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 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 2n3/2/3√3, but rectangle structures can have areas at least n2/(6+√6).
Here are the maximum areas so far:
|6||2.414||EF, CP, DL|
|7||3.148||EF, JD, CP, DL, GS|
|20||20.472||EF, CP, BF|
Here are the best areas that people managed to contain with 40 rigid segments:
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: