It is well-known that the tent made of n segments with the largest area is half a 2n-gon. This month's problem is to investigate the rigid tent made from n segments with the largest area. What are the best results for various n? Can you give lower or upper bounds for the largest possible area? Since I turn 40 this month, I'll give a special prize of $10 to the person who can enclose the largest area with 40 rigid unit segments.

George L. Sicherman asks what the largest rigid tent is using n segments with no triangles. Can this be done at all?

Guenter Stertenbrink wonders what the largest rigid structures are with no line to connect to. I think the answers are usually collections of equilateral triangles, though for n=15 this is not the case!

Guenter Stertenbrink also gives an application of this month's problem. A bacteria farmer wants to enclose the largest combined area against shore using n equal length planks that can only be fastened at the ends.

Claudio Baiocchi notes that for large n, trapezoid structures can have area at most 2n^{3/2}/3√3, but rectangle structures can have areas at least n^{2}/(6+√6).

Here are the maximum areas so far:

n | Largest Area | Author | Picture |
---|---|---|---|

2 | 0.500 | EF | |

4 | 1.299 | EF | |

5 | 1.732 | CW, DL | |

6 | 2.414 | EF, CP, DL | |

7 | 3.148 | EF, JD, CP, DL, GS | |

8 | 4.116 | ||

9 | 5.095 | ||

10 | 6.080 | ||

11 | 7.069 | ||

12 | 8.096 | CP | |

13 | 9.628 | ||

14 | 11.148 | ||

15 | 12.661 | ||

16 | 14.171 | ||

17 | 15.679 | ||

18 | 17.184 | ||

19 | 18.689 | ||

20 | 20.472 | EF, CP, BF | |

21 | 22.472 | ||

22 | 24.472 | ||

23 | 26.472 | ||

24 | 28.472 | ||

25 | 30.472 | ||

26 | 32.472 | ||

27 | 34.472 | ||

28 | 36.518 | BJ | |

29 | 39.016 | ||

30 | 41.513 | ||

31 | 44.011 | ||

32 | 46.509 | ||

33 | 49.007 | ||

34 | 51.505 | ||

35 | 54.004 | ||

36 | 56.502 | ||

37 | 59.084 | GS | |

38 | 61.937 | BF, CP | |

39 | 64.937 | ||

40 | 67.937 | ||

41 | 70.937 | ||

42 | 73.937 | ||

43 | 76.937 | ||

44 | 79.937 | ||

45 | 82.937 | ||

46 | 85.937 | ||

47 | 88.937 | ||

48 | 91.937 | ||

49 | 94.937 | ||

50 | 97.937 |

Here are the best areas that people managed to contain with 40 rigid segments:

Name | Best Area |
---|---|

Bertram Felgenhauer | 67.93 |

Brian J | 67.93 |

Dan Dima | 67.93 |

Dave Langers | 67.93 |

Corey Plover | 67.93 |

Guenter Stertenbrink | 67.54 |

Emilio Schiavi | 63.91 |

Erich Friedman | 63.80 |

Andrew Bayly | 63.80 |

David Cantrell | 63.74 |

Gary Gerken | 58.80 |

George L. Sicherman | 56.14 |

Joseph DeVincentis | 53.68 |

The best rigid tent for n=40 is the last picture shown above. Bertram Felgenhauer wins the $10 prize. Here are some of the other solutions I received:

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/29/05.