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
[编辑]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
[编辑]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。[1][2]
顶点坐标
[编辑]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
[编辑]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 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
[编辑]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
[编辑]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
[编辑]- 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
[编辑]- 埃里克·韦斯坦因. Hypercube. MathWorld.
- Olshevsky, George, Measure polytope at Glossary for Hyperspace.
- Multi-dimensional Glossary: hypercube Garrett Jones