查看“︁立方體半形”︁的源代码

{{NoteTA|G1=Math}}
{{distinguish|超半方形}}
{{Infobox polyhedron
| name = 立方體半形
| polyhedron = 立方體半形
| imagename = Hemicube.svg
| original = [[立方體]] （[[多面體半形|半形體]]）
| en_name=hemicube
| Type = {{en-link|抽象多胞形|Abstract polytope}}<br/>{{en-link|射影多面體|projective polyhedron}}
| Coxeter_diagram = 
| Face = 3
| Edge = 6
| Vertice = 4
| Genu = 
| Face_type = 正方形
| Vertice_type = 4.4.4
| Vertice_configuration = 
| Schläfli = {4,3}/2<br/>{4,3}<sub>3</sub>
| Wythoff = 
| Face_configuration = 
| Conway = 
| Symmetry_group = ''S''<sub>4</sub>, 24階
| Index_references = 
| dual = [[八面體半形]]<ref name= "Regular Map Petrial tetrahedron"/>
| Rotation_group = 
| Dihedral_angle = 
| Properties = [[可定向性|不可定向]]、 [[歐拉示性數]]為1
| 3d_image = 
| vfigimage = Cube_vertfig.png
| dual_image = Hemi-octahedron2.png
| net_image = 
}}
在抽象[[幾何學]]中，'''立方體半形'''是一種僅由一半數量的[[立方體]]面構成的抽象多面體。這個抽象多面體與立方體類似，它們的每個頂點都是3個正方形的公共頂點，然而立方體有6個面，而立方體半形僅有3個面；同時，這個立體無法嵌入在三維歐幾里得空間中<ref name="Mark Mixer 2009">{{Cite web | url=http://mathserver.neu.edu/~ramras/Tapas/MixerNotes09.pdf | title=Introduction to abstract polytopes | author=Mark Mixer | date=2009-05-19 | publisher=Northeastern University | access-date=2021-07-31 | archive-date=2021-08-06 | archive-url=https://web.archive.org/web/20210806085418/http://mathserver.neu.edu/~ramras/Tapas/MixerNotes09.pdf }}</ref>。在拓樸學上，其可以視為[[正四面體]]的[[皮特里對偶]]<ref name="phdthesis pellicer2007graficas">{{Citation
|title=Gráficas cpr y polytopos abstractos regulares
|author=Pellicer, D
|year=2007
|publisher=Universidad Nacional Autónoma de México, Mexico City, Mexico
|url=https://www.matem.unam.mx/~roli/docencia/tesis/Thesis_Daniel.pdf
|accessdate=2021-07-31
|archive-date=2021-08-06
|archive-url=https://web.archive.org/web/20210806085213/https://www.matem.unam.mx/~roli/docencia/tesis/Thesis_Daniel.pdf
}}</ref>。

== 性質 ==
立方體半形由3個[[面 (幾何)|面]]、6條[[邊 (幾何)|邊]]和4個[[頂點 (幾何)|頂點]]組成，每個面都是[[正方形]]，且每個頂點都是3個正方形的公共頂點，在施萊夫利符號中可以用{4,3}/2或{4,3}<sub>3</sub>來表示，其中{4,3}代表且每個頂點都是3個正方形的公共頂點<ref name="article sequin2007hyperseeing">{{Cite journal
  |title=Hyperseeing the regular Hendecachoron
  |author=Séquin, Carlo H and Lanier, Jaron
  |journal=Proc ISAMA
  |pages=159–166
  |year=2007
  |url=http://graphics.berkeley.edu/papers/Sequin-HRH-2007-05/
  |access-date=2021-08-04
  |archive-date=2021-08-04
  |archive-url=https://web.archive.org/web/20210804075509/http://graphics.berkeley.edu/papers/Sequin-HRH-2007-05/
  }}</ref>，然而{4,3}代表正常的立方體，即[[正六面體]]，因此用[[除以二|「/2」]]符號來表示所有元素都僅有立方體的一半數量<ref name="article hartley2003classification">{{Cite journal
|title=The Classification of Rank 4 Locally Projective Polytopes and Their Quotients
|author=Hartley, Michael I
|journal=arXiv preprint math/0310429
|url=https://arxiv.org/pdf/math/0310429.pdf
|year=2003
|access-date=2021-07-31
|archive-date=2021-08-06
|archive-url=https://web.archive.org/web/20210806090202/https://arxiv.org/pdf/math/0310429.pdf
}}</ref><ref name="Abstract Regular Polytopes 2002">{{citation | last1 = McMullen | first1 = Peter | first2 = Egon | last2 = Schulte | chapter = 6C. Projective Regular Polytopes | title = Abstract Regular Polytopes | edition = 1st | publisher = Cambridge University Press | isbn = 0-521-81496-0 |date=December 2002 | pages = [https://books.google.com/books?id=JfmlMYe6MJgC&pg=PA162 162–165] }}</ref>。
:[[File:Hemicube.png|250px]]
立方體半形的對偶多面體為[[正八面體半形]]，這個在更高維度的類比結構中同樣成立，即<math>n+1</math>維超方形半形（[[施萊夫利符號]]：<math>{\left\{4,3^{n-1}\right\} }_{n+1}</math>）的對偶多胞形為<math>n+1</math>維正軸形半形（施萊夫利符號：<math>{\left\{3^{n-1},4\right\} }_{n+1}</math>）。<ref name="article hartley2003classification"/><ref name="Abstract Regular Polytopes 2002"/>

特別地，這個立體的每個面皆與相鄰面共用2條邊，且每個面都包含了立體中所有頂點。一般而言，多胞形的面可以透過其點集來決定<ref name="article kaibel2002computing">{{Cite journal
|title=Computing the face lattice of a polytope from its vertex-facet incidences
|author=Kaibel, Volker and Pfetsch, Marc E
|journal=Computational Geometry
|volume=23
|number=3
|pages=281–290
|year=2002
|url=https://www2.mathematik.tu-darmstadt.de/~pfetsch/Publications/facelattice.pdf
|publisher=Elsevier
|access-date=2021-07-31
|archive-date=2021-08-06
|archive-url=https://web.archive.org/web/20210806085343/https://www2.mathematik.tu-darmstadt.de/~pfetsch/Publications/facelattice.pdf
}}</ref>，也就是說，一般不會存在2個相異面點集合相同的情況，因此這個立體是面無法僅從點集來確定的抽象多面體的例子之一。
:[[File:Tetrahedron 3 petrie polygons.png|160px]]

=== 構造 ===
立方體半形可從有公共頂點的半個立方體（即三個面，下圖的I、II、III）開始構造。此形狀的邊界為一個六邊形，然後下一步是將此六條邊分成三組對邊（下圖的4、5、6），將每對邊（沿同一方向，例如順時針）黏合，就得到立方體半形<ref name="article sequin2007hyperseeing"/>。這樣的構建方式使用了正四面體的骨架<ref name="Carlo H. Séquin">{{Cite web | url=https://people.eecs.berkeley.edu/~sequin/SCULPTS/sequin.html | title=Sculpture designs and math models | author=Carlo H. Séquin | publisher=University of California, Berkeley | quote=[http://people.eecs.berkeley.edu/%7Esequin/SCULPTS/CHS_miniSculpts/RibbedSculptures/Ribbed_HemiCube_A_.JPG "Ribbed Hemicube" (June 2007) - 5"] | access-date=2021-07-31 | archive-date=2021-10-22 | archive-url=https://web.archive.org/web/20211022032755/https://people.eecs.berkeley.edu/~sequin/SCULPTS/sequin.html }}</ref>，同時其構成的面不會共面<ref name="article sequin2007hyperseeing"/>，其與正四面體的皮特里多邊形相同，其骨架在圖論中對應到四面體圖，可以視為''K''<sub>4</sub>[[完全圖]]嵌入於[[射影平面]]上的結果。<ref name="article sequin2007hyperseeing"/>
{| class="wikitable" width="300"
|- align="center"
| [[Image:Hemicube2.PNG|150px]]<br/>立方體半形
| [[Image:3-simplex_graph.svg|150px]]<br/>''K''<sub>4</sub>[[完全圖]]
| [[Image:Petrial_tetrahedron.gif|150px]]<br/>皮特里四面體
|}

=== 具象化 ===
立方體半形可被視為是{{En-link|射影多面體|projective polyhedron}} （可視為由三個[[四邊形]]構成的[[實射影平面]][[鑲嵌_(幾何學)|鑲嵌]]）<ref name="phdthesis helfand2013constructions">{{Citation
  |title=Constructions of k-orbit Abstract Polytopes
  |author=Helfand, Ilanit
  |year=2013
  |publisher=Northeastern University}}</ref>。要將其視覺化，可以透過將射影平面構築為一個半球體，並過半球體的邊界連接[[對蹠點]]，同時確保連接的部分能將半球體平均分割成三等份。

立方體半形和[[超半方形|半立方體]]不同，立方體半形是一個{{En-link|射影多面體|projective polyhedron}}，且無法嵌入在三維歐幾里得空間中<ref name="Mark Mixer 2009"/>；而半立方體是一個位於三維歐幾里德空間中的普通多面體。 雖然它們的頂點數皆為立方體的一半，立方體'''半形'''可以視為立方體的[[商空間]]，而'''半'''立方體則不是，半立方體只有頂點為立方體頂點的[[子集]]。

== 皮特里四面體 ==
{{Infobox polyhedron
| name = 皮特里四面體
| polyhedron = 皮特里四面體
| imagename = Petrial_tetrahedron.gif
| caption = 以不同顏色表示每個面
| Type = 皮特里對偶<br/>[[正則地區圖]]
| WikidataID = Q107723233
| Schläfli = {3,3}<sup>{{pi}}</sup><br/>{4,3}<sub>3</sub>
| Face = 3
| Edge = 6
| Vertice = 4
| Symmetry_group = T<sub>d</sub>, [3,3], *332
| dual = [[八面體半形]]
| Dihedral_angle = (不存在)
| Properties = [[扭歪多邊形|扭歪]]、[[正則地區圖|正則]]
}}
皮特里四面體是[[正四面體]]的[[皮特里對偶]]<ref name= "Regular Map Petrial tetrahedron">{{cite web | url = http://www.weddslist.com/rmdb/map.php?a=N1.1p | title = The hemicube | publisher = Regular Map database - map details | accessdate = 2021-07-24 | archive-date = 2019-05-02 | archive-url = https://web.archive.org/web/20190502092501/http://www.weddslist.com/rmdb/map.php?a=N1.1p }}</ref><ref name="Regular Map tetrahedron" >{{cite web | url = http://www.weddslist.com/rmdb/map.php?a=R0.1 | title = The tetrahedron | publisher = Regular Map database - map details | accessdate = 2021-07-24 | archive-date = 2021-08-23 | archive-url = https://web.archive.org/web/20210823170239/http://www.weddslist.com/rmdb/map.php?a=R0.1 }}</ref>。在拓樸學上，這個結構與立方體半形同構，並可以視為立方體半形的一種具象化方式<ref name="article sequin2007hyperseeing"/>。相對的立方體半形的皮特里對偶為[[正四面體]]，這意味著其皮特里多邊形可以與[[超半方形|半立方體]]（此例對應[[正四面體]]）的面對應<ref name="article bracho2014finite">{{Cite journal
  |title=A Finite Chiral 4-Polytope in <math>\mathbb{R}^4</math>
  |author=Bracho, Javier and Hubard, Isabel and Pellicer, Daniel
  |journal=Discrete & Computational Geometry
  |volume=52
  |number=4
  |pages=799--805
  |year=2014
  |publisher=Springer}}</ref>。也就是說，立方體半形和正四面體互為[[皮特里對偶]]。<ref name= "Regular Map Petrial tetrahedron"/><ref name="Regular Map tetrahedron"/>

皮特里四面體由3個面、6條邊和4個頂點組成，其中，3個面皆為[[正四面體]]的[[皮特里多邊形]]。[[正四面體]]的[[皮特里多邊形]]是一個扭歪四邊形。<ref>{{citation|title=Geometry at Work|series=MAA Notes|volume=53|first=Catherine A.|last=Gorini|publisher=Cambridge University Press|year=2000|isbn=9780883851647|page=181|url=https://books.google.com/books?id=Eb6uSLa2k6IC&pg=PA181}}</ref>由於皮特里四面體由扭歪四邊形組成<ref>{{citation|title=Abstract Regular Polytopes|volume=92|series=Encyclopedia of Mathematics and its Applications|first1=Peter|last1=McMullen|first2=Egon|last2=Schulte|publisher=Cambridge University Press|year=2002|isbn=9780521814966|page=192|url=https://books.google.com/books?id=JfmlMYe6MJgC&pg=PA192}}</ref>，因此無法確立其封閉範圍，故無法計算其表面積和體積。<ref name="book barnard2014aro">{{Citation
  |title=Aro -- Healing Touching Lives -- Theories, Techniques and Therapies: The Techniques and Therapies of Aro-Healing
  |author=Barnard, L.
  |isbn=9781483631646
  |url=https://books.google.com.tw/books?id=brZNBAAAQBAJ
  |year=2014
  |publisher=Xlibris UK
  |accessdate=2021-07-31
  |archive-date=2021-07-31
  |archive-url=https://web.archive.org/web/20210731072011/https://books.google.com.tw/books?id=brZNBAAAQBAJ
  }}</ref>

皮特里四面體是一個不可定向且歐拉示性數為1的幾何結構<ref name= "Regular Map Petrial tetrahedron"/>。
{| class="wikitable" width="300"
|- align=center
| [[Image:Skeleton_4b,_Petrie,_stick,_size_m,_2-fold_square.png|150px]]<br/>[[正四面體]]的[[皮特里多邊形]]
| [[Image:Face_of_petrial_tetrahedron.gif|150px]]<br/>構成皮特里立方體的[[扭歪四邊形]]面
|}
皮特里四面體的頂點、邊和面數皆為立方體的一半，因此皮特里四面體可以被立方體（的表面）[[覆疊空間|二重覆蓋]]<ref name= "Regular Map Petrial tetrahedron"/>。皮特里四面體的對偶多面體為[[八面體半形]]<ref name= "Regular Map Petrial tetrahedron"/>。皮特里四面體可以[[截半 (幾何)|截半]]為[[截半立方體半形]]<ref name= "Regular Map Petrial tetrahedron"/><ref name= "Regular Map Hemi-cuboctahedron" >{{cite web | url = http://www.weddslist.com/rmdb/map.php?a=Q1.1 | title = Hemi-cuboctahedron | publisher = Regular Map database - map details | accessdate = 2021-07-24 | archive-date = 2021-01-26 | archive-url = https://web.archive.org/web/20210126031706/http://www.weddslist.com/rmdb/map.php?a=Q1.1 }}</ref>。
{| class="wikitable" width="300"
|- align=center
| [[Image:Tetrahedron 3 petrie polygons.png|150px]]<br/>皮特里四面體
| [[Image:Hemicube.svg|150px]]<br/>以正則地區圖表示的皮特里四面體
| [[Image:Hemi-octahedron2.png|150px]]<br/>皮特里四面體的對偶多面體以正則地區圖表示
|}

== 相關多面體 ==
立方體半形是正多面體的半形體之一，其他也是正多面體的半形之結構有<ref name="Abstract Regular Polytopes 2002"/>：
{| class="wikitable" width="300"
|- align=center
| [[Image:Hemicube.svg|150px]]<br/>'''立方體半形'''
| [[Image:Hemi-octahedron2.png|150px]]<br/>[[八面體半形]]
| [[Image:Hemi-dodecahedron.png|150px]]<br/>[[十二面體半形]]
| [[Image:Hemi-icosahedron.png|150px]]<br/>[[二十面體半形]]
|}
立方體半形與皮特里四面體拓樸同構，其可以視為是正多面體的皮特里對偶之一。其他也是正多面體的皮特里對偶之幾何結構有：<ref name= "Regular Map genus 0">{{cite web | url = http://www.weddslist.com/rmdb/man.php?m=o0 | title = Regular maps in the orientable surface of genus 0 | publisher = Regular Map database - map details | accessdate = 2021-07-31 | archive-date = 2021-10-19 | archive-url = https://web.archive.org/web/20211019042151/http://www.weddslist.com/rmdb/man.php?m=o0 }}</ref>
{| class="wikitable" width="300"
|- align=center
| [[Image:Petrial_tetrahedron.gif|150px]]<br/>'''皮特里四面體'''
| [[Image:Petrial_cube.gif|150px]]<br/>[[皮特里立方體]]
| [[Image:Petrial_octahedron.gif|150px]]<br/>[[皮特里八面體]]
| [[Image:Petrial_dodecahedron.gif|150px]]<br/>[[皮特里十二面體]]
| [[Image:Petrial_icosahedron.gif|150px]]<br/>[[皮特里二十面體]]
|}

== 參考資料 ==
{{Reflist|2}}

==外部連結==
* [http://www.weddslist.com/rmdb/map.php?a=N1.1p 立方體半形] {{Wayback|url=http://www.weddslist.com/rmdb/map.php?a=N1.1p |date=20190502092501 }}
{{多面體半形}}

[[Category:射影多面體]]
[[Category:多面體半形]]
[[Category:抽象正多面體]]