W. Randolph Franklin

Here is a sampling of how 2D geometry is essentially different from 3D.

2D | 3D | |
---|---|---|

1. | There are an infinite number of regular polygons, | but only a finite number of regular polyhedra. |

2. | Given two equal-area polygons, one can be dissected into a finite number of pieces, then reassembled into the other. | This is usually not true for pairs of equal-volume polyhedra. |

3. | There exists a square that is decomposable into smaller, different, squares. | There is no cube that is decomposable into smaller, different, cubes. |

4. | All polygons are decomposable into triangles by adding only interior edges. | Not all polyhedra are decomposable into tetrahedra by adding only interior faces. |

5. | For polygons, all such decompositions have the same number of triangles. | Some polyhedra can be decomposed different ways into different numbers of tetrahedra. |

6. | Every polygon has every interior point visible from some vertex. | Some polyhedra have interior points not visible from any vertex. |

7. | A 2-D Voronoi diagram's complexity is linear. | A 3-D Voronoi diagram's complexity can be quadratic. |

8. | Given two convex polygons, there exists an edge of one that separates them. | Given two polyhedra, it is possible that none of their faces separate them. |

9. | Rotations commute. | Rotations usually don't commute. |

10. | For each edge of a polygon, consider the half-plane of points on the inside side of that edge. Then, the polygon's interior can be expressed as a Boolean expression in those half planes, with each half plane used only once. | Not in 3D. |

Email: wrfATecse.rpi.edu

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Copyright © 1994-2002, Wm. Randolph Franklin. You may use my material for non-profit research and education, provided that you credit me, and link back to my home page.