使用者:ItMarki/六維超正方體

六維超正方體
類型	正六維多胞體
家族	超方形
維度	六維
對偶多胞形	六維正軸體
識別
名稱	六維超正方體
鮑爾斯縮寫	ax
數學表示法
考克斯特符號
施萊夫利符號	{4,34}
性質
五維胞	12個五維超正方體
四維胞	60個超正方體
胞	160個立方體
面	192個正方形
邊	192
頂點	64
特殊面或截面
皮特里多邊形	正十二邊形
對稱性
對稱群	B6, [34,4]
特性
凸

在幾何學中，六維超正方體（英語：6-cube、hexeract）是一個正六維多胞體，由64個頂點、192個邊、240個正方形面、160個立方體胞、60個四維超正方體胞和12個五維超正方體胞組成。它的施萊夫利符號是{4,3⁴}，代表每個四維胞周圍有3個五維超正方體。

相關多胞體

六維超正方體是超方形系列的一員。它的對偶多面體六維正軸體，而六維正軸體是正軸形系列的一員。

對六維超正方體進行交錯（去除交替頂點）後，結果是另一個均勻多胞形，名為六維超半方形（超半方形系列的一員），有12個五維超半方形胞和32個五維正六胞體胞。

排佈

以下列出六維超正方體的排佈矩陣。列和行對應頂點、邊、面、胞、四維胞和五維胞。對角線元素代表整個六維超正方體中每種元素有多少個。其他數字代表該列的元素中有多少個該行的元素。

${\begin{bmatrix}{\begin{matrix}64&6&15&20&15&6\\2&192&5&10&10&5\\4&4&240&4&6&4\\8&12&6&160&3&3\\16&32&24&8&60&2\\32&80&80&40&10&12\end{matrix}}\end{bmatrix}}$

頂點坐標

一中心為原點、邊長為2的六維超正方體的頂點坐標為

(±1,±1,±1,±1,±1,±1)

而其內部由所有點(x₀, x₁, x₂, x₃, x₄, x₅)組成，其中−1 < x_i < 1。

構造

六維超正方體有三個考克斯特群，一個是 There are three Coxeter groups associated with the 6-cube, one regular, with the C₆ or [4,3,3,3,3] Coxeter group, and a half symmetry (D₆) or [3^3,1,1] Coxeter group. The lowest symmetry construction is based on hyperrectangles or proprisms, cartesian products of lower dimensional hypercubes.

Name	Coxeter	Schläfli	Symmetry	Order
Regular 6-cube		{4,3,3,3,3}	[4,3,3,3,3]	46080
Quasiregular 6-cube			[3,3,3,31,1]	23040
hyperrectangle		{4,3,3,3}×{}	[4,3,3,3,2]	7680
		{4,3,3}×{4}	[4,3,3,2,4]	3072
		{4,3}2	[4,3,2,4,3]	2304
		{4,3,3}×{}2	[4,3,3,2,2]	1536
		{4,3}×{4}×{}	[4,3,2,4,2]	768
		{4}3	[4,2,4,2,4]	512
		{4,3}×{}3	[4,3,2,2,2]	384
		{4}2×{}2	[4,2,4,2,2]	256
		{4}×{}4	[4,2,2,2,2]	128
		{}6	[2,2,2,2,2]	64

Projections

orthographic projections

Coxeter plane	B6	B5	B4
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	Other	B3	B2
Graph
Dihedral symmetry	[2]	[6]	[4]
Coxeter plane		A5	A3
Graph
Dihedral symmetry		[6]	[4]

3D Projections
6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D.	6-cube quasicrystal structure orthographically projected to 3D using the golden ratio.
A 3D perspective projection of an hexeract undergoing a triple rotation about the X-W1, Y-W2 and Z-W3 orthogonal planes.

Related polytopes

The 64 vertices of a 6-cube also represent a regular skew 4-polytope {4,3,4 | 4}. Its net can be seen as a 4×4×4 matrix of 64 cubes, a periodic subset of the cubic honeycomb, {4,3,4}, in 3-dimensions. It has 192 edges, and 192 square faces. Opposite faces fold together into a 4-cycle. Each fold direction adds 1 dimension, raising it into 6-space.

The 6-cube is 6th in a series of hypercube: Template:Hypercube polytopes

This polytope is one of 63 uniform 6-polytopes generated from the B₆ Coxeter plane, including the regular 6-cube or 6-orthoplex.

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
• Klitzing, Richard. 6D uniform polytopes (polypeta) o3o3o3o3o4x - ax. bendwavy.org. 

External links

• 埃里克·韋斯坦因. Hypercube. MathWorld. 
• Olshevsky, George, Measure polytope at Glossary for Hyperspace.
• Multi-dimensional Glossary: hypercube Garrett Jones