Solving Coverage with Sets: Min Info Needed?

[axelmorack]

A question about sets??

I have a number of weird shaped flat objects. I am interested in covering as much of the floor as I can. After placing the objects on the floor, the only info I have is:

Choosing any two objects on the floor, the overlap between them is at a minimum possible.

What can I say about the coverage? Do I have too little information to say anything?

 

[SteveL27]

Too little info, but the general subjects you're interested in are tilings and tesselations.

http://en.wikipedia.org/wiki/Tessellation

http://en.wikipedia.org/wiki/Tiling_by_regular_polygons

An interesting historical aside is the discovery of new classes of tesselations in the 70's by an amateur mathematician, a housewife with a high school education. Her name is Marjorie Rice. She read a Scientific American article on tesselations and started working with them in her spare time. She'd work out her formulas on her kitchen counter and cover up her work whenever anyone came into the kitchen.

She eventually developed her own system of notation, sent her results off to the local university, and was recognized for having made brand new mathematical discoveries.

http://en.wikipedia.org/wiki/Marjorie_Rice

Her website is here.

http://tessellations.home.comcast.net/~tessellations/

You didn't mention if your flat objects are all the same shape or not. That's going to make a huge difference in being able to solve the problem.

 

[axelmorack]

Too little info, but the general subjects you're interested in are tilings and tesselations.

http://en.wikipedia.org/wiki/Tessellation

http://en.wikipedia.org/wiki/Tiling_by_regular_polygons

I suspected I have too little information. However, I am not tiling because the objects do overlap. Thanks
The objects are different shapes, The only guarantee is what is stated about the overlap in pairs.

 

[theorem4.5.9]

Perhaps you are interested in covering theorems? You may start with the 5-r covering theorem (it's one of the most basic and easier to understand), then maybe the Vitali or Besicovitch covering theorem, though they get very technical.

Though your phrasing makes me think that you have some kind of optimization in mind, in which case if there isn't any symmetry or regularity in your shapes, then there wouldn't be a quick solution.

 

[CellCoree]

I would first clarify what is meant by "coverage" in this context. Is it the total area covered by the objects on the floor, or the percentage of the floor that is covered? Once that is established, I would suggest using mathematical modeling and set theory to solve this problem. By representing the objects as sets and using concepts such as intersection and union, we can determine the exact coverage of the floor. Additionally, we can explore different arrangements and combinations of the objects to optimize the coverage. Therefore, I would say that with the information provided, we can make meaningful conclusions about the coverage, but further analysis and experimentation may be needed to find the optimal solution.