查看“︁小十二面二十面體”︁的源代码

{{NoteTA|G1=Math}}
{{Infobox polyhedron
| name = 小十二面二十面體
| polyhedron = 小十二面二十面體
| en_name = small dodecicosahedron<br/>small dodekicosahedron
| imagename = Small_dodecicosahedron.png
| Type = [[均勻星形多面體]]
| Face = 32
| Edge = 120
| Vertice = 60
| Face_type =20個[[正六邊形]]<br/>12個[[正十邊形]]
| Vertice_type = 6.10.6/5.10/9 
| Vertice_configuration = 
| Coxeter_diagram = 
| schläfli = 
| Wythoff = 
| Face_configuration = 
| Conway = 
| Symmetry_group = I<sub>h</sub>, [5,3], *532
| Index_references = [[均勻多面體|U]]<sub>50</sub>, [[哈罗德·斯科特·麦克唐纳·考克斯特|C]]<sub>64</sub>, [[溫尼爾多面體模型列表|W]]<sub>90</sub>
| dual = {{link-wd|Q7543046}}
| Rotation_group = 
| Dihedral_angle = 
| Properties = 
| 3d_image = Small_dodecicosahedron.stl
| vfigimage = Small_dodecicosahedron_vertfig.png
| dual_image = DU50_small_dodecicosacron.png
| net_image = 
}}
'''小十二面二十面體'''是一種[[星形均勻多面體]]，由20個[[正六邊形]]和12個[[正十邊形]]組成<ref name="dmccooey">{{cite web | url = http://dmccooey.com/polyhedra/SmallDodecicosahedron.html | title = Versi-Quasi-Regular Polyhedra: Small Dodecicosahedron | author = David I. McCooey | access-date = 2022-08-23 | archive-date = 2022-02-14 | archive-url = https://web.archive.org/web/20220214093903/http://dmccooey.com/polyhedra/SmallDodecicosahedron.html | dead-url = no }}</ref>，索引為U<sub>50</sub>，[[對偶多面體]]為{{link-wd|Q7543046}}<ref name="mathworld">{{cite mathworld | urlname=SmallDodecicosahedron | title = Small Dodecicosahedron}}</ref>，具有{{link-en|二十面體群對稱性|Icosahedral symmetry}}<ref name="MathConsult">{{Cite web|url=https://www.mathconsult.ch/static/unipoly/50.html|title=50: small dodecicosahedron|last=Maeder|first=Roman|website=MathConsult|access-date=2022-08-23|archive-date=2022-08-23|archive-url=https://web.archive.org/web/20220823082521/https://www.mathconsult.ch/static/unipoly/50.html|dead-url=no}}</ref><ref name="dmccooey"/><ref name="Paul Bourke. 2004">{{Cite web | url = http://paulbourke.net/geometry/polyhedra/ | title = Uniform Polyhedra (80) | publisher = Math Consult AG | author = Paul Bourke | date = October 2004 | access-date = 2019-09-27 | archive-url = https://www.webcitation.org/6JK8SwSwv?url=http://paulbourke.net/geometry/polyhedra/ | archive-date = 2013-09-02 | dead-url = no }}</ref>，並且可以視為[[小二十面化截半二十面體]]的{{link-en|刻面|Faceting}}多面體<ref name="Richard Klitzing">{{cite web | url = https://bendwavy.org/klitzing/incmats/siddy.htm | title = small dodekicosahedron, siddy | author = Richard Klitzing | website = bendwavy.org | access-date = 2022-08-23 | archive-date = 2021-09-24 | archive-url = https://web.archive.org/web/20210924121158/https://www.bendwavy.org/klitzing/incmats/siddy.htm | dead-url = no }}</ref>。

== 性質 ==
小十二面二十面體共由32個[[面 (幾何)|面]]、120條[[邊 (幾何)|邊]]和60個[[頂點 (幾何)|頂點]]組成<ref name="MathConsult"/>。在其32個面中，有20個面是[[正六邊形]]面、12個面是[[正十邊形]]面<ref name="dmccooey"/>，其中的20個正六邊形面又可以再分成10個一般的正六邊形面（施萊夫利符號：{{mset|6}}）和10個反向相接的正六邊形面（施萊夫利符號：{{mset|6/5}}）；其12個正十邊形面又可以再分成6個一般的正十邊形面（施萊夫利符號：{{mset|10}}）和6個反向相接的正十邊形面（施萊夫利符號：{{mset|10/9}}）<ref name="kaleido">{{cite web | url = http://harel.org.il/zvi/kaleido/data/55.html | title = Kaleido Data: Uniform Polyhedron #55, small dodecicosahedron | author = Zvi Har'El | website = harel.org.il | date = 2006-11-14 | accessdate = 2022-08-14 | archive-date = 2022-08-23 | archive-url = https://web.archive.org/web/20220823082518/http://harel.org.il/zvi/kaleido/data/55.html | dead-url = no }}</ref>。在其60個頂點中，每個頂點都是兩個正十邊形面和兩個正六邊形面的公共頂點，並且這些面在構成頂角的多面角時，以正十邊形、正六邊形、反向相接的正十邊形和反向相接的正六邊形的順序排列，在頂點圖中可以用{{nowrap|(5.6.5/3.6)}}<ref name="article kovivc2012classification">{{cite journal
  |title=Classification of uniform polyhedraby their symmetry-type graphs
  |author=Kovič, J
  |journal=Int. J. Open Problems Compt. Math
  |volume=5
  |url=http://www.ijopcm.org/Vol/vol.5/IJOPCM(vol.5.4.8.D.12).pdf
  |number=4
  |year=2012
  |access-date=2022-08-23
  |archive-date=2022-08-14
  |archive-url=https://web.archive.org/web/20220814151106/http://www.ijopcm.org/Vol/vol.5/IJOPCM(vol.5.4.8.D.12).pdf
  |dead-url=no
}}</ref>或{{nowrap|(10.6.10/9.6/5)}}<ref name="kaleido"/><ref name="MathConsult"/>來表示。

=== 尺寸 ===
若小十二面二十面體的邊長為單位長，則其外接球半徑為：<ref name="mathworld"/><ref name="dmccooey"/>
:<math>R=\frac{\sqrt{34+6\sqrt{5}}}{4}\approx 1.72148932</math>

邊長為單位長的小十二面二十面體，[[中分球]]半徑為：<ref name="dmccooey"/>
:<math>R=\frac{\sqrt{6\left(5+\sqrt{5}\right)}}{4}\approx 1.647278207</math>

=== 二面角 ===
小十二面二十面體共有兩種[[二面角]]，皆為六邊形和十邊形的二面角，但根據所在位置的不同，其角度也不同。這兩種二面角分別位於五角星坑洞的位置以及三角形坑洞的位置。<ref name="Richard Klitzing"/>

其中，位於五角星坑洞位置的六邊形和十邊形的二面角，其角度約為79.18768度：<ref name="Richard Klitzing"/><ref name="dmccooey"/>
:<math>\arccos\left(\frac{\sqrt{15\left(5-2\sqrt{5}\right)}}{15}\right)\approx 1.382085796 \approx 79.187683036^\circ</math>

而位於五角星坑洞位置的六邊形和十邊形的二面角，其角度約為79.18768度：<ref name="Richard Klitzing"/><ref name="dmccooey"/>
:<math>\arccos\left(\frac{\sqrt{15\left(5+2\sqrt{5}\right)}}{15}\right)\approx 0.65235813978 \approx 37.377368141^\circ</math>

== 相關多面體 ==
小十二面二十面體與[[大星形截角十二面体]]共用相同的頂點佈局。其也與[[小雙三角十二面截半二十面體]]和[[小二十面化截半二十面體]]共用相同的邊佈局。<ref name="Richard Klitzing"/>
{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
| [[Image:Great stellated truncated dodecahedron.png|100px]]<BR>[[大星形截角十二面体]]
| [[Image:Small icosicosidodecahedron.png|100px]]<BR>[[小二十面化截半二十面體]]
| [[Image:Small ditrigonal dodecicosidodecahedron.png|100px]]<BR>[[小雙三角十二面截半二十面體]]
| [[Image:Small dodecicosahedron.png|100px]]<BR>'''小十二面二十面體'''
|}
== 參見 ==
*{{link-en|均勻多面體列表|List_of_uniform_polyhedra}}

== 參考文獻 ==
{{Reflist}}

{{均勻多面體導航}}

[[Category:星形多面体]]
[[Category:均勻多面體]]
[[Category:非凸多面體]]