://bulatov.org/polyhedra/dual/ud23.html|title=Small Rhombihexacron|author=Vladimir Bulatov|access-date=2021-09-03|archive-date=2020-02-23|archive-url=https://web.archive.org/web/20200223125026/http://www.bulatov.org/polyhedra/dual/ud23.html|dead-url=no}}</ref>。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 name="dmccooey,SmallRhombihexacron">{{cite web | author = David I. McCooey | url = http://dmccooey.com/polyhedra/SmallRhombihexacron.html | title = Versi-Quasi-Regular Duals: Small Rhombihexacron | publisher = dmccooey.com | access-date = 2019-09-07 | archive-date = 2021-09-03 | archive-url = https://web.archive.org/web/20210903101005/http://dmccooey.com/polyhedra/SmallRhombihexacron.html | dead-url = no }}</ref>。

每個反平行四邊形具有2個<math>\arccos(\frac{1}{4}+\frac{1}{2}\sqrt{2})\approx 16.842\,116\,236\,30^{\circ}</math>的角以及2個<math>\arccos(-\frac{1}{2}+\frac{1}{4}\sqrt{2})\approx 98.421\,058\,118\,15^{\circ}</math>的角，其對角線交角為<math>\arccos(\frac{1}{4}+\frac{1}{8}\sqrt{2})\approx 64.736\,825\,645\,55^{\circ}</math>。每個邊相鄰的兩個反平行四邊形間的[[二面角]]皆為<math>\arccos(\frac{-7-4\sqrt{2}}{17})\approx 138.117\,959\,055\,51^{\circ}</math>。<ref name="dmccooey,SmallRhombihexacron"/>

== 相關多面體 ==
=== 反平行四邊形二十四面體 ===
{{main|反平行四邊形二十四面體}}
小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 (幾何)|頂點]]<ref name="dmccooey,SmallRhombihexacron"/>，這些特性與[[反平行四邊形二十四面體]]相同<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 = 2021-09-03 | 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>，但他們的對偶多面體不同：小反平行四邊形二十四面體的[[對偶多面體]]為[[小斜方立方體]]<ref name="s/s426"/>；而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 | 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>。

另外一個差別在於其頂點間的相聯方式，小反平行四邊形二十四面體的[[頂點圖]]為八邊形，而[[反平行四邊形二十四面體]]頂點圖為八角星，根據對偶多面體的定義，對偶多面體的面形狀會與原始立體的頂點圖相同<ref>{{cite mathworld|urlname=DualPolyhedron|title=Dual Polyhedron}}</ref>，[[小斜方立方體]]由正方形和八邊形組成<ref>{{cite mathworld|urlname=SmallRhombihexahedron|title=Small Rhombihexahedron}}</ref>，而[[大斜方立方體]]由三角形和八角星組成<ref>{{cite mathworld|urlname=GreatRhombihexahedron|title=Great Rhombihexahedron}}</ref>而導致了這些差異。
{|class="wikitable"
!多面體
|[[File:DU18_small_rhombihexacron.png|150px]]<br/>小反平行四邊形二十四面體
|[[File:DU21_great_rhombihexacron.png|150px]]<br/>[[反平行四邊形二十四面體]]
|-
!組成的面
|[[File:DU18 facets.png|150px]]<br/>[[反平行四邊形]]
|[[File:DU21 facets.png|150px]]<br/>[[反平行四邊形]]
|-
!對偶多面體
|[[File:Small_rhombihexahedron.png|150px]]<br/>[[小斜方立方體]]
|[[File:Great_rhombihexahedron.png|150px]]<br/>[[大斜方立方體]]
|}

=== 小六角星化二十四面體 ===
小反平行四邊形二十四面體的外觀與小六角星化二十四面體相同，但組成面的形狀不同：小反平行四邊形二十四面體的面由反平行四邊形組成，而小六角星化二十四面體的面由凹[[鳶形]]組成。<ref name="w7_dual_13">{{Cite web | url = https://gratrix.net/polyhedra/uniform/w7_dual_13.html | title = Dual 13: small hexacronic icositetrahedron | publisher = gratrix.net | access-date = 2021-09-03 | archive-date = 2021-04-01 | archive-url = https://web.archive.org/web/20210401061230/http://gratrix.net/polyhedra/uniform/w7_dual_13.html | dead-url = no }}</ref>
{|class="wikitable"
!多面體
|[[File:DU18_small_rhombihexacron.png|150px]]<br/>小反平行四邊形二十四面體
|[[File:DU13_small_hexacronic_icositetrahedron.png|150px]]<br/>{{link-wd|Q7543070}}
|-
!組成的面
|[[File:DU18 facets.png|150px]]<br/>[[反平行四邊形]]
|[[File:DU13 facets.png|150px]]<br/>凹鳶形
|-
!對偶多面體
|[[File:Small_rhombihexahedron.png|150px]]<br/>[[小斜方立方體]]
|[[File:Small_cubicuboctahedron.png|150px]]<br/>[[小立方立方八面體]]
|}

== 參見 ==
*[[反平行四邊形二十四面體]]

== 參考文獻 ==
{{Reflist}}

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