Dear Dr. Math,
I've noticed that hexagons show up in a lot of different places. Now that I've started looking for them, I see them everywhere! What's the deal with hexagons?
Jules, Canton OH
I know I'm not supposed to play favorites with mathematical objects, but I have to confess, the hexagon is probably my favorite shape. (Sorry, rhombicuboctahedron.)
While I suspect that you may be experiencing a fair amount of confirmation bias, it's true that hexagons do make an appearance in a large variety of different contexts. (It's also possible that you have the "hexagon madness" and are seeing them when they're not actually there. You might want to get that checked out.)
Part of the reason hexagons are so ubiquitous is that they have so many useful properties, probably even more than "familiar" shapes like squares and trapezoids. Primarily, I'm referring to regular hexagons--hexagons with 6 equal sides--like this guy:
Probably these are the ones you're seeing, Jules. Next time, I'll talk a little about the properties of irregular hexagons and why you might expect to see those, too.
First of all, a regular hexagon has the property that its opposite sides are parallel to each other, making it an ideal shape for a nut or bolt, because it fits nicely into a wrench:
Squares, octagons, and some other n-gons [those with even n] have the same property, but with a hexagonal nut or bolt, you can grab it at a variety of different angles, which is useful if you're putting together Ikea furniture in a tiny Manhattan apartment, for example. Also, since the exterior angles of an n-gon add up to 360° and there are n of them, each one measures 360/n. Therefore, more sides aren't really so good, because as the number of sides gets larger, the sharpness of the corners decreases, allowing for a greater possibility of slippage. A hexagon seems to be a nice compromise between a 2-gon and an -gon for these purposes.
Another important property of regular hexagons (that I'm sure you're aware of if you've ever looked at the floor of a public bathroom) is that they tile the plane. In other words, by putting a bunch of identical hexagons together as tiles, we can cover an entire plane surface:
There are other tilings, of course, with squares or triangles, but this one has some very appealing aspects. (For one, it's made of hexagons!) It turns out that among all possible tilings of the plane of shapes with a fixed area, the hexagonal one has the smallest possible perimeter.
One way to think about this is that the perimeter:area ratio goes down as a shape gets closer to being a circle. So, if we're using n-gons to tile the plane, we want n to be as big as possible. On the other hand, we have to be able to glue them together so that at each point of intersection, the angles add up to 360°. By the same reasoning as before, we can show that the interior
angles on an n-gon are each 180 - 360/n, and since there have to be at least 3 of these angles meeting at each corner, the greatest this angle can be is 120°. In this case, , so 360/n = 60; therefore, n=6. Hexagon!
Now, why does all of that matter? Well, say you weren't cutting these tiles out of a piece of ceramic but instead were building up walls to section an area into a number of chambers. If the material in those walls was really expensive for you to produce, it would bee in your best interests to make the chambers in the shape of a regular hexagon:
I don't know how bees managed to figure this out and yet here we are still living in rectangular grids like chumps.
A slightly different, but related, property of the regular hexagonal tiling is that it shows up if you're trying to pack together some circles:
See the hexagons?
Once again, this way of packing circles has the property of being optimal, in the sense that it leaves the least amount of empty space between circles. In fact, using a little trigonometry, we can even work out the efficiency of this packing:
The triangle in the picture is equilateral with all sides equal to 2*r, where r is the radius of the circles we're packing. If we split one in half (where the two black circles intersect), we'll get a right triangle with hypoteneuse 2*r and one leg r. Therefore, by the Pythagorean Theorem, if h is the height, then . So , and therefore, . That means the area of the triangle is .
Inside each triangle are three pieces of a circle, which together make up half of a circle of radius r. Thus, the area of the circular pieces is . This means the ratio of circular area to total area of the triangle is , approximately 0.91. Since the whole plane is made up of these triangles, the proportion of circle-area to total-area is the same, meaning the circles take up about 91% of the space. Pretty efficient, and fun at parties, too!
To Be Continued...