# Maximal Unit Lengths in 3D

Maximal Unit Lengths in 3D

Put points in 3D space. What is the maximal number of unit lengths? For four points, it is obvious the maximal number is six unit lengths, using a tetrahedron. With five points, two joined tetrahedra give nine unit lengths.

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This Demonstration gives the following best-known solutions up to 20 points.

points | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

lengths | 9 | 12 | 15 | 18 | 22 | 27 | 31 | 35 | 39 | 44 | 48 | 52 | 57 | 62 | 66 | 71 |

solutions | 1 | 2 | 8 | 26 | 5 | 1 | 1 | 1 | 2 | 1 | 4 | 8+ | 1+ | 1+ | 2+ | 1+ |

If you can find missed solutions matching or surpassing the best known maximum values, please send them to the author.