I hadn’t really planned to, but it’s both Pi Day and the birthday of Albert Einstein. As a fan of both the number and the man, it seems I should post something.
But I’ve written a lot about pi and Einstein, so — especially not having planned anything — I don’t have anything to say about either right now. In any event, I’m more inclined to celebrate Tau Day when we have double the pi(e). I do have something that’s maybe kind of vaguely of pi-ish. It’s something I was going to mention when I wrote about Well World.
It’s a little thing about hexagons.
Which are cool in a number of ways. There’s the whole honeycomb thing, for one; I’ll get back to that.
A thing I always thought was really cool about hexagons is that you can create them with just a compass and straight edge.
There’s a whole sub-field of geometry (started by those ancient Greeks, of course) that involves what you can — or cannot — do with just those two tools. Some of it is quite involved.
But hexagons are easy.
Drawing a hexagon involves a few easy steps:
- Using the compass, draw a circle.
- Without changing the compass, start at any point on the circle and draw another circle.
- At the two points where the second circle intersects the first, place the compass and draw two more circles. This creates two new intersection points.
- At the two new points, draw two new circles. This creates a single new intersection point where both these circles meet.
- (Optional) Draw that last circle (for symmetry).
- Use the straightedge to connect the six secondary circle center points.
That’s the technique I learned long ago. There’s a shorter version, using just two secondary circles, that you can find on the Wiki page (it’s animated and very clear).
This works because a hexagon contains six equilateral triangles (like pie slices), which means each face has the same length as the radius.
Therefore, the same compass setting that creates the circle’s radius is the same as the circle’s chord that is the hexagon face.
In the diagram above, you see the basic dimensions of a hexagon. The only one that isn’t obvious is the u parameter — the (minimum) length from the center to any face.
But it’s easy enough to calculate. It’s just sin(60º) ~0.866r. (The length from the center to any point is obviously just r.)
Hexagons are cool because they’re a shape that can tile a plane.
That’s another whole subfield of geometry: tiling (tessellation). That one also gets intense. (Let’s face it, most math fields do.)
But as simple obvious shapes go for tiling, it’s pretty much squares, rectangles, triangles, and hexagons. And with triangles, the tiles have to change their orientation. Only rectangles (and squares) and hexagons can tile without changing orientation.
(I mentioned on my blog anniversary an example using octagons, but small squares are required to fill in the gaps.)
And then there are bees and their honeycomb (which is delicious):
We’ve copied that design as a way to make structural elements that are rigid and strong, but also lightweight.
Bees use the design because of its strength and rigidity. And because the thin walls leave lots of room for honey.
Similar designs are sometimes used in fuel tanks to protect the tanks from crushing (and the fuel from sloshing).
All in all, hexagons are pretty awesome!
So whip out your compass and a ruler and draw some hexagons to celebrate Pi Day and Albert’s birthday!
I’ll give you something to do while you enjoy your virus-mandated social isolation. Seize this unique opportunity for self-education!
It’ll do you more good than binging on Netflix.
Stay hexed, my friends!
March 14th, 2020 at 3:14 pm
What makes hexagons sort of, kind of, vaguely pi-ish is that they divide the circle into six segments. The actual circumference of a circle of 2π (radians), which is 6.28 (my friend tau, again).
So, six faces in a hexagon, six segments of the circle, and 6+ radian segments.
I’m only off by 0.28 or so…
March 14th, 2020 at 7:38 pm
Heh, talk about synchronicity. Mathematician John Baez, on his Azimuth blog, has a new post that touches on “squaring the circle” (with a compass and straightedge). It’s a problem that, he writes, was “known to mathematicians since 414 BC.”
It was finally resolved in 1882 as being impossible (because circles are a little bit magical — which is to say transcendental).