# 六维空间

## 幾何

### 六維多胞形

（ 根據對稱性的考斯特平面正交投影 ）
A6 BC6 D6 E6

221英语2 21 polytope

122英语1 22 polytope

### 五維球面

${\displaystyle S^{5}=\left\{x\in \mathbb {R} ^{6}:\|x\|=r\right\}.}$

${\displaystyle V_{6}={\frac {\pi ^{3}r^{6}}{6}}}$

### 六維球面

${\displaystyle S^{6}=\left\{x\in \mathbb {R} ^{7}:\|x\|=r\right\}.}$

${\displaystyle V_{7}={\frac {16\pi ^{3}r^{7}}{105}}}$

## 應用

### 電磁學

${\displaystyle \partial \mathbf {F} =\mathbf {J} \,}$

## 理論背景

### 四維空間中的雙重向量

2

=6個組件，並且可以最通用地寫成

${\displaystyle \mathbf {B} =B_{12}\mathbf {e} _{12}+B_{13}\mathbf {e} _{13}+B_{14}\mathbf {e} _{14}+B_{23}\mathbf {e} _{23}+B_{24}\mathbf {e} _{24}+B_{34}\mathbf {e} _{34}}$

### 六維向量空間

${\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4}+a_{5}b_{5}+a_{6}b_{6}.}$

${\displaystyle \left|\mathbf {a} \right\vert ={\sqrt {\mathbf {a} \cdot \mathbf {a} }}={\sqrt {{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}+{a_{4}}^{2}+{a_{5}}^{2}+{a_{6}}^{2}}}.}$

${\displaystyle {\sqrt {1+1+1+1+1+1}}={\sqrt {6}}=2.4495,}$

## 腳註

1. ^ Arthur Besier. Perspectives of Modern Physics. McGraw-Hill. 1969.
2. ^ Lounesto (2001), pp. 109–110
3. ^ Aharony (2000)
4. ^ Lounesto (2001), pp. 86-89
5. ^ Josiah Willard Gibbs, Edwin Bidwell Wilson. Vector analysis: a text-book for the use of students of mathematics and physics. Yale University Press. 1901: 481 ff.