查看“︁欠四面十二面體”︁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 = 欠四面十二面體
| polyhedron = 欠四面十二面體
| image = [[File:Tetrahedral_self-dual_hexadecahedron.png|240px]]<br/>自身對偶形式<br/>[[File:Tetrahedrally_stellated_icosahedron.png|240px]]<br/>星形化正二十面體形式<br/>[[File:Tetrahedrally_diminished_regular_dodecahedron.png|240px]]<br/>欠缺四個頂角的正十二面體形式
| Type = [[凸多面體]]
| Coxeter_diagram = 
| Face = 16
| Edge = 30
| Vertice = 16
| Face_List = 4個[[三角形]]<br/>12個[[四邊形]]
| Vertice_type = 
| Vertice_configuration = {{math|3.4.4.4<br/>4.4.4}}
| Wythoff = 
| Face_configuration = 
| Symmetry_group = {{math|{{link-en|四面體群對稱性|Tetrahedral symmetry|T}}, [3,3]<sup>+</sup>, (332),}} order 12
| Index_references = 
| Rotation_group = 
| Dihedral_angle = 
| Properties = 凸
| 3d_image = Self-dual form tetrahedrally diminished dodecahedron.stl
| vfigimage = 
| dual_image = Green self-dual form tetrahedrally diminished dodecahedron.svg
| net_image = Net of self-dual form tetrahedrally diminished dodecahedron.svg
}}
'''[[欠_(多面體變換)#添加稜的欠|欠]]四面十二面體'''（tetrahedrally diminished dodecahedron）又稱'''四面星形化二十面體'''（tetrahedrally stellated icosahedron）或'''螺旋四面體'''（propello tetrahedron）<ref name="article hart2000sculpture">{{cite journal
  |title=Sculpture based on Propellorized Polyhedra
  |author=Hart, George W
  |journal=Proceedings of MOSAIC
  |pages=61-70
  |url=http://www.georgehart.com/propello/propello.html
  |year=2000
  |access-date=2023-01-10
  |archive-date=2017-11-03
  |archive-url=https://web.archive.org/web/20171103221228/http://georgehart.com/propello/propello.html
  |dead-url=no
  }}</ref>是一種拓樸自身對偶的[[十六面體]]，由4個正三角形[[面 (幾何)|面]]、12個[[全等]]的四邊形面、30條邊和16個頂點組成。<ref name="Tetrahedrally Stellated Icosahedron"/>

欠四面十二面體並非是少4個面的[[十二面體]]，其名稱是來自[[#欠缺四個頂角的正十二面體形式|欠缺四個頂角的正十二面體形式]]的欠四面十二面體，之所以稱為欠四面十二面體是因為其在四個頂角處各[[欠_(多面體變換)#添加稜的欠|欠缺]]了四面體狀的結構，因此稱為欠四面十二面體。

== 形式 ==
欠四面十二面體有三種形式，一種是多爾曼盧克對偶構造的[[自身對偶]]形式；一種是欠缺四個頂角的正十二面體形式；還有一種是星形化的正二十面體形式。<ref name="www.polyhedra-world.nc">{{cite web
 | url=http://www.polyhedra-world.nc/tetra_di_.htm
 | title=tetrahedrally truncated dodecahedron and stellated icosahedron
 | author=Martin Kraus
 | website=polyhedra-world.nc
 | access-date=2023-01-10
 | archive-date=2023-01-10
 | archive-url=https://web.archive.org/web/20230110165559/http://www.polyhedra-world.nc/tetra_di_.htm
 | dead-url=no }}</ref>

=== 自身對偶形式 ===
在自身對偶形式中，欠四面十二面體是302404種自身對偶的十六面體中，1476種至少具有2階對稱性中唯一具有四面體對稱性的立體。<ref>{{cite web | url=http://dmccooey.com/polyhedra/SelfDualHexadecahedron1.html | title=Symmetric Canonical Self-Dual Hexadecahedra: Self-Dual Hexadecahedron #1 (canonical) | author=David I. McCooey | year=2015 | access-date=2023-01-10 | archive-date=2023-05-21 | archive-url=https://web.archive.org/web/20230521082842/http://dmccooey.com/polyhedra/SelfDualHexadecahedron1.html | dead-url=no }}</ref>
<gallery widths="180px" heights="180px">
File:Tetrahedral_self-dual_hexadecahedron.png|自身對偶形式的欠四面十二面體
File:Self-dual form tetrahedrally diminished dodecahedron.stl|自身對偶形式的欠四面十二面體的3D模型
File:Net of self-dual form tetrahedrally diminished dodecahedron.svg|自身對偶形式的欠四面十二面體的展開圖</gallery>

=== 欠缺四個頂角的正十二面體形式 ===
在[[欠 (多面體變換)|欠缺]]四個頂角的正十二面體形式中，其移除了4個正十二面體的頂角，並將相鄰的五邊形面切割成梯形。<ref name="www.polyhedra-world.nc"/>這種形式的欠四面十二面體有兩種二面角，分別為梯形和梯形的二面角，以及梯形和三角形的二面角。其中，梯形和梯形的二面角為：<ref name="teddoe">{{cite web | url = https://www.bendwavy.org/klitzing/incmats/tet-dim-doe.htm | title = teddoe, tetrahedrally diminished dodecahedron | author = Richard Klitzing | website = bendwavy.org | access-date = 2023-01-10 | archive-date = 2023-01-11 | archive-url = https://web.archive.org/web/20230111140737/https://bendwavy.org/klitzing/incmats/tet-dim-doe.htm | dead-url = no }}</ref>
:<math>\arccos{-\frac{1}{\sqrt{5}}} \approx 116.565051^\circ</math>

而梯形和三角形的[[二面角]]為：
:<math>\arccos{\sqrt{\frac{5+2\sqrt{5}}{15}}} \approx 142.622632^\circ</math>
<gallery widths="180px" heights="180px">
File:Tetrahedrally diminished regular dodecahedron.png|欠缺四個頂角的正十二面體
File:Tetrahedrally diminished dodecahedron.stl|欠缺四個頂角的正十二面體的3D模型
File:Tetrahedrally diminished regular dodecahedron net.png|欠缺四個頂角的正十二面體的展開圖</gallery>

=== 星形化正二十面體形式 ===
在星形化正二十面體形式中，其是32種以四面提群對稱性定義的星形化正二十面體之一，並具有[[鳶形]]面。<ref>{{cite web|url=http://www.georgehart.com/virtual-polyhedra/stellations-icosahedron-tetrahedral.html|title=Tetrahedral Stellations of the Icosahedron|author=George W. Hart|year=1996|access-date=2023-01-10|archive-date=2023-05-26|archive-url=https://web.archive.org/web/20230526191524/https://www.georgehart.com/virtual-polyhedra/stellations-icosahedron-tetrahedral.html|dead-url=no}}</ref>這種欠四面十二面體是[[五複合四面體]]中，少一個四面體之幾何結構的星狀核<ref>{{cite web | url=http://www.mi.sanu.ac.rs/vismath/zefiro2008/__generation_of_icosahedron_by_5tetrahedra.htm | title=Generation of an icosahedron by the intersection of five tetrahedra: geometrical and crystallographic features of the intermediate polyhedra | author=Livio Zefiro | website=www.mi.sanu.ac.rs | access-date=2023-01-10 | archive-date=2016-01-23 | archive-url=https://web.archive.org/web/20160123033638/http://www.mi.sanu.ac.rs/vismath/zefiro2008/__generation_of_icosahedron_by_5tetrahedra.htm | dead-url=no }}</ref>。在康威多面體表示法中，其可以用pT來表示，代表通過{{link-en|喬治·W·哈特|George W. Hart}}的螺旋變換（propeller operator）的[[正四面體]]。<ref>{{cite web|url=http://www.georgehart.com/virtual-polyhedra/conway_notation.html|title=Conway Notation for Polyhedra|author=George W. Hart|year=1996|quote=pT is the tetrahedrally stellated icosahedron|access-date=2023-01-10|archive-date=2014-11-29|archive-url=https://web.archive.org/web/20141129085342/http://www.georgehart.com/virtual-polyhedra/conway_notation.html|dead-url=no}}</ref>

假設中交球的半徑为1，則存在一个邊長比為0.849:1.057的典型形式，其鳶形面保持等腰。
<gallery widths="180px" heights="180px">
File:Tetrahedrally_stellated_icosahedron.png|星形化正二十面體形式
File:Tetrahedrally stellated icosahedron.stl|星形化正二十面體形式的3D模型
File:Tetrahedrally stellated icosahedron net.png|星形化正二十面體形式的展開圖</gallery>

== 性質 ==

=== 構造 ===
欠四面十二面體具有手性四面體群對稱性，因此其可以構造自四個面星形化的五角十二面體群對稱性之[[扭稜四面體]]<ref name="Tetrahedrally Stellated Icosahedron">{{cite web|url=http://www.georgehart.com/virtual-polyhedra/tetrahedrally_stellated_icosahedron.html|title=Tetrahedrally Stellated Icosahedron|author=George W. Hart|year=1996|access-date=2023-01-10|archive-date=2022-11-27|archive-url=https://web.archive.org/web/20221127085318/https://www.georgehart.com/virtual-polyhedra/tetrahedrally_stellated_icosahedron.html|dead-url=no}}</ref>或構造自欠缺4個頂點的[[五角十二面體]]。

=== 頂點座標 ===
自身對偶形式的欠四面十二面體頂點座標為：<ref>{{cite web | url=http://dmccooey.com/polyhedra/SelfDualHexadecahedron1.txt | title=data of Self-Dual Hexadecahedron #1 (canonical) | author=David I. McCooey | year=2015 | access-date=2023-01-10 | archive-date=2021-02-23 | archive-url=https://web.archive.org/web/20210223171347/http://dmccooey.com/polyhedra/SelfDualHexadecahedron1.txt | dead-url=no }}</ref>
:<math>\left( C_1,\,\pm C_0,\,\pm 1\right)</math>
:<math>\left( -C_1,\,\mp C_0,\,\pm 1\right)</math>
:<math>\left( 1,\,\pm C_1,\,\pm C_0\right)</math>
:<math>\left( -1,\,\mp C_1,\,\pm C_0\right)</math>
:<math>\left( C_0,\,\pm 1,\,\pm C_1\right)</math>
:<math>\left( -C_0,\,\mp 1,\,\pm C_1\right)</math>
:<math>\left( C_2,\,\mp C_2,\,\pm C_2\right)</math>
:<math>\left( -C_2,\,\pm C_2,\,\pm C_2\right)</math>
其中：
:<math>C_0=\frac{\sqrt[3]{4\left(11+3\sqrt{69}\right) }-\sqrt[3]{4\left( 3\sqrt{69}-11\right) }-1}{3}</math>
::≈0.139680581996
:<math>C_1=\frac{\sqrt[3]{4\left(25+3\sqrt{69}\right)}+\sqrt[3]{4\left(25-3\sqrt{69}\right)}-5}{3}</math>
::≈0.509755332493
:<math>C_2=\frac{\sqrt[3]{4\left(371+33\sqrt{69}\right)}+\sqrt[3]{4\left(371-33\sqrt{69}\right)}-1}{33}</math>
::≈0.606267870861

== 相關幾何體 ==
欠四面十二面體是雙曲均勻堆砌體{{link-en|部分欠缺二十面體堆砌|Partially diminished icosahedral honeycomb}}（施萊夫利符號：pd{3,5,3}）的頂點圖<ref name="Richard Klitzing pd{3,5,3}">{{cite web | url = https://www.bendwavy.org/klitzing/incmats/pt353.htm | title = pd{3,5,3} | author = Richard Klitzing | website = bendwavy.org | access-date = 2023-01-10 }}</ref>，每個頂點都是12個五角反棱柱和4個正十二面體的公共頂點。<ref>Wendy Y. Krieger, Walls and bridges: The view from six dimensions, ''Symmetry: Culture and Science'' Volume 16, Number 2, pages 171–192 (2005) [http://symmetry.hu/content/aus_journal_content_abs_2005_16_2.html] {{Webarchive|url=https://web.archive.org/web/20131007060826/http://symmetry.hu/content/aus_journal_content_abs_2005_16_2.html |date=2013-10-07 }}</ref>
<gallery>
File:Partial truncation order-3 icosahedral honeycomb verf.png|最為頂點圖以施萊格爾投影呈現的欠四面十二面體
File:H3_353-pd_center_ultrawide2.png|{{link-en|部分欠缺二十面體堆砌|Partially diminished icosahedral honeycomb}}
</gallery>

== 參見 ==
*[[四階十二面體]]

== 參考文獻 ==
{{Reflist}}

[[Category:多面體]]