Today I don’t have anything big. I just wanted to point people to “Spherical Cycloids”, a post on the WordPress blog The Inner Frame. Cycloids are somewhat familiar, at least to kids who grew up in the United States in my age cohort, because you get them out of spirographs. You make them by taking some point within a rotatable object. Roll that object along the path, normally one that’s defined by another shape. You can get wonderful and strange and exotic-looking curves, many with a hypnotic regularity.
The Inner Frame made a variation of this. It’s got cycloids drawn on the surface of the sphere. This immediately adds a new level of strangeness and wonder to the curves. The pictures are lovely and hypnotic. Folks with 3-D printers can probably also make some grand exotic candleholders from the pattern, too.
Robert Austin, of the RobertLovesPi blog, got to thinking about one of those interesting mathematics problems. It starts with the equations that describe the volume and the surface area of a sphere.
If the sphere has radius r, then the surface area of the sphere is 4πr2. And the volume is (4/3)πr3. What’s interesting about this is that there’s a relationship between these two expressions. The first is the derivative of the second. The derivative is one of the earliest things one learns in calculus. It describes how much a quantity changes with a tiny change in something it depends on.
And this got him to thinking about the surface area of a cube. Call the length of a cube’s side s. Its surface is six squares, each of them with a side of length s. So the surface area of each of the six squares is s2, which is obvious when you remember we call raising something to the second power “squaring”. Its total surface area then is 6s2. But its volume is is s3. This is why we even call raising something to the third power “cubing”. And the derivative of s3 is 3s2. (If you don’t know calculus, but you suspect you see a pattern here, you’re learning calculus. If you’re not sure about the pattern, let me tell you that the derivative of s4 would be 4s3, and the derivative of (1/3)s2 would be (2/3)s.)
There’s an obvious flaw there, and Austin’s aware of it. But it got him pondering different ways to characterize how big a cube is. He can find one that makes the relationship between volume and surface area work out like he expects. But the question remains, why that? And what about other shapes?
I think that’s an interesting discussion to have, and mean to think about it some more myself. And I wanted to point people who’d be interested over there to join in.
If you asked someone to say what mathematicians do, there are, I think, three answers you’d get. One would be “they write out lots of decimal places”. That’s fair enough; that’s what numerical mathematics is about. One would be “they write out complicated problems in calculus”. That’s also fair enough; say “analysis” instead of “calculus” and you’re not far off. The other answer I’d expect is “they draw really complicated shapes”. And that’s geometry. All fair enough; this is stuff real mathematicians do.
Geometry has always been with us. You may hear jokes about never using algebra or calculus or such in real life. You never hear that about geometry, though. The study of shapes and how they fill space is so obviously useful that you sound like a fool saying you never use it. That would be like claiming you never use floors.
There are different kinds of geometry, though. The geometry we learn in school first is usually plane geometry, that is, how shapes on a two-dimensional surface like a sheet of paper or a computer screen work. Here we see squares and triangles and trapezoids and theorems with names like “side-angle-side congruence”. The geometry we learn as infants, and perhaps again in high school, is solid geometry, how shapes in three-dimensional spaces work. Here we see spheres and cubes and cones and something called “ellipsoids”. And there’s spherical geometry, the way shapes on the surface of a sphere work. This gives us great circle routes and loxodromes and tales of land surveyors trying to work out what Vermont’s northern border should be.
Continue reading “A Summer 2015 Mathematics A To Z: hypersphere”