# 四维空间

（重定向自四维

## 作为空间的第四维数

$\mathbf{x} = (p, q, r, s)$

$\| \mathbf{x} \| = \sqrt{p^{2} + q^{2} + r^{2} + s^{2}}$

### 向量

#### 向量运算

$\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{pmatrix}.$

$\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}; \mathbf{e}_4 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix},$

$\mathbf{a} = a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3 + a_4\mathbf{e}_4.$

$\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 + a_4 b_4.$

$\left| \mathbf{a} \right| = \sqrt{\mathbf{a} \cdot \mathbf{a} } = \sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2},$

$\theta = \arccos{\frac{\mathbf{a} \cdot \mathbf{b}}{\left|\mathbf{a}\right| \left|\mathbf{b}\right|}}.$

$\mathbf{a} \wedge \mathbf{b} = (a_1b_2 - a_2b_1)\mathbf{e}_{12} + (a_1b_3 - a_3b_1)\mathbf{e}_{13} + (a_1b_4 - a_4b_1)\mathbf{e}_{14} + (a_2b_3 - a_3b_2)\mathbf{e}_{23} + (a_2b_4 - a_4b_2)\mathbf{e}_{24} + (a_3b_4 - a_4b_3)\mathbf{e}_{34}.$

## 幾何

### 超球體

$\mathbf V = 2 \pi^2 R^3$

## 參考文献

1. ^ C Møller. The Theory of Relativity. Oxford UK: Clarendon Press. 1952: p. 93. ISBN 0198512562.
2. ^ Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., p. 119.
3. ^ Michio Kaku (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension, Part I, chapter 3, The Man Who "Saw" the Fourth Dimension (about tesseracts in years 1870 - 1910). ISBN 0-19-286189-1.
4. ^ Google Books Flatland: A Romance of Many Dimensions. By Edwin A. Abbott, Published by Filiquarian Publishing, LLC., 2007. ISBN 1-59986-928-4, 9781599869285, 148 pages
5. ^ Google Books Spaceland: A Novel of the Fourth Dimension. By Rudy Rucker, Published by Tom Doherty Associates, LLC, 2002. ISBN 0-7653-0366-3, 9780765303660, 304 pages
6. ^ Ray d'Inverno (1992), Introducing Einstein's Relativity, Clarendon Press, chp. 22.8 Geometry of 3-spaces of constant curvature, p.319ff, ISBN 0-19-859653-7