The spectrum of a square or triangle is {1,2,3,4, . . . }. The spectrum of a circle is {1}. This month's problem is "Which sets of positive integers are the spectrum of some shape?"

Connecting the center of an equilateral triangle or square to its vertices by curved but symmetric lines gives shapes with spectra 2N, 3N, and 4N. This was noticed by Joseph DeVincentis.

Joseph DeVincentis found an infinite family of chevrons with spectra {4,3n+4,3n+6,3n+8, . . . } and {4n,4n+2,4n+4, . . . } for all n.

Mike Reid found an infinite family of polyominoes that have all but finitely many integers in their spectra.

m\n | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|

1 | (Ed Pegg) | ? | ? | ? | ? | ||

2 | (Joe DeVincentis) | (Mike Reid) | ? | (George Sicherman) | ? | (George Sicherman) | |

3 | ? | ? | ? | ? | ? | ||

4 | ? | (Mike Reid) | ? | (George Sicherman) |

Let N={1,2,3,4, . . . }. Here are some other shapes and their spectra:

Shape | Spectrum | Author |
---|---|---|

{1,2,3} | Erich Friedman | |

{1,2,4} | Erich Friedman | |

{1,2,5} | Erich Friedman | |

{1,3,4} | Mike Reid | |

{1,3,9} | Mike Reid | |

{2,3,4} | Erich Friedman | |

{2,6,10} | George Sicherman | |

{1,2,3,4} | George Sicherman | |

{1,2,3,5} | Erich Friedman | |

{1,2,3,6} | Erich Friedman | |

{1,2,3,7} | Erich Friedman | |

{1,2,4,6} | Erich Friedman | |

{1,2,3,4,16} | Karl Scherer | |

{1,2,9,14,18} | Ed Pegg |

Karl Scherer found this next tile, though George Sicherman improved its spectrum to {1,2,3,4,5,6,8,10,11,12,16,18,20,24,26,48}:

Shape | Spectrum | Author |
---|---|---|

N | Erich Friedman | |

N–{1} | Mike Reid | |

N–{1,5} | Mike Reid | |

N–{1,3,5,7,9} | Mike Reid | |

N–{1,3,5,7,9,11,13,17} | Mike Reid |

Shape | Spectrum | Author |
---|---|---|

2N | Erich Friedman | |

{1,3} ∪ 2N | Dave Barlow | |

2N–{2} | Erich Friedman |

Shape | Spectrum | Author |
---|---|---|

3N | Erich Friedman | |

{1,2} ∪ 3N | Erich Friedman |

Shape | Spectrum | Author |
---|---|---|

4N | Erich Friedman | |

2k+4N | Erich Friedman |

Shape | Spectrum | Author |
---|---|---|

3+5N | George Sicherman |

Shape | Spectrum | Author |
---|---|---|

{n≠5 (mod 6)} ? | Mike Reid |

Shape | Spectrum | Author |
---|---|---|

8N | Mike Reid | |

{6,10,14,18, . . . } ∪ {24,32,40, . . . } | Mike Reid |

Shape | Spectrum | Author |
---|---|---|

N-{1,9,odd primes} | Mike Reid | |

{4,8,18,24} U {7,9,11, . . . } ∪ {32,36,40, . . . } ? | Joe DeVincentis George Sicherman | |

{30,42,46,50,54, . . . } ? | Mike Reid George Sicherman | |

{2,4,6,8,10,11,12,13,14,16,18,20,21,22, . . . } | Mike Reid | |

{2,5,6,7,8,11,14,15,16,17,18,19,21+} ? | Mike Reid George Sicherman | |

{8,12,13,16,18,20,23,24,28,32,33, . . . } | Mike Reid George Sicherman | |

12N^{2} | John Wallace Dave Barlow |

George Sicherman found a shape whose spectrum is apparently multiples of 4, together with a quadratically increasing sequence {2, 18, 66, 138, 234, ...}.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/16/12.