查看“︁星形截角立方体”︁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 = 星形截角立方体
| polyhedron = 星形截角立方体
| imagename = Stellated truncated hexahedron.png
| Type = [[均匀星形多面體]]
| Face = 14
| Edge = 36
| Vertice = 24
| Genu = 
| Face_type = 8個[[三角形]]{3}<br/>6個[[八角星]]{<sup>8</sup>/<sub>3</sub>}
| Vertice_type = 3.<sup>8</sup>/<sub>3</sub>.<sup>8</sup>/<sub>3</sub>
| Coxeter_diagram = {{CDD|node_1|4|rat|d3|node_1|3|node}}
| Schläfli = t{<sup>4</sup>/<sub>3</sub>,3}
| Wythoff = 2 3 &#124; <sup>4</sup>/<sub>3</sub><br/>2 <sup>3</sup>/<sub>2</sub> &#124; <sup>4</sup>/<sub>3</sub>
| Face_configuration = 8{3}+6{<sup>8</sup>/<sub>3</sub>}
| Symmetry_group = O<sub>h</sub>, [4,3], *432
| Index_references = [[均勻多面體|U]]<sub>19</sub>, [[哈羅德·斯科特·麥克唐納·考克斯特|C]]<sub>66</sub>, [[溫尼爾多面體模型列表|W]]<sub>92</sub>
| Rotation_group = 
| Dihedral_angle = 
| Properties = 非凸、半正、點可遞
| vfigimage = Stellated truncated hexahedron vertfig.png
| dual_image = DU19 great triakisoctahedron.png
}}
在[[幾何學]]中，'''星形截角立方體'''是一種[[鳶形二十四面體]]的星形多面體，由互相相交的三角形和八角星組成，其索引為U<sub>19</sub>，對偶多面體是[[大三角化八面體]]<ref>{{Cite web | url = http://archive.lib.msu.edu/crcmath/math/math/g/g309.htm | author = Eric W. Weisstein | title=Great Triakis Octahedron, The Dual of the Stellated Truncated Hexahedron. | publisher = [[密西根州立大學]][[圖書館]] | archiveurl = https://web.archive.org/web/20160319132133/http://archive.lib.msu.edu/crcmath/math/math/g/g309.htm | archivedate =2016-03-19}}</ref>。

== 性質 ==
星形截角立方體共有14個面、36條邊和24個[[頂點 (幾何)|頂點]]<ref>{{cite web | url = http://bulatov.org/polyhedra/uniform/u24.html | title = stellated truncated hexahedron | publisher = bulatov.org | archiveurl = https://web.archive.org/web/20160326183208/http://bulatov.org/polyhedra/uniform/u24.html | archivedate = 2016-03-26}}</ref>，在14個面中，有8個[[正三角形]]和6個[[八角星]]，且每個頂點都是2個八角星和1個三角形的公共頂點，[[頂點圖]]可以用<sup>8</sup>/<sub>3</sub>.<sup>8</sup>/<sub>3</sub>.3表示<ref>{{Cite web | url = https://www.mathconsult.ch/static/unipoly/19.html | title = Uniform Polyhedra 19: stellated truncated hexahedron|publisher=mathconsult | archiveurl = https://web.archive.org/web/20160327052413/http://www.mathconsult.ch/static/unipoly/19.html | archivedate = 2016-03-27}}</ref>。在[[施萊夫利符號|施萊夫利符號中]]計為 t{<sup>4</sup>/<sub>3</sub>,3}，其代表著經過[[截角 (幾何)|截角變換]]的<math>\left\{\frac{4}{3},3\right\}</math>圖形。[[考克斯特記號]]中可以用{{CDD|node_1|4|rat|d3|node_1|3|node}}表示。

=== 結構 ===
星形截角立方體的結構可以視為[[立方體]]透過一種名為「[[星形截角]]」的[[多面體]]變換構造<ref>{{Cite web|url=http://winkervsbecks.github.io/truncation/|title=Truncation|accessdate=2017-03-24|archive-date=2019-04-14|archive-url=https://web.archive.org/web/20190414015548/http://winkervsbecks.github.io/truncation/|dead-url=no}}</ref><ref name="software3d">{{Cite web|url=http://www.software3d.com/StelTruncHexa.php|title=Stellated Truncated Hexahedron|website=software3d.com|accessdate=2017-03-24|archive-date=2019-10-28|archive-url=https://web.archive.org/web/20191028200558/https://www.software3d.com/StelTruncHexa.php|dead-url=no}}</ref>。

將立方體截角變換，截到截面交會後繼續截，但將交會的部分切去，並反向延長，然後持續截更深並延長直到延長的面也交會，並繼續截更深同時也繼續增加延長的面，直到延長的面交錯並凸出。
{|
|[[File:Cube_truncation_0.00.png|100px]]
|[[File:Cube_truncation_0.25.png|100px]]
|[[File:Cube_truncation_0.50.png|100px]]
|[[File:Cube truncation 0.75.png|100px]]
|-
|[[File:Cube truncation 0.75.png|100px]]
|[[File:Cube_truncation_1.00.png|100px]]
|[[File:Cube_truncation_1.25.png|100px]]
|[[File:Cube truncation 1.50.png|100px]]
|-
|[[File:Cube truncation 1.50.png|100px]]
|[[File:Cube truncation 1.75.png|100px]]
|[[File:Cube_truncation_2.25.png|100px]]
|[[File:Cube_truncation_2.50.png|100px]]
|}

其也可以視為在立方體的正方形面角落擺上[[三角錐|直角三角錐]]，因此每個頂點旁都會被擺上三個直角三角錐<ref name="software3d"/>。
{| class = wikitable
|[[File:Cube_truncation_0.00.png|100px]]
|[[File:Cube_truncation_2.25.png|100px]]
|[[File:Cube_truncation_2.50.png|100px]]
|}

=== 二面角 ===
星形截角立方體有兩種[[二面角]]，包括了八角星-三角形二面角和八角星-八角星二面角。其中八角星-八角星二面角為[[直角]]；八角星-三角形二面角為[[三的平方根]]倒數之[[反餘弦]]值<ref name="dmccooey">{{cite web|url = http://dmccooey.com/polyhedra/StellatedTruncatedHexahedron.html|title=Self-Intersecting Truncated Regular Polyhedra: Stellated Truncated Hexahedron |publisher=dmccooey.com| archiveurl = https://web.archive.org/web/20160324081057/http://dmccooey.com/polyhedra/StellatedTruncatedHexahedron.html | archivedate =2016-03-24}}</ref>：
:<math>\cos^{-1} (\frac{\sqrt{3}}{3}) \approx 0.9553166 \approx 54.7356 ^{\circ}</math>

=== 尺寸 ===
若邊長為1，則星形截角立方體的[[外接球]][[半徑]]為<ref name="dmccooey"/>：
:<math>	\frac{ \sqrt{ 7 - 4 \sqrt{2}} }{2} \approx 0.57947082551833871527 </math>
體積<math>V</math>與表面積<math>A</math>為：<ref name="bendwavy stellated truncated hexahedron">{{cite klitzing|quith.htm|rooturl=incmats| title = quasitruncated hexahedron : quith  | access-date = 2021-09-05 }}</ref>
:<math>V=\frac{21-14\sqrt{2}}{3}\approx 0.400337</math>
:<math>A=-12+12\sqrt{2}+2\sqrt{3}\approx 8.434664</math>

=== 頂點座標 ===
若星形截角立方體的邊長為1個單位長，則[[頂點 (幾何)|頂點]][[座標]]為<ref name="dmccooeydata">{{Cite web|url=http://dmccooey.com/polyhedra/StellatedTruncatedHexahedron.txt|title=Data of Stellated Truncated Hexahedron | publisher = dmccooey.com | archiveurl = https://web.archive.org/web/20160902145438/http://dmccooey.com/polyhedra/StellatedTruncatedHexahedron.txt| archivedate =2016-09-01}}</ref>：
:<math>( \pm \frac{\sqrt{2}-1}{2}, \pm \frac{1}{2}, \pm \frac{\sqrt{2}-1}{2})</math>
:<math>( \pm \frac{\sqrt{2}-1}{2}, \pm \frac{\sqrt{2}-1}{2}, \pm \frac{1}{2})</math>
:<math>( \pm \frac{1}{2}, \pm \frac{\sqrt{2}-1}{2}, \pm \frac{\sqrt{2}-1}{2})</math>

== 正交投影 ==
{|class=wikitable width=600
|+ 星形截角立方體的正交投影
|-
! 建立於
! 八角星面
! 正三角形面
! 八角星-八角星<br/>交棱
|-
!圖像
| colspan=3 | [[File:Stellated_truncated_hexahedron_ortho_wireframes.png|480px]]
|}

== 相關多面體 ==
星形截角立方體與另外三種多面體有著相同的{{link-en|頂點布局|vertex arrangement}}，他們分別為[[小斜方截半立方體]]、[[小立方立方八面體]]和[[小斜方立方體]]。

{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
| [[Image:Small rhombicuboctahedron.png|100px]]<BR>[[小斜方截半立方體]]
| [[Image:Small cubicuboctahedron.png|100px]]<BR>[[小立方立方八面體]]
| [[Image:Small rhombihexahedron.png|100px]]<BR>[[小斜方立方體]]
| [[Image:Stellated truncated hexahedron.png|100px]]<BR>星形截角立方體
|}

=== 對偶複合體 ===
星形截角立方體與其對偶的複合體為'''複合星形截角立方體大三角化八面體'''。其共有38個面、72條邊和38個頂點，其尤拉示性數為4，虧格為-1，有6個非凸面<ref>{{Cite web|url=http://bulatov.org/polyhedra/uniform_compounds/uc24.html|title=compound of stellated truncated hexahedron and great triakisoctahedron|publisher=bulatov.org | archiveurl = https://web.archive.org/web/20150906140734/http://bulatov.org/polyhedra/uniform_compounds/uc24.html | archivedate =2016-09-06}}</ref>。

== 參見 ==
* [[截角_(幾何)#廣義截角|星形截角]]
* [[立方體]]

== 參考文獻 ==
{{refbegin|2}}
{{reflist}}
{{refend}}

== 外部連結 ==
* {{mathworld | urlname = StellatedTruncatedHexahedron| title = 星形截角立方体}}
* {{Cite web|url=https://www.facebook.com/pages/%E6%98%9F%E5%BD%A2%E6%88%AA%E8%A7%92%E7%AB%8B%E6%96%B9%E4%BD%93/1452594044958357?rf=102905829751325 |title=星形截角立方体的臉書專頁 |publisher=[[Facebook]] |archiveurl=https://archive.today/20160902152228/https://www.facebook.com/pages/Stellated-truncated-hexahedron/102905829751325?rf=1452594044958357 |archivedate=2016-09-02 |deadurl=no }}
{{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]]