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If we join two circles on a plane, each will have exactly one neighbor.

Neighboring Circles - 1

Given three or more circles, we can build a chain where each circle has exactly two neighbors.

Neighboring Circles - 2

There are also arrangements where each circle has exactly three neighbors, like the one shown here.

Neighboring Circles - 3

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?

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up vote 11 down vote accepted

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):

enter image description here

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:

enter image description here

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With an infinite number of circles, the simplest way is to have a line of circles nestled on top of another line of circles.

I believe it is impossible with a finite number of circles. First, let's start with a standard trapezoid, 2 circles on 3 circles. The bottom middle circle has 4 neighbors. The problem are the other two bottom circles. At this point, when adding a circle you can only connect to 2 other circles, and then the new circle needs 2 more connections as well.

For circles of different sizes, use the simple double line, but have 1 line be smaller than the other. The line will curve and meet itself.

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1: no, but on a cylinder, yes. two touching staggered lines around the circumference

2: yes, arrange (eg)5 in a circle and then larger ones outside each touching 2 inner circles and two neighbours

additionally 5 neighbours is possible if you allow a circle of negative radius, draw splayed dodecahedron

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Can you explain this a bit more? I do not fully understand the answer or where the 1,2,5 came from. If you want multiple lines in a spoiler tag, please use <br>. – wythagoras Jan 9 at 13:04
1  
Actually, for multiple lines in a spoiler, end each line with double space. – JonTheMon Jan 9 at 15:42
    
1, 2 per part1 part 2, 5 was a number 5 sorry for creating confusion, – Jasen Jan 9 at 23:16

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