Problem of the Month (April 2005)

Start with a horizontal line called the ground. We say a tent is a collection of unit line segments that are only joined to the top half of the ground and each other at their endpoints. We call a tent rigid if its framework of segments is rigid in the plane as long as the connections to the ground are fixed. We define the area of a tent to be the area enclosed by the tent and ground.

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.


ANSWERS

Clinton Weaver improved my area for n=5 to A = √3 = 1.732+.

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 2n3/2/3√3, but rectangle structures can have areas at least n2/(6+√6).

Here are the maximum areas so far:

Largest Areas of Rigid Tents Using n Unit Segments
nLargest AreaAuthorPicture
20.500EF
41.299EF
51.732CW, DL
62.414EF, CP, DL
73.148EF, JD, CP, DL, GS
84.116
95.095
106.080
117.069
128.096CP
139.628
1411.148
1512.661
1614.171
1715.679
1817.184
1918.689
2020.472EF, CP, BF
2122.472
2224.472
2326.472
2428.472
2530.472
2632.472
2734.472
2836.518BJ
2939.016
3041.513
3144.011
3246.509
3349.007
3451.505
3554.004
3656.502
3759.084GS
3861.937BF, CP
3964.937
4067.937
4170.937
4273.937
4376.937
4479.937
4582.937
4685.937
4788.937
4891.937
4994.937
5097.937

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

Largest Areas of Rigid Tent With n=40
NameBest Area
Bertram Felgenhauer67.93
Brian J67.93
Dan Dima67.93
Dave Langers67.93
Corey Plover67.93
Guenter Stertenbrink67.54
Emilio Schiavi63.91
Erich Friedman63.80
Andrew Bayly63.80
David Cantrell63.74
Gary Gerken58.80
George L. Sicherman56.14
Joseph DeVincentis53.68

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.