查看“︁四角錐柱”︁的源代码

{{noteTA
|G1=Math
|1=zh:Johnson多面體; zh-hans:约翰逊多面体; zh-hant:詹森多面體;
|2=zh-hans:台塔; zh-hant:帳塔;
|3=zh-hans:丸塔; zh-hant:罩帳;
|4=zh-hans:约翰逊; zh-hant:詹森;
}}
{{Infobox polyhedron
| name =正四角錐柱
| polyhedron = 正四角錐柱
| imagename = elongated_square_pyramid.png
| Type = [[棱錐柱]]<br/>[[詹森多面體]] <br/> [[正三角錐柱|J<sub>7</sub>]] - '''J<sub>8</sub>''' - [[正五角錐柱|J<sub>9</sub>]]
| Face = 9
| Edge = 16
| Vertice = 9
| Face_type = 4個[[三角形]]<br/>5個[[正方形]]
| Vertice_configuration = 4個(4<sup>3</sup>)<br/>1個(3<sup>4</sup>)<br/>4個(3<sup>2</sup>.4<sup>2</sup>)
| Conway = P4+Y4
| Symmetry_group ={{link-en|循環對稱性|cyclic symmetries|''C''<sub>4v</sub>}}, [4], (*44)
| dual = [[正四角錐柱|四角錐柱]]（自身對偶）
| Rotation_group = ''C''<sub>4</sub>, [4]<sup>+</sup>, (44)
| Properties = 凸、demi-regular
| 3d_image = Pirámide_cuadrada_elongada.stl
| dual_image = elongated_square_pyramid.png
| net_image = Elongated_Square_Pyramid_Net.svg
}}
在[[幾何學]]中，'''四角錐柱'''是底面為[[四邊形]]的[[角錐柱]]，其可以視為將底面全等的四角錐與四角柱疊合所形成的立體，又稱為'''方尖碑'''（Obelisk）<ref name="mathworld">{{cite mathworld|urlname=ElongatedSquarePyramid|title=Elongated Square Pyramid}}</ref>。若底面為[[正方形]]則稱為正四角錐柱，等邊的正四角錐柱是一種[[詹森多面體]]<ref name="mathworld"/>。四角錐柱具有9個面、16個邊、和9個頂點，每個四角錐柱皆為一個[[九面體]]。

== 詹森多面體 ==
考慮一個正四角錐柱，若每一個面皆為正多邊形，則為92種[[Johnson多面體]]（'''J<sub>8</sub>'''）中的其中一個，也是[[角錐柱]]的一種，可由詹森多面體中的[[正四角錐]]與[[柏拉圖立體]]中的[[立方體]]於相等大小的[[正方形]][[面 (幾何)|面]]接合而組成，這種立體又稱為'''側錐立方體'''（augmented cube）<ref name="polyhedra.tessera.li">{{cite web | url=https://polyhedra.tessera.li/elongated-square-pyramid/info | title=Elongated square pyramid | website=polyhedra.tessera.li | access-date=2022-09-07 | archive-date=2022-09-07 | archive-url=https://web.archive.org/web/20220907133405/https://polyhedra.tessera.li/elongated-square-pyramid/info | dead-url=no }}</ref>。詹森多面體是[[凸多面體]]，面皆由正多邊形組成但不屬於均勻多面體，共有92種。這些立體最早在1966年由{{tsl|en|Norman Johnson (mathematician)|諾曼·詹森}}（Norman Johnson）命名並給予描述<ref>{{citation
 | last = Johnson | first = Norman W.
 | doi = 10.4153/cjm-1966-021-8
 | journal = {{link-en|Canadian Journal of Mathematics}}
 | mr = 0185507
 | pages = 169–200
 | title = Convex polyhedra with regular faces
 | volume = 18
 | year = 1966
 | zbl = 0132.14603
  }}</ref>。

=== 性質 ===
正四角錐柱共由9個[[面 (幾何)|面]]、16條[[邊 (幾何)|邊]]、和9個[[頂點 (幾何)|頂點]]組成<ref name="dmccooey">{{Cite web| url = http://dmccooey.com/polyhedra/ElongatedSquarePyramid.html| title = Johnson Solids: Elongated Square Pyramid| author = David I. McCooey| access-date = 2022-09-07| archive-date = 2022-09-07| archive-url = https://web.archive.org/web/20220907134021/http://dmccooey.com/polyhedra/ElongatedSquarePyramid.html| dead-url = no}}</ref><ref>{{cite web | url=https://www.qfbox.info/4d/J8 | title=The Elongated Square Pyramid | website=qfbox.info | access-date=2022-09-07 | archive-date=2022-10-05 | archive-url=https://web.archive.org/web/20221005062252/https://www.qfbox.info/4d/J8 | dead-url=no }}</ref><ref name="polyhedra.tessera.li"/><ref name="bulatov">{{Cite web|url=http://bulatov.org/polyhedra/johnson/j052.html|title=elongated square pyramid|author=V. Bulatov|access-date=2022-09-07|archive-date=2022-05-22|archive-url=https://web.archive.org/web/20220522173805/http://bulatov.org/polyhedra/johnson/j052.html|dead-url=no}}</ref>。在其9個面中，有4個正三角形面和5個正方形面<ref name="dmccooey"/>。在其9個頂點中，有3種頂點，分別是4個三角形的公共頂點，在[[頂點圖]]中可以用[3<sup>4</sup>]來表示<ref name="Richard Klitzing">{{cite web | url = https://bendwavy.org/klitzing/incmats/esquipy.htm | title = elongated square pyramid, esquipy | author = Richard Klitzing | website = bendwavy.org | access-date = 2022-09-07 | archive-date = 2022-11-14 | archive-url = https://web.archive.org/web/20221114153741/https://www.bendwavy.org/klitzing/incmats/esquipy.htm | dead-url = no }}</ref>，這種頂點有1個<ref name="polyhedra.tessera.li"/>、另一種頂點為2個三角形和2個正方形的公共頂點，在[[頂點圖]]中可以用[3<sup>2</sup>,4<sup>2</sup>]來表示<ref name="Richard Klitzing"/>，這種頂點有4個<ref name="polyhedra.tessera.li"/>、還有一種頂點是3個正方形的公共頂點，在[[頂點圖]]中可以用[4<sup>3</sup>]來表示<ref name="Richard Klitzing"/>，這種頂點有4個<ref name="polyhedra.tessera.li"/>。

==== 體積與表面積 ====
若一個正四角錐柱邊長為<math>L</math>，則其[[體積]]<math>V</math>與表面積<math>A</math>為：<ref name="pye">{{cite journal |last=Sapiña |first=R. |title=Area and volume of the Johnson solid J<sub>8</sub> |url=https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J8/calculadora-area-volumen-formulas.html |issn=2659-9899 |access-date=2020-08-28 |language=es |journal=Problemas y Ecuaciones |archive-date=2020-09-30 |archive-url=https://web.archive.org/web/20200930062311/https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J8/calculadora-area-volumen-formulas.html |dead-url=no }}</ref>
:<math>V =  L^3 \left( 1 + \frac{\sqrt{2}}{6}\right)\approx L^3\cdot 1.23570226</math>
:<math>A =  L^2 \cdot \left( 5 + \sqrt{3} \right) \approx L^2\cdot 6.732050808</math>
這樣的正四角錐柱整體的高<math>H</math>為：
:<math>H =  L\cdot \left( 1 + \frac{\sqrt{2}}{2}\right) \approx L\cdot 1.707106781</math>

==== 二面角 ====
正四角錐柱共有3種[[二面角]]，分別為三角形和正方形的二面角、三角形和三方形的二面角以及正方形和正方形的二面角<ref name="Richard Klitzing"/>。其中正方形和正方形的二面角為直角，即90度角。<ref name="Richard Klitzing"/>
:<math>\angle</math>正方形<math>,</math>正方形<math>\,=90 ^\circ</math>

而三角形和正方形的二面角為負根號三分之二的反餘弦值，約為144.7356度：<ref name="Richard Klitzing"/>
:<math>\angle</math>三角形<math>,</math>正方形<math>\,=\arccos\left(-\sqrt{\frac{2}{3}}\right)\approx 2.52611294 \approx 144.735610 ^\circ</math>

三角形和三方形的二面角為負三分之一的反餘弦值，約為109.471度：<ref name="Richard Klitzing"/>
:<math>\angle</math>三角形<math>,</math>三方形<math>\,=\arccos\left(-\frac{1}{3}\right)\approx 1.9106332 \approx 109.471221 ^\circ</math>

==== 頂點座標 ====
若一個正四角錐柱邊長為單位長，則其頂點座標為：<ref>{{Cite web | url = http://dmccooey.com/polyhedra/ElongatedSquarePyramid.txt | title = Data of Elongated Square Pyramid | author = David I. McCooey | access-date = 2022-09-07 | archive-date = 2022-09-07 | archive-url = https://web.archive.org/web/20220907135815/http://dmccooey.com/polyhedra/ElongatedSquarePyramid.txt | dead-url = no }}</ref>
:<math>\left(\pm\frac{1}{2},\,\pm\frac{1}{2},\,\pm\frac{1}{2}\right)</math>
:<math>\left(0,\,0,\,\frac{1+\sqrt{2}}{2}\right)</math>

=== 對偶多面體 ===
四角錐柱的對偶多面體為四角錐台錐，由4個三角形、1個正方形和4個梯形組成。
{| class=wikitable width=320
|- valign=top
!四角錐柱的對偶多面體
!對偶多面體的展開圖
|- valign=top
|[[File:Dual elongated square pyramid.png|160px]]
|[[File:Dual elongated square pyramid net.png|160px]]
|}

== 相關多面體 ==
四角錐柱是以四邊形為底的角錐柱。<ref name="mathworld"/>以其他多邊形為底的角錐柱有：
{| class=wikitable
|+ [[棱锥柱|棱锥柱体]]
|-
![[二角錐柱]]
![[三角錐柱]]
!四角錐柱
![[五角錐柱]]
![[六角錐柱]]
![[七角錐柱]]
!...
![[圓錐柱]]
|- valign=top
|[[File:Elongated digonal pyramid.png|100px]]
|[[File:Elongated trigonal pyramid.png|100px]]
|[[File:Elongated tetragonal pyramid.png|100px]]
|[[File:Elongated Pentagonale piramide.png|100px]]
|[[File:Elongated hexagonal pyramid.png|100px]]
|[[File:Elongated heptagonal pyramid.png|100px]]
|
|[[File:Elongated cone.png|100px]]
|}
四角錐柱可以視為側錐四角柱，其為底面為四邊形之柱體對應的側錐柱體，其他的側錐柱體有：
{| class=wikitable
|+ 側錐柱體
|-
!3
!4
!5
!6
!7
|- valign=top align=center
|[[File:Green augmented triangular prism.svg|100px]]
|[[File:Green augmented square prism.svg|100px]]
|[[File:Green augmented pentagonal prism.svg|100px]]
|[[File:Green augmented hexagonal prism.svg|100px]]
|[[File:Green augmented heptagonal prism.svg|100px]]
|- align=center
|[[側錐三角柱]]
|'''側錐四角柱'''
|[[側錐五角柱]]
|[[側錐六角柱]]
|[[側錐七角柱]]
|}

此外，正四角錐柱可以與正四面體共同填滿空間<ref>{{Cite web|url=http://woodenpolyhedra.web.fc2.com/J8.html|title=J8 honeycomb|website=woodenpolyhedra.web.fc2.com|access-date=2022-09-07|archive-date=2022-08-07|archive-url=https://web.archive.org/web/20220807051038/http://woodenpolyhedra.web.fc2.com/J8.html|dead-url=no}}</ref>。

==參見==
*[[詹森多面體]]
*[[正方體]]
*[[正四角錐]]

== 參考文獻 ==
{{Reflist}}

{{錐體與柱體}}
{{詹森多面體}}
{{Convex polyhedron navigator}}

[[Category:约翰逊多面体]]