Circles Surrounding Circles – Part I

After grappling with history, tough concepts and philosophy in recent weeks, I’ve decided to post something a little more trivial this week, just some clean Mathematics in two parts.

Note: While I was researching this topic, I stumbled upon this page. It presented exercises exploring exactly what I wanted to explore. So these posts have basically turned into my commentary on the exercises provided there. I have done some original work, but I thought I’d give the credit where it’s due.

If you’re anything like me, you will have found yourself at one point arranging loose change by laying the coins out flat on a surface, packing them as tight as you can together. If you try this for seven, say 2 pence coins, you might find yourself arranging them like so.

seven_circles_ba4a5fd8b0a4a751dc704e843912f6ef

As you can see, the outer six circles each touch their neighbors while still touching the inner circle. It is intuitive that on a flat surface it is impossible to arrange seven circles around the inner circle such that each of the outside circles touch the inner circle. This begs the question, why six? What properties of the circle are coming into play here?

To answer this, consider the fact that as each circle is identical they all have the same radius. This means we can draw an equilateral triangle whose points are the centers of three circles and whose sides pass through the points of intersections of these circles as shown below.

2requal
We know this triangle is equilateral as each side is equal to 2r, with r being the radius of the circles.

One of the properties of equilateral triangles is that each angle is equal to 60 degrees. Another basic geometric property is that 360 degrees exist around a point. Combining these two properties and considering the center of the inner circle we can work out that a maximum of six equilateral triangles can be placed around the center of the inner circle (As 6 * 60 = 360). This is shown below.

Hexagon
Forgive me, I made these images myself…

We can see from the diagram above that the six equilateral triangle are tightly packed into a ‘hexagon’ shape. It is not possible to fit another equilateral triangle around the center of the inner circle, much in the same way it is not possible to fit another coin around the inner coin. This relationship provides some geometric insight on the problem.

However this result doesn’t suffice as a conclusion to the question of ‘why?’. As it still doesn’t provide a huge insight into the relationship between the circles. So let us take a mathematical approach and abstract the problem a little, to see if this sheds any light on the underlying process of what’s really going on.

In part II, we’ll consider a similar problem to the one here. However, instead of only considering the case where all the circles are of the same size, we’ll ask the question: ‘What if the inner circle could be a different size to the outer circles?’

EDIT: Here’s the link to part II.

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