User:ItMarki/六维超正方体

六维超正方体
類型六維多胞體英语6-polytope
家族超方形
維度六維
對偶多胞形六維正軸體英语6-orthoplex
識別
名稱六维超正方体
鮑爾斯縮寫
verse-and-dimensions的wikiaBowers acronym
ax
數學表示法
考克斯特符號
英语Coxeter-Dynkin diagram
node_1 4 node 3 node 3 node 3 node 3 node 
施萊夫利符號{4,34}
性質
五維12個五維超正方体
四維60個超正方体
160個立方体
192個正方形
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

几何学中,六维超正方体(英語:6-cubehexeract)是一个正六维多胞体,由64个顶点、192个、240个正方形、160个立方体、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英语6-orthoplex,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英语uniform polytope,名为六维超半方形英语6-demicube超半方形系列的一员),有12个五维超半方形英语5-demicube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英语configuration (polytope)。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。[1][2]

 

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 < xi < 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 [33,1,1] Coxeter group. The lowest symmetry construction is based on hyperrectangle英语hyperrectangles or proprism英语proprisms, cartesian products of lower dimensional hypercubes.

Name Coxeter英语Coxeter diagram Schläfli Symmetry英语Coxeter notation 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英语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 projection英语orthographic projections
Coxeter plane英语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.

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-polytope英语uniform 6-polytopes generated from the B6 Coxeter plane英语Coxeter plane, including the regular 6-cube or 6-orthoplex英语6-orthoplex.

Template:Hexeract family

References

  1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. ^ Coxeter, Complex Regular Polytopes, p.117
  • Coxeter, H.S.M. Regular Polytopes英语Regular Polytopes (book), (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. 

Template:Polytopes