查看“︁六維正七胞體”︁的源代码

{{NoteTA|G1=Math}}
{{Infobox polytope
| name = 正七胞體
| imagename = 6-simplex_t0.svg
| polytope = 正七胞體
| Type = [[正圖形|正]]{{link-en|六維多胞體|6-polytope}}<br/>[[七胞體]]
| group_type = [[單純形]]
| Dimension = 六維
| dim1 = 五維
| count1 = 7個[[五維正六胞體]][[Image:5-simplex_t0.svg|25px]]
| dim = 四維
| count = 21個[[正五胞體]][[Image:4-simplex_t0.svg|25px]]
| Cell = 35個[[正四面體]][[Image:3-simplex_t0.svg|25px]]
| Face = 35個[[正三角形]][[Image:2-simplex_t0.svg|25px]]
| Edge = 21
| Vertice = 7
| Vertice_type = [[五維正六胞體]]<br/>[[Image:5-simplex_t0.svg|50px]]
| Schläfli = {3,3,3,3,3}<br/>{3<sup>5</sup>}
| Euler = 0
| Bowers_acronym = hop
| Coxeter_diagram = {{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}
| Petrie = [[正七邊形]]
| Symmetry_group =  A<sub>6</sub> [3<sup>5</sup>], 5040階
| dual = 正七胞體（自身對偶）
| Properties = 
}}
在幾何學中，'''六維正七胞體'''（heptapeton<ref name="french2014hidden">{{cite book
  |title=The Hidden Geometry of Life: The Science and Spirituality of Nature
  |author=French, K.L.
  |isbn=9781780288451
  |series=Gateway series
  |url=https://books.google.com.tw/books?id=ixPLCQAAQBAJ
  |year=2014
  |publisher=Watkins Media Limited}}</ref>{{rp|127}}）是一種[[對偶多面體|自身對偶]]的[[正圖形|正]]{{link-en|六維多胞體|6-polytope}}<ref name="klitzing hop">{{cite web | url = https://bendwavy.org/klitzing/incmats/hop.htm | title = heptapeton | author = Klitzing, Richard | publisher = bendwavy.org | access-date = 2022-06-02 | archive-date = 2021-09-30 | archive-url = https://web.archive.org/web/20210930064120/https://bendwavy.org/klitzing/incmats/hop.htm }}</ref>，是六維空間中的[[單純形]]<ref name="2019/v14i230122">{{cite journal
 |author = Ufuoma, Okoh and Ikhile, Agun
 |date = 2019-06
 |pages = 1-20
 |title = On Simplicial Polytopic Numbers
 |journal = Asian Research Journal of Mathematics
 |doi = 10.9734/arjom/2019/v14i230122}}</ref>，又稱為6-單純形（6-simplex）<ref>{{cite web|url=https://www.slac.stanford.edu/slac/sass/talks/JoshuaFittingTheUnknown.pdf|title=Fitting The Unknown|author=Joshua Lande|date=2010-09-01|publisher=slac.stanford.edu|access-date=2022-06-02|archive-date=2015-10-09|archive-url=https://web.archive.org/web/20151009042854/https://www.slac.stanford.edu/slac/sass/talks/JoshuaFittingTheUnknown.pdf}}</ref>，由7個[[五維正六胞體]]組成，其二面角為cos<sup>−1</sup>(1/6)約為80.41°。<ref name="klitzing hop"/>

== 性質 ==
六維正七胞體共有7個[[頂點 (幾何)|頂點]]、21條[[邊 (幾何)|邊]]、35個[[正三角形|三角形]]的[[面 (幾何)|面]]、35個[[正四面體|四面體]]的[[胞 (幾何)|胞]]、21個[[四維正五胞體]]的{{link-wd|Q4637223|四維胞}}和7個[[五維正六胞體]]的{{link-wd|Q18028552|五維胞}}組成<ref name="ferretti2015algebraic">{{cite journal
  |title=The algebraic formulation: Why and how to use it
  |author=Ferretti, Elena
  |journal=Curved and Layered Structures
  |volume=2
  |number=1
  |year=2015
  |publisher=De Gruyter Open}}</ref>，其中五維正六胞體為六維正七胞體的維面。对于一个边长为a的六維正七胞体，其超胞积是<math>\cfrac{\sqrt{7}a^6}{5760}</math>，表胞积是<math>\cfrac{7\sqrt{3}a^5}{480}</math>，高是<math>\cfrac{\sqrt{21}a}{6}</math>。
若一个六維正七胞体的棱长为1，则其外接六維超球的半径为<math>\frac{\sqrt{21}}{7}</math>，內切六維超球的半径为<math>\frac{\sqrt{21}}{42}</math>。<ref name="klitzing hop"/>

=== 作為一種排佈 ===
六維正七胞體的{{link-en|排佈 (多胞形)|Configuration_(polytope)|排佈矩陣}}為：<ref name="klitzing hop"/>
:<math>\begin{bmatrix}\begin{matrix}7 & 6 & 15 & 20 & 15 & 6 \\ 2 & 21 & 5 & 10 & 10 & 5 \\ 3 & 3 & 35 & 4 & 6 & 4 \\ 4 & 6 & 4 & 35 & 3 & 3 \\ 5 & 10 & 10 & 5 & 21 & 2 \\ 6 & 15 & 20 & 15 & 6 & 7 \end{matrix}\end{bmatrix}</math>
行和列對應於六維正七胞體的[[頂點 (幾何)|頂點]]、[[邊 (幾何)|邊]]、[[面 (幾何)|面]]、[[胞 (幾何)|胞]]、{{link-wd|Q4637223|四維胞}}、{{link-wd|Q18028552|五維胞}}。對角線上的數字表示該元素在六維正七胞體中的數量。非對角線的數量表示對應行所代表的元素上有多少列所代表的元素交於該處。由於六維正七胞體是一種自身對偶的多胞體，因此這個排佈矩陣旋轉180度後會相同。{{#tag:ref|Coxeter 1973<ref>{{cite book |first=H.S.M. |last=Coxeter |chapter=Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) |title=Regular Polytopes (book) |url=https://archive.org/details/regularpolytopes00coxe_869 |publisher=Dover |edition=3rd |year=1973 |isbn=0-486-61480-8 |pages=[https://archive.org/details/regularpolytopes00coxe_869/page/n319 296] }}</ref>, §1.8 Configurations}}<ref>{{cite book |first=H.S.M. |last=Coxeter |title=Regular Complex Polytopes |url=https://books.google.com/books?id=9BY9AAAAIAAJ&pg=PA117 |pages=117 |edition=2nd |publisher=Cambridge University Press |year=1991 |isbn=9780521394901}}</ref>

=== 頂點座標 ===
若一個六維正七胞體幾何中心位於原點，且邊長為2單位長，則其頂點座標為：

:<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)</math>
:<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)</math>
:<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(-\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>

== 圖像 ==
{| class=wikitable
|+ 正投影圖
|- style="text-align:center;"
!A<sub>k</sub>考克斯特平面
!A<sub>6</sub>
!A<sub>5</sub>
!A<sub>4</sub>
|- style="text-align:center;"
!圖像
|[[File:6-simplex t0.svg|150px]]
|[[File:6-simplex t0_A5.svg|150px]]
|[[File:6-simplex t0_A4.svg|150px]]
|- style="text-align:center;"
![[二面體群|二面體群對稱性]]
|[7]
|[6]
|[5]
|- style="text-align:center;"
!A<sub>k</sub>考克斯特平面
!A<sub>3</sub>
!A<sub>2</sub>
|- style="text-align:center;"
!圖像
|[[File:6-simplex t0_A3.svg|150px]]
|[[File:6-simplex t0_A2.svg|150px]]
|- style="text-align:center;"
!二面體群對稱性
|[4]
|[3]
|}

== 參考文獻 ==
{{reflist}}

{{正多胞形}}

[[Category:多胞体]]