Friday, February 27, 2009

Wholly Whexagons!

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

Dear Jules,

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 $\infty$-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, $180 - \frac{360}{n} = 120$, 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 $h^2 + r^2 = (2r)^2 = 4r^2$. So $h^2 = 3r^2$, and therefore, $h = \sqrt{3r^2} = r\sqrt{3}$. That means the area of the triangle is $\frac{1}{2} *\text{base}*\text{height}$$= \frac{1}{2} * (2r)*r\sqrt{3} = r^2 \sqrt{3}$.

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 $\frac{\pi r^2}{2}$. This means the ratio of circular area to total area of the triangle is $(\frac{\pi r^2}{2})/(r^2 \sqrt{3}) = \frac{\pi}{2\sqrt{3}}$, 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...

-DrM

Daniel Grossberg said...

Don't forget to mention that hexagonal packing is exceptionally common at the atomic level. It is the densest form of packing (on a plane). Picture those oranges as atoms, and it is obvious.

Another interesting, and related, property is that a hexagon is THE shape that is formed when you circumscribe a circle with other like-sized circles. I.E., if you take one circle, you can surround it perfectly with 6 circles on the outside. No other number of circles will surround it tightly.

drmath said...

Good points, Daniel! Of course, where things really heat up is in three dimensions, where once again the hexagon makes an appearance in the problem of optimally packing spheres. Is there no limit to the awesomeness of the hexagon?

DU said...

Squares, octagons, and other n-gons have the same property [of having parallel opposite sides]...

But only for even n. You wouldn't want a triangular (n=3) or pentagonal (n=5) bolt.

drmath said...

Right, good catch, DU. I should have been more clear about that.

As a side note, I believe pentagonal nuts/bolts do get used sometimes--for example, on fire hydrants--exactly because you can't open them with a standard wrench.

DU said...

Whoa, I never noticed that. I'll have to check it out.

John T said...

Wonderful! I am building a brick patio with a central design that is based on a hexagon. I'd love to hear some tips on geometric constructions that would be easy to lay out physically.

To lay out my hexagon-based shape on paper I draw a large circle and one diameter line. Using a protractor from the center off the diameter I draw two additional diameters at 60 degrees off.

Next I draw in chords where the diameters intersect the circle to form the hexagon.

Since the result of my shape is a flower I next form circles whose center is the midpoint of each chord (hexagon side) that intersect at the corners of the hexagon.

I potentially see some trouble with these last steps in the real world, where I'd be using less accurate tools and methods, but would like to end up as perfect a result as possible.

I'd love to hear more from you about how people used to lay out complex but regular geometries for floors and architecture using simple tools.

drmath said...

Hmmm... I'm not an expert in this sort of thing, but one idea I have is to exploit the fact that a hexagon is composed of six equilateral triangles, for which there are perhaps better construction techniques.

For example, starting with two points a fixed distance d apart, you can use a compass to draw a circle of radius d at each point. Taking (either) point of intersection of the two circles gives you an equilateral triangle with side length d. No need for a protractor. If you repeat the process six times, you should get a nice regular hexagon.

I'm interested to hear other people's thoughts, though. Let me know how it works out for you!

Anonymous said...

Dr M
from a pre-blog era...

Pappus of Alexandria. ca 290-350. Greek geometer.

"Bees. . . by virtue of a certain geometrical forethought . . . know that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material in constructing each".

mike

drmath said...

Nice, Mike! Now why don't we have that same "geometrical forethought"?

Anonymous said...

Thank you for this awesome post. I also love Hexagons. They are very common in board games. I'm currently working on a android video game that takes a lot of inspiration from D&D and board games. I'm going to use hexagons to lay out tiles for the battles. The beauty is that instead of squares where you have 4 places to move/attack to/from you instead have 6 increasing the fun by 50%.