How is it that 6 circles of equal diameter surround a 7th perfectly? Is there a mathematical proof of this?

Answer 1

The hexagon argument above shows why that works. It's fairly standard. Proving that that's the most efficient way to pack them in takes a little more work. A more detailed account of this question may be found at:

http://en.wikipedia.org/wiki/Kissing_number
http://mathworld.wolfram.com/KissingNumber.html

Unlike what someone asserted, it is NOT required that the surrounding circles be tightly packed. That would help, but in three dimensions or more, that may not be possible. In 1 dimension, the two surrounding intervals cannot even touch at all!

I'm not sure about the answer that alleges that any convex shape has the "7 penny property." I would disagree. A square is convex, and it has some sort of "8 penny property."

Answer 2

Imagine 6 circles of diameter 2x and then a 7th cirlce of the same diameter, 2x.

The radius of the circle is x.

Since it is given that 6 circles of equal diameter surround the 7th one perfectly, follow exactly what they say.

As it says, the circles SURROUND it, so the circles must be touching each other including the one it's surrounding.
This is to say that the diameters connect to each other and form a REGULAR hexagon of perimeter 12x.

Then here comes the fun part. Since the circle is completely surrounded, the centriod of both the hexagon and the circle are the same. Find the centroids yourself, its a lot of work. A suggestion is to give real values for your diameters. Also use a Cartesian plane.

Answer 3

Linlyons describes it pretty well, I think. I'd start with three equal circles and notice that, when you connect the centers of the three circles, you get an equilateral triangle. Add four more circles, and you'll get the hexagon. So, there are 60° angles everywhere, and six 60° angles makes 360°, a perfect fit.

On the "7-penny property" mentioned by Philo: Heh, first time I've heard of that. It is unclear to me how this applies to squares.

Answer 4

Consider the cross sections of a stack of cylinders forming a rough triangular prism as follows:

...O
..OO
.OOO
OOOO

The middle cylinder on the row with 3 cylinders is surrounded perfectly by the 6 that touch it. But you want to know why.

Basically, the reason that it works is because when one cylinder rests on top of two others, it naturally falls to the lowest spot it can -- putting its center directly above the midpoint of the centers of the circles upon which it is resting. Thus when the bottom row is level and the circle touch, the circles on top of them will need to have exactly the same radius to be in the lowest position and still touch each other as well.

I hope this helps!

Answer 5

It turns out that ANY convex shape will have the 7-penny property. My teacher, the late Paul Kelly of UC Santa Barbara, proved it when he was a graduate student, and HIS teacher included the proof as a HOMEWORK PROBLEM in the textbook he was currently writing. Dr. Kelly later used that book in his classes, and I'd give you the title but it's in one of 50 boxes of books I haven't unpacked since I retired and moved. Research convex sets if you want to find a proof.