# blade

A blade is a term often used to describe a basis entity in the space defined by a geometric algebra. Since a geometric algebra is a multi-graded space, the basis entities also have multiple^{} grades. To distinguish the various graded entities, the blades are often prefixed by their grade. For example a grade-$k$ basis entity would be called a $k$-blade.

The number of linearly independent^{} $k$-blades in a particular geometric algebra is dependent on the number of dimensions^{} of the manifold on which the algebra is defined. For an $n$-dimensional manifold, the number of $k$-blades is given by the binomial coefficient^{}.

$${N}_{k}=\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)$$ |

The total number of basis blades of all grades in a geometric algebra defined on an $n$-manifold is then:

$$N=\sum _{k=0}^{n}{N}_{k}={2}^{n}$$ |

Title | blade |
---|---|

Canonical name | Blade |

Date of creation | 2013-03-22 15:58:40 |

Last modified on | 2013-03-22 15:58:40 |

Owner | PhysBrain (974) |

Last modified by | PhysBrain (974) |

Numerical id | 5 |

Author | PhysBrain (974) |

Entry type | Definition |

Classification | msc 15A03 |

Classification | msc 15A75 |

Classification | msc 15A66 |

Related topic | Basis |

Related topic | UnitVector |