If we join two circles on a plane, each will have exactly one neighbor.
Given three or more circles, we can build a chain where each circle has exactly two neighbors.
There are also arrangements where each circle has exactly three neighbors, like the one shown here.
Is it possible to arrange a finite number of equally sized, non-overlapping circles on a plane such that each circle has exactly four neighbors?
What if the circles can have different sizes?
Answer
It is not possible when the radii are equal. Consider the topmost circle, C. In order to pack four neighbors around C, none of which are above C, you have to use the below arrangement (where C is the gray circle):
This means there is a circle to the immediate right and left of C. Repeating the same argument over and over shows that there is an infinite line of circles at the same level as C, contradicting the finiteness requirement.
When you allow differing radii, the below arrangement works:
No comments:
Post a Comment