Tricky question of circles

Sep 2017
2
0
US
Circle is a shape that cannot be tesselated. But how to optimize the number of circles fitted in a bounded region. Is there any relationship between the number of circles and the dimension of the regions. Design a rack with minimum area or perimeter to fit in 50 coke cans. Restrict the problem to racks in the shape of regular polygons. What kind of mathematical model can be used to make prediction?
 
Jun 2015
915
269
England
This is known as Malfatti's problem.

Some references

Excursions in geometry : Ogilvy : Oxford University Press

Unsolved and Unsolvable Problems in Geometry : Meschowski : Oliver and Boyd

The Seven Circles Theorem and Other New Theorems : Evelyn, Money-Coutts and Tyrrell :Stacey International
 
Jun 2015
915
269
England
You may like to know that Malfatti's problem is the dual or negative of yours.

He wanted to cut cylinders out of regualr blocks of marble, with minimum wastage.

You want to stack cylinders into minimum blocks.
 

skipjack

Forum Staff
Dec 2006
21,473
2,466
But how to optimize the number of circles fitted in a bounded region.
Circles that needn't all have the same size? If so, what is being optimized?

Design a rack with minimum area or perimeter to fit in 50 coke cans. Restrict the problem to racks in the shape of regular polygons.
I'll assume each can has radius 1.

If the rack has the shape of an equilateral triangle large enough to contain a row of ten cans along one side, it can hold up to 55 cans and has perimeter 64.4 approximately.

If the rack is square and uses square packing, it needs to have a side length of 16, and so will have a perimeter of 64. It will hold up to 64 cans. See this article and also this article if square packing needn't be used.

If the rack has the shape of a regular hexagon and uses hexagonal packing, it needs to be large enough to contain a row of five cans along one side, it can hold up to 61 cans and I'll let you calculate its perimeter.