2004/08/24

There are some constructions with equal polyspheres, which look like polyhedrons with regular faces. In the picture one of the used pieces is blue colored. The table below shows the properties of those figures and the number of spheres needed.

Name | Made of | Faces | Volume | n=3 |
---|---|---|---|---|

Tetrahedron | triangle(n) +..+ triangle(1) | 4 triangles | n(n+1)(n+2)/6 | |

Square Pyramid | square(n) +..+ square(1) | 4 triangles, 1 square |
n(n+1)(2n+1)/6 | |

Octahedron | square pyramid(n) + square pyramid(n-1) |
8 triangles | n(2n^2+1)/3 | |

Cubeoctahedron | octahedron(2n-1) - 6*square pyramid(n-1) |
8 triangles, 6 squares |
(2n-1)(5n^2-5n+3)/3 | |

Truncated Tetrahedron | tetrahedron(3n-2) - 4*tetrahedron(n-1) |
4 triangles, 4 hexagons |
n(23n^2-27n+10)/6 | |

Truncated Octahedron | octahedron(3n-2) - 6*square pyramid(n-1) |
6 squares, 8 hexagons |
16n^3-33n^2+24n-6 |

Click on the numbers to see the layers and some pictures of the constructions.

Name/Size | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|

Tetrahedron | 4 | 10 | 20 | 35 | 56 | 84 | 120 | 165 | 220 |

Y4, D5,... | D5 | Y4, J4 | Y4 | C3, D3, Y4, D5 | D5 | Y4, D5 | |||

Square Pyramid | 5 | 14 | 30 | 55 | 91 | 140 | 204 | 285 | 385 |

L3 | J4, Y4, D5 | C3, L3, Y4 | |||||||

Octahedron | 6 | 19 | 44 | 85 | 146 | 231 | 344 | 489 | 670 |

L3 | J4, Y4 | D5 | |||||||

Cubeoctahedron | 13 | 55 | 147 | 309 | 561 | 923 | 1415 | 2057 | 2869 |

D5 | C3, L3 | ||||||||

Truncated Tetrahedron | 16 | 68 | 180 | 375 | 676 | 1106 | 1688 | 2445 | 3400 |

Y4 | C4, J4, Y4 | C3, D3, I3, C4, J4, Y4, D5 | |||||||

Truncated Octahedron | 38 | 201 | 586 | 1289 | 2406 | 4033 | 6266 | 9201 | 12934 |

C3, L3 |

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