User:ItMarki/六维超正方体

六维超正方体
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	六维超正方体
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個五維超正方体
四維胞	60個超正方体
胞	160個立方体
面	192個正方形
邊	192
頂點	64
特殊面或截面
皮特里多边形	正十二边形
對稱性
對稱群	B6, [34,4]
特性
凸

在几何学中，六维超正方体（英語：6-cube、hexeract）是一个正六维多胞体，由64个顶点、192个边、240个正方形面、160个立方体胞、60个四維超正方體胞和12个五维超正方体胞组成。它的施莱夫利符号是{4,3⁴}，代表每个四维胞周围有3个五维超正方体。

相关多胞体

六维超正方体是超方形系列的一员。它的對偶多面體六維正軸體，而六維正軸體是正轴形系列的一员。

对六维超正方体进行交错（去除交替顶点）后，结果是另一个均勻多胞形，名为六维超半方形（超半方形系列的一员），有12个五维超半方形胞和32个五維正六胞體胞。

排佈

以下列出六维超正方体的排佈矩阵。行和列对应顶点、边、面、胞、四维胞和五维胞。对角线元素代表整个六维超正方体中每种元素有多少个。其他数字代表该行的元素中有多少个该列的元素。

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

顶点坐标

一中心为原点、边长为2的六维超正方体的顶点坐标为

(±1,±1,±1,±1,±1,±1)

而其内部由所有点(x₀, x₁, x₂, x₃, x₄, x₅)组成，其中−1 < x_i < 1。

构造

六维超正方体有三个考克斯特群，一个是 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·韦斯坦因. Hypercube. MathWorld. 
• Olshevsky, George, Measure polytope at Glossary for Hyperspace.
• Multi-dimensional Glossary: hypercube Garrett Jones