查看“︁七角反棱柱”︁的源代码

{{NoteTA|G1=Math}}
{{Infobox polyhedron
| name = 正七角反棱柱
| polyhedron = 正七角反棱柱
| imagename = Antiprism 7.png
| Type = [[反棱柱]]<br/>[[柱狀均勻多面體]]
| Coxeter_diagram = {{CDD|node_h|2x|node_h|14|node}}<BR>{{CDD|node_h|2x|node_h|7|node_h}}
| Face = 16
| Edge = 28
| Vertice = 14
| Face_type = 14個[[三角形]] <br> 2個[[正七邊形]]
| Vertice_type = 3.3.3.7
| Schläfli = sr{2,7}
| Face_configuration = 14{3}+2{7}
| Wythoff = &#124; 2 2 7
| Symmetry_group = [[Dihedral symmetry in three dimensions|D<sub>7d</sub>]], [2<sup>+</sup>,14], (2*7), order 28
| Index_references = [[Uniform polyhedron|U]]<sub>77(e)</sub>
| dual = [[七方偏方面體]]
| Rotation_group = D<sub>7</sub>, [7,2]<sup>+</sup>, (722), order 14
| Properties = [[凸集|凸]] 
| 3d_image = 
| vfigimage = Heptagonal antiprism vertfig.png
| dual_image = Heptagonal_trapezohedron.png
| net_image = <!-- 檔案不存在 Net of heptagonal antiprism.png -->
}}
在幾何學中，'''七角反棱柱'''又稱為'''反七角柱'''或'''七角反柱'''是指底為[[七邊形]]的[[反棱柱]]，側面由[[三角形]]組成，若每一個面皆為正多邊形則稱為[[正七角反棱柱]]。每個七角反棱柱皆含有16個面<ref name="pugh">{{citation|title=Polyhedra: A Visual Approach|first=Anthony|last=Pugh|publisher=University of California Press|year=1976|isbn=9780520030565|pages=21, 27, 62|url=http://books.google.com/books?id=IDDxpYQTR7kC&pg=PA21|accessdate=2014-06-22|archive-date=2014-07-09|archive-url=https://web.archive.org/web/20140709030631/http://books.google.com/books?id=IDDxpYQTR7kC&pg=PA21|dead-url=no}}.</ref><ref>[http://m.wolframalpha.com/input/?i=heptagonal+antiprism+vertices&lk=1 heptagonal antiprism vertices] {{Wayback|url=http://m.wolframalpha.com/input/?i=heptagonal+antiprism+vertices&lk=1 |date=20160304110933 }} wolframalpha.com [2014-06-22]</ref><ref>[http://www.korthalsaltes.com/model.php?name_en=heptagonal%20antiprism net of heptagonal antiprism] {{Wayback|url=http://www.korthalsaltes.com/model.php?name_en=heptagonal%20antiprism |date=20140626133601 }} korthalsaltes.com [2014-06-22]</ref>，是一種[[十六面體]]。

'''正七角反棱柱'''是基底為[[正七邊形]]的七角反棱柱，其可視為一種半正多面體，[[施萊夫利符號]]s{2,7}表示其可以藉由七邊形二面體透過扭稜變換構造。其具有D<sub>7</sub>對稱群<ref>Melnyk, Theodor William, Osvald Knop, and William Robert Smith. "Extremal arrangements of points and unit charges on a sphere: equilibrium configurations revisited." Canadian Journal of Chemistry 55.10 (1977): 1745-1761.</ref>，其在{{en-link|威佐夫符號|Wythoff symbol}}中用&#124; 2 2 7表示<ref>[http://home.cc.umanitoba.ca/~budzelaa/Polyhedra/Prisms.html Heptagonal prisms and antiprisms] {{Wayback|url=http://home.cc.umanitoba.ca/~budzelaa/Polyhedra/Prisms.html |date=20180630133145 }} umanitoba[[.ca]] [2014-6-22]</ref>。

== 正七角反棱柱 ==
<gallery>
File:Antiprism 7.png|正七角反棱柱<ref>[http://www.antiprism.com/examples/150_named_models/560_polygons/ant7_Med.jpg.26.html {7}-antiprism] {{Wayback|url=http://www.antiprism.com/examples/150_named_models/560_polygons/ant7_Med.jpg.26.html |date=20190218172844 }} antiprism.com [2014-6-22]</ref>
</gallery>
當底面為正七邊形時，會具備一些特別的性質

當基底邊長為a的時候：
*[[頂點 (幾何)|頂點]]數目：14<ref>[http://dmccooey.com/polyhedra/HeptagonalAntiprism.html Heptagonal Antiprism] {{Wayback|url=http://dmccooey.com/polyhedra/HeptagonalAntiprism.html |date=20160304120922 }} dmccooey.com [2014-6-22]</ref>
*[[邊 (幾何)|邊]]數目：28
*[[面 (幾何)|面]]數目：16
*[[對偶多面體]]：[[七方偏方面體]]
高：<math>\frac{1}{2}a\sqrt{4-\sec{\frac{\pi}{14}}\sec{\frac{\pi}{14}}}</math><br />
表面積：<math>\frac{7}{2}a^{2}(\cot{\frac{\pi}{7}}+\sqrt{3})</math><br />
體積：<math>\frac{7}{12}a^{3}(\cot{\frac{\pi}{14}}+\cot{\frac{\pi}{7}})\sqrt{1-\frac{1}{4}\sec{\frac{\pi}{14}}\sec{\frac{\pi}{14}}}</math>

== 相關多面體與鑲嵌 ==
{| class="wikitable"
|+ 半正七邊形二面體球面多面體
|- align=center
!colspan=7|{{link-en|球面對稱群列表|List of spherical symmetry groups|對稱群}}：[[二面體群|[7,2]]], (*722)
![7,2]<sup>+</sup>, (722)
|- align=center
!{{CDD|node_1|7|node|2|node}}
!{{CDD|node_1|7|node_1|2|node}}
!{{CDD|node|7|node_1|2|node}}
!{{CDD|node|7|node_1|2|node_1}}
!{{CDD|node|7|node|2|node_1}}
!{{CDD|node_1|7|node|2|node_1}}
!{{CDD|node_1|7|node_1|2|node_1}}
!{{CDD|node_h|7|node_h|2x|node_h}}
|- align=center
| [[File:Heptagon.svg|50px]]
| [[File:Regular_tetradecagon.svg|50px]]
| [[File:Heptagon.svg|50px]]
| [[File:Spherical heptagonal prism.svg|50px]]
| [[File:Spherical heptagonal hosohedron.svg|50px]]
| 
| [[File:Spherical truncated hexagonal prism.png|50px]]
| [[File:Spherical_heptagonal_antiprism.svg|50px]]
|- align=center
|{{AnyLink|多邊形二面體|{7,2}|七邊形二面體}}
|{{AnyLink|十四邊形|t{7,2}|十四邊形二面體}}
|{{AnyLink|七邊形|r{7,2}|七邊形二面體}}
|[[正七角柱|2t{7,2}=t{2,7}]]
|{{AnyLink|多面形|2r{7,2}{{=}}{2,7}|七面形}}
|{{AnyLink|正七角柱|rr{7,2}|小斜方截半七邊形二面體}}
|{{AnyLink|正七角柱|tr{7,2}|大斜方截半七邊形二面體}}
|{{AnyLink|正七角反稜柱|sr{7,2}|扭稜七邊形二面體}}

|-
!colspan=10|半正對偶
|- align=center
!{{CDD|node_f1|7|node|2|node}}
!{{CDD|node_f1|7|node_f1|2|node}}
!{{CDD|node|7|node_f1|2|node}}
!{{CDD|node|7|node_f1|2|node_f1}}
!{{CDD|node|7|node|2|node_f1}}
!{{CDD|node_f1|7|node|2|node_f1}}
!{{CDD|node_f1|7|node_f1|2|node_f1}}
!{{CDD|node_fh|7|node_fh|2x|node_fh}}
|- align=center
| [[File:Spherical heptagonal hosohedron.svg|50px]]
| [[File:Spherical dodecagonal hosohedron.svg|50px]]
| [[File:Spherical heptagonal hosohedron.svg|50px]]
| [[File:Spherical hexagonal bipyramid.svg|50px]]
| [[File:Heptagon.svg|50px]]
| [[File:Spherical heptagonal bipyramid.svg|50px]]
| [[File:Spherical dodecagonal bipyramid.svg|50px]]
| [[File:Spherical heptagonal trapezohedron.svg|50px]]
|- align=center
|{{AnyLink|多面形|V7<sup>2</sup>|七面形}}
|{{AnyLink|多面形|V14<sup>2</sup>|十四面形}}
|{{AnyLink|多面形|V7<sup>2</sup>|七面形}}
|[[雙七角錐|V4.4.7]]
|{{AnyLink|多邊形二面體|V2<sup>7</sup>|七邊形二面體}}
|[[雙七角錐|V4.4.7]]
|{{AnyLink|雙錐體|V4.4.14|雙十四角錐}}
|[[七方偏方面體|V3.3.3.7]]
|}
{|class="wikitable"
|+ 半正反棱柱系列
|-
![[二角反棱柱|2]]
![[三角反棱柱|3]]
![[四角反棱柱|4]]
![[五角反棱柱|5]]
![[六角反棱柱|6]]
!7
![[八角反棱柱|8]]
![[九角反棱柱|9]]
![[十角反棱柱|10]]
![[十一角反棱柱|11]]
![[十二角反棱柱|12]]
![[反棱柱|''n'']]
|-
!s{2,4}<BR>sr{2,2}
!s{2,6}<BR>sr{2,3}
!s{2,8}<BR>sr{2,4}
!s{2,10}<BR>sr{2,5}
!s{2,12}<BR>sr{2,6}
!s{2,14}<BR>sr{2,7}
!s{2,16}<BR>sr{2,8}
!s{2,18}<BR>sr{2,9}
!s{2,20}<BR>sr{2,10}
!s{2,22}<BR>sr{2,11}
!s{2,24}<BR>sr{2,12}
!s{2,2''n''}<BR>sr{2,n}
|-
!{{CDD|node_h|2x|node_h|4|node}}<BR>{{CDD|node_h|2x|node_h|2x|node_h}}
!{{CDD|node_h|2x|node_h|6|node}}<BR>{{CDD|node_h|2x|node_h|3|node_h}}
!{{CDD|node_h|2x|node_h|8|node}}<BR>{{CDD|node_h|2x|node_h|4|node_h}}
!{{CDD|node_h|2x|node_h|1x|0x|node}}<BR>{{CDD|node_h|2x|node_h|5|node_h}}
!{{CDD|node_h|2x|node_h|1x|2x|node}}<BR>{{CDD|node_h|2x|node_h|6|node_h}}
!{{CDD|node_h|2x|node_h|1x|4|node}}<BR>{{CDD|node_h|2x|node_h|7|node_h}}
!{{CDD|node_h|2x|node_h|1x|6|node}}<BR>{{CDD|node_h|2x|node_h|8|node_h}}
!{{CDD|node_h|2x|node_h|1x|8|node}}<BR>{{CDD|node_h|2x|node_h|9|node_h}}
!{{CDD|node_h|2x|node_h|2x|0x|node}}<BR>{{CDD|node_h|2x|node_h|10|node_h}}
!{{CDD|node_h|2x|node_h|2x|2x|node}}<BR>{{CDD|node_h|2x|node_h|11|node_h}}
!{{CDD|node_h|2x|node_h|2x|4|node}}<BR>{{CDD|node_h|2x|node_h|12|node_h}}
!{{CDD|node_h|2x|node_h|2x|n|node}}<BR>{{CDD|node_h|2x|node_h|n|node_h}}
|-
|[[File:Digonal_antiprism.png|40px]]
|[[Image:Trigonal antiprism.png|40px]]
|[[Image:Square antiprism.png|40px]]
|[[Image:Pentagonal antiprism.png|40px]]
|[[File:Hexagonal antiprism.png|40px]]
|[[File:Antiprism 7.png|40px]]
|[[Image:Octagonal antiprism.png|40px]]
|[[File:Enneagonal antiprism.png|40px]]
|[[Image:Decagonal antiprism.png|40px]]
|[[File:Hendecagonal antiprism.png|40px]]
|[[Image:Dodecagonal antiprism.png|40px]]
|-
!colspan=11|作為球面鑲嵌
|-
|[[Image:Spherical digonal antiprism with digonal face.svg|40px]]
|[[Image:Spherical trigonal antiprism.svg|40px]]
|[[Image:Spherical square antiprism.svg|40px]]
|[[Image:Spherical pentagonal antiprism.svg|40px]]
|[[Image:Spherical hexagonal antiprism.svg|40px]]
|[[Image:Spherical heptagonal antiprism.svg|40px]]
|[[Image:Spherical octagonal antiprism.svg|40px]]
|
|
|
|
|}

== 在其他領域中 ==

== 參見 ==
*[[反棱柱]]
*[[偏方面體]]
== 參考文獻 ==
<references/>
*Fowler, P. W., T. Tarnai, and Zs Gáspár. "From circle packing to covering on a sphere with antipodal constraints." Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 458.2025 (2002): 2275-2287.
*[http://pages.rediff.com/heptagonal-antiprism/1266821 heptagonal antiprism]{{Wayback|url=http://pages.rediff.com/heptagonal-antiprism/1266821 |date=20190510161723 }} rediff.com [2014-6-22]
==外部連結==
*[http://mathworld.wolfram.com/Antiprism.html]{{Wayback|url=http://mathworld.wolfram.com/Antiprism.html |date=20190502103253 }}
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra]{{Wayback|url=http://www.georgehart.com/virtual-polyhedra/vp.html |date=20080223132026 }} www.georgehart.com: The Encyclopedia of Polyhedra
** [[VRML]] [http://www.georgehart.com/virtual-polyhedra/vrml/heptagonal_antiprism.wrl model]{{Wayback|url=http://www.georgehart.com/virtual-polyhedra/vrml/heptagonal_antiprism.wrl |date=20141023102605 }}
** [http://www.georgehart.com/virtual-polyhedra/conway_notation.html Conway Notation for Polyhedra]{{Wayback|url=http://www.georgehart.com/virtual-polyhedra/conway_notation.html |date=20141129085342 }} Try: "A7"
{{反棱柱}}
{{錐體與柱體}}

[[Category:均勻多面體]]