The -hypersphere (often simply called the -sphere) is a generalization of the Circle () and
Sphere () to dimensions . It is therefore defined as the set of -tuples of points
(, , ..., ) such that

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

Strangely enough, the hyper-Surface Area and Content reach Maxima
and then decrease towards 0 as increases. The point of Maximal hyper-Surface Area satisfies

(9) |

(10) |

0 | 1 | 1 | 0 |

1 | 2 | 1 | 2 |

2 | |||

3 | |||

4 | |||

5 | |||

6 | |||

7 | |||

8 | |||

9 | |||

10 |

In 4-D, the generalization of Spherical Coordinates is defined by

(11) | |||

(12) | |||

(13) | |||

(14) |

The equation for a 4-sphere is

(15) |

(16) |

(17) |

(18) |

**References**

Conway, J. H. and Sloane, N. J. A. *Sphere Packings, Lattices, and Groups, 2nd ed.*
New York: Springer-Verlag, p. 9, 1993.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, p. 58, 1983.

Peterson, I. *The Mathematical Tourist: Snapshots of Modern Mathematics.* New York: W. H. Freeman, pp. 96-101, 1988.

© 1996-9

1999-05-25