The Shapes of Friends

If you exist but have no friends, no circle of friends, are you a simply a dot?

I was asked this question recently, by a friend, and my answer was this:
You are correct.
And, I went on, if you have only one friend, then you have a line of friend.
Not the plural, I pointed out, since there is only a single friendship.

So, my friend asked, is it a triangle of friends if you have two?

You are correct again, I said, the type of triangle depending
on each friendship’s strength and length.
If you have an equally strong bond with each, and each is of equal duration,
then yours is an equilateral-triangle of friends.

And if one is stronger than the other? my friend asked.

It varies by degrees, I said, strength being indicated by an angle’s degree.
Interestingly, if you are represented by a point at the vertex of a 90 degree angle
where one arm extends to each friend, regardless of length of either friendship,
then you can be sure if you have a friendship triangle
where the length of one friendship can be assigned the number 3 and the other the number 4,
then the square of first friendship plus the square of second friendship
equals the square of the friendship triangle’s hypotenuse, in this case 5:
3 squared + 4 squared = 5 squared, or 9 + 16 = 25.
This, I explained is called a Pythagorean friendship, a very special friendship triangle.

And if you have three friends, said my friend, then you have a friendship square.

Exactly, I said. By George, I think you’ve got it.

But then, said my friend, realizing the enormity of the situation,
in order to have a circle of friends, you would have to have a heck of a lot of friends.

Very true, I agreed. To say you have a small or large circle of friends is to use fuzzy labels.
To say you have a circle of friends, if indeed your choice of the word is not
an exaggeration, is to say that you have a number of friendships approaching infinity.
Now if you say, I have what is roughly a circle of friends, you may be correct.
But most of us have a number of friends that, if not zero or one, would form
various polygon shapes: If five friends, you would have a pentagon of friends,
if six, a hexagon, if seven, a heptagon, if eight an octagon, and so on.

A perfect circle of friends, my friend said, would be a terrible thing at Christmas time,
if you’re the kind of person, like me, who enjoys mailing cards to every friend.

Yes, I agreed, you would be addressing cards until doomsday, and you would still
not have sent them to everyone on your list for even one single year.

Which, I added, would be very good for the post office in terms of stamps income.

But potentially not so good, said my friend, if you have to actually deliver the mail.

True, I agreed, although most of us have not enough friends
to approach anywhere near an imperfect circle of them.

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