查看“︁扭棱四面體”︁的源代码

{{NoteTA|G1=Math}}
{{Infobox polyhedron
| name =扭棱四面體
| polyhedron = 扭棱四面體
| imagename =Uniform polyhedron-33-s012.svg
| Type = 凸多面體
| Face = 20
| Edge = 30
| Vertice = 12
| Face_type = 8個正三角形<br/>12個等腰三角形
| Coxeter_diagram = {{CDD|node_h|3|node_h|4|node}} (五角十二面體對稱) [[File:Uniform polyhedron-43-h01.svg|24px]]<br />{{CDD|node_h|3|node_h|3|node_h}} (四面體對稱) [[File:Uniform polyhedron-33-s012.svg|24px]]
| Vertice_type = 
| Vertice_configuration = 
| Schläfli = 
| Wythoff = 
| Symmetry_group = [[Pyritohedral symmetry|T<sub>h</sub>]], [4,3<sup>+</sup>], (3*2), order 24
| Index_references = 
| dual = [[五角十二面體]]
| Rotation_group = [[Tetrahedral symmetry|T<sub>d</sub>]], [3,3]<sup>+</sup>, (332), order 12
| Properties = 凸
| 3d_image = 
| vfigimage = 
| dual_image = Polyhedron_pyritohedron_from_yellow.png
| net_image = Pseudoicosahedron_flat.png
}}
在[[幾何學]]中，'''扭棱四面體'''是指[[正四面體]]經過[[扭棱]]變換後所形成的多面體。其拓樸結構與[[正二十面體]]等價。一般會將這種立體的面分為3組，一組是原始四面體的面，另一組是來自[[原像 (幾何)|原像]]之[[頂點圖]]的面，另一組是扭棱變換過程中所形成的面。若兩組面構成的三角形不全等，其結果立體將會變成一個外觀與[[正二十面體]]非常相似，但不相同的立體，因此又稱'''偽正二十面體'''<ref name="Baez">{{cite web|url=http://math.ucr.edu/home/baez/golden.html|title=Fool's Gold|author=John Baez|date=September 11, 2011|access-date=2021-08-14|archive-date=2018-05-19|archive-url=https://web.archive.org/web/20180519002756/http://math.ucr.edu/home/baez/golden.html|dead-url=no}}</ref>，其具備五角十二面體（黃鐵礦晶型）對稱性<ref>{{cite web | url = http://www.polytope.net/hedrondude/sym.htm | title=Symmetries Up To Three Dimensions |archive-date=2021-08-14 |archive-url=https://webcache.googleusercontent.com/search?q=cache:http://www.polytope.net/hedrondude/sym.htm}}</ref>。部分文獻將這種立體稱為扭棱八面體（snub octahedron）<ref name="kappraff2001connections">{{cite book
  |title=Connections: The Geometric Bridge Between Art & Science (2nd Edition)
  |author=Kappraff, J.
  |isbn=9789814491327
  |series=Series On Knots And Everything
  |url=https://books.google.com.tw/books?id=YwLVCgAAQBAJ
  |year=2001
  |page=475
  |publisher=World Scientific Publishing Company
  |access-date=2021-08-14
  |archive-date=2021-08-14
  |archive-url=https://web.archive.org/web/20210814091527/https://books.google.com.tw/books?id=YwLVCgAAQBAJ
  |dead-url=no
  }}</ref>、扭棱截半四面體（或扭棱四-四面體，snub tetratetrahedron）<ref name="orchidpalms">{{Cite web | url = http://www.orchidpalms.com/polyhedra/uniform/twisters/twisters.htm | title = Polyhedral "Twisters" | publisher = orchidpalms.com | author = Jim McNeill | access-date = 2021-08-14 | archive-date = 2019-03-11 | archive-url = https://web.archive.org/web/20190311141122/http://www.orchidpalms.com/polyhedra/uniform/twisters/twisters.htm | dead-url = no }}</ref>。部分礦石的晶體結構會結晶成這種形狀。<ref name="Who Discovered the Icosahedron"/>

這個立體是[[五角十二面體]]的[[對偶多面體]]。<ref>{{Cite journal|location=Chester, England|url=https://www2.physics.ox.ac.uk/sites/default/files/CrystalStructure_2Dand3DPointGroups.pdf|title=Crystallographic and noncrystallographic point groups|isbn=9780792365907|publisher=International Union of Crystallography|edition=1|volume=A|series=International Tables for Crystallography|page=pp. 763–795|date=2006-10-01|journal=International Tables for Crystallography: Space-group symmetry|accessdate=2021-08-14|editor=Th. Hahn|doi=10.1107/97809553602060000100|archive-date=2021-08-14|archive-url=https://web.archive.org/web/20210814090000/https://www2.physics.ox.ac.uk/sites/default/files/CrystalStructure_2Dand3DPointGroups.pdf|dead-url=no}}</ref>

== 性質 ==
扭棱四面體是一種[[二十面體]]，由20個三角形組成。扭棱四面體可以視為四面體經過扭棱變換所形成的立體，在扭稜的過程中會形成3種面，一種是原始四面體的面、另一種是來自原像[[頂點圖]]的面、還有一種是扭棱變換過程中所形成的面。若這三種面皆全等，整個立體將與[[正二十面體]]無異。<ref>{{cite journal|journal=The Mathematical Gazette|volume=82|issue=494|language=en|issn=0025-5572|date=1998-07|pages=203–214|doi=10.2307/3620403|url=https://www.cambridge.org/core/product/identifier/S0025557200161765/type/journal_article|title=Have you seen this number?|accessdate=2021-08-16|author=John Sharp}}</ref><ref>{{Cite web|author=George W. Hart|year=1996|title=Symmetry Planes|url=https://www.georgehart.com/virtual-polyhedra/symmetry_planes.html|access-date=2021-08-14|archive-date=2021-08-16|archive-url=https://web.archive.org/web/20210816104228/https://www.georgehart.com/virtual-polyhedra/symmetry_planes.html|dead-url=no}}</ref>
{|class=wikitable width=200px
|[[File:P1-P5.gif|200px]]<br/>四面體扭棱成扭棱四面體的過程， 其中藍色的面代表原始四面體的面； 紅色的面代表來自原像[[頂點圖]]的面； 白色的面代表扭棱變換過程中所形成的面
|}

=== 拓樸結構 ===
扭棱四面體的拓樸結構與正二十面體等價<ref name="Who Discovered the Icosahedron">{{cite conference
 | author = John Baez
 | date = September 11, 2009
 | title = Who Discovered the Icosahedron?
 | conference = Special Session on History and Philosophy of Mathematics, 2009 Fall Western Section Meeting of the AMS
 | conferenceurl = http://www.ams.org/meetings/sectional/2163_program_ss4.html#title
 | url = https://math.ucr.edu/home/baez/icosahedron/
 | access-date = 2021-08-14
 | archive-date = 2020-05-29
 | archive-url = https://web.archive.org/web/20200529211429/http://www.math.ucr.edu/home/baez/icosahedron/
 | dead-url = no
 }}</ref>。若將四面體扭棱過程中所形成的面兩兩合併為1個菱形，則其拓樸結構與[[截半立方體]]相同。<ref name="orchidpalms"/>
:[[File:Icosahedron_in_cuboctahedron.png|350px]]

=== 頂點座標 ===
這種立體的頂點座標可以用<math>(\pm 2, \pm 1, 0)</math>的循環排列來構造，這個頂點排構建方式又可視為是交錯截角的截角八面體，其與[[耶森二十面體]]相同，但頂點間相連方式不同<ref name=jessen>{{cite journal
 | author = Børge Jessen
 | issue = 2
 | journal = Nordisk Matematisk Tidskrift
 | jstor = 24524998
 | mr = 0226494
 | pages = 90–96
 | title = Orthogonal icosahedra
 | volume = 15
 | year = 1967}}</ref>。而若取<math>(\pm \varphi, \pm 1, 0)</math>則會變為正二十面體，其中<math>\varphi</math>為[[黃金比例]]。<ref name=Baez/>

== 耶森二十面體 ==
{{main|耶森二十面體}}
[[耶森二十面體]]是一個與扭棱四面體相同頂點排列方式的立體，但耶森二十面體頂點間的相連方式與扭棱四面體不同。耶森二十面體是非凸多面體，並具有直角的二面角。<ref name=grunbaum>{{cite book
 | author = Branko Grünbaum
 | contribution = Acoptic polyhedra
 | contribution-url = https://sites.math.washington.edu/~grunbaum/BG225.Acoptic%20polyhedra.pdf
 | doi = 10.1090/conm/223/03137
 | location = Providence, Rhode Island
 | mr = 1661382
 | pages = 163–199
 | publisher = American Mathematical Society
 | series = Contemporary Mathematics
 | title = Advances in Discrete and Computational Geometry (South Hadley, MA, 1996)
 | volume = 223
 | year = 1999
 | access-date = 2021-08-14
 | archive-date = 2021-03-31
 | archive-url = https://web.archive.org/web/20210331174406/https://sites.math.washington.edu/~grunbaum/BG225.Acoptic%20polyhedra.pdf
 | dead-url = no
 }}</ref>
:[[File:Jessen icosahedron with snub icosahedron.png|350px]]

== 對偶多面體 ==
{{main|五角十二面體}}
[[File:Polyhedron pyritohedron transparent max.png|thumb|五角十二面體]]
這種立體因為外觀與[[正二十面體]]十分類似，但不是[[正多面體]]因此又被稱為'''偽正二十面體'''。其對偶多面體也非常類似正二十面體的對偶多面體——[[正十二面體]]，然而其也不是正多面體。這種立體的[[對偶多面體]]為[[五角十二面體]]，是一種由12個不等邊[[五邊形]]組成的[[十二面體]]，具有四面體群對稱性。其與[[正十二面體]]類似，皆是由12個全等的五邊形組成，且每個頂點都是3個五邊形的公共頂點<ref>[http://www.galleries.com/minerals/property/crystal.htm#dodecahe Crystal Habit] {{Wayback|url=http://www.galleries.com/minerals/property/crystal.htm#dodecahe |date=20210816104230 }}. Galleries.com. Retrieved on 2016-12-02.</ref>，但由於其面不是正多邊形，其頂點的排佈未能達到五摺對稱性，因此不屬於正多面體。部分的化學物質或礦石<ref name="article 中村慶三郎1942朝鮮コバルト鑛床調査概報">{{Cite journal
  |title=朝鮮コバルト鑛床調査概報
  |author=中村慶三郎
  |journal=地学雑誌
  |volume=54
  |number=6
  |pages=211--230
  |year=1942
  |publisher=公益社団法人 東京地学協会}}</ref>其晶體形狀是這種形狀，例如[[黄铁矿]]和部分的[[天然氣水合物]]<ref>{{Cite web | url=https://sa.ylib.com/MagArticle.aspx?Unit=easylearn&id=2141 | title=天然氣水合物能替代石油嗎？ | publisher=科學人雜誌 - 遠流 | quote=天然氣水合物常見的兩種籠狀結構為五角十二面體 | access-date=2021-08-14 | archive-date=2021-08-16 | archive-url=https://web.archive.org/web/20210816104230/https://sa.ylib.com/MagArticle.aspx?Unit=easylearn&id=2141 | dead-url=no }}</ref>。其英文名稱Pyritohedron是來自[[黄铁矿]]的英文pyrite以及多面體的字尾-hedron命名的。<ref name="stonetrust Pyrite">{{Cite Web | url=http://www.stonetrust.com/p53/Pyrite.htm | title=Pyrite | publisher=stonetrust | access-date=2019-11-04 | archive-url=https://web.archive.org/web/20190223202302/http://www.stonetrust.com/p53/Pyrite.htm | archive-date=2019-02-23 | dead-url=no }}</ref>

== 相關多面體 ==
{| class=wikitable
|+扭稜立體
|- align=center
!原像
|[[File:Tetrahedron.png|150px]]<br/>[[正四面體]]
|[[File:Hexahedron.png|150px]]<br/>[[立方體]]
|[[File:Octahedron.png|150px]]<br/>[[正八面體]]
|[[File:Dodecahedron.png|150px]]<br/>[[正十二面體]]
|[[File:Icosahedron.png|150px]]<br/>[[正二十面體]]
|- align=center
!rowspan=2|扭稜
|rowspan=2|[[File:Snub tetrahedron.png|150px]]<br/>[[扭棱四面體]]<br/>sr{3,3}
|colspan=2 style="border-bottom: none;"|[[File:Snub_hexahedron.png|150px]]
|colspan=2 style="border-bottom: none;"|[[File:Snub_dodecahedron_ccw.png|150px]]
|- align=center
|style="border-top: none; border-right: none;"|[[扭棱立方体]]<br/>sr{4,3}
|style="border-top: none; border-left: none;"|[[扭棱八面體]]<br/>sr{3,4}
|style="border-top: none; border-right: none;"|[[扭棱十二面体]]<br/>sr{5,3}
|style="border-top: none; border-left: none;"|[[扭棱二十面体]]<br/>sr{3,5}
|- align=center
!完全扭稜
|[[File:Holosnub tetrahedron.svg|150px]]<br/>完全扭稜四面體<br/>β{3,3}
|[[File:Holosnub cube.svg|150px]]<br/>完全扭稜立方體<br/>β{4,3}
|[[File:Holosnub octahedron.png|150px]]<br/>[[二複合二十面體]]<br/>β{3,4}
|[[File:Holosnub dodecahedron.svg|150px]]<br/>完全扭稜十二面體<br/>β{5,3}
|[[File:Small_snub_icosicosidodecahedron.png|150px]]<br/>[[完全扭稜二十面體]]<br/>β{3,5}
|}

== 參見 ==
*[[扭稜鍥形體]]
*[[五角十二面體]]
*[[正二十面體]]

== 參考文獻 ==
{{Reflist|2}}

== 外部連結 ==
*[https://demonstrations.wolfram.com/SnubTetrahedron/ 扭棱四面體的各種變體] {{Wayback|url=https://demonstrations.wolfram.com/SnubTetrahedron/ |date=20210816102950 }}

{{阿基米德立體}}
[[Category:均勻多面體]]