I’ve quit explaining math on Twitter and moved my activities of that sort to Mathstodon. This is a branch of Mastodon, a federated social network that is run by its own users—not by an unpredictable self-centered billionaire.
It feels a lot like the internet of the late 80’s or early 90’s, with people pitching in to build things they themselves use, not serving as cogs in the giant machine of surveillance capitalism. I invite you to join us!
I plan to write things there, polish them up a bit and put them here. An example is my post about modes. Today I had fun solving a puzzle about sundials. My goal was to use the minimum amount of math.
Colin Beveridge posed this puzzle:
“Suppose I planted a metre-long straight stick vertically in the ground and traced the locus of the end of its shadow. What shape would it make? Happy to assume a locally flat Earth if it makes things easier.”
Equivalently: what curve is traced out by the shadow of the tip of a sundial during one day, if the shadow lands on flat ground?
The answer is: a hyperbola—or in one very special case a straight line!
To see this, work in Earth-centered coordinates and treat the Sun as a point S moving in a circle over the course of a day. Treat the ground as a plane P. Sunlight traces out a line L going from S to the sundial’s tip T and hitting this plane P at some point X.
As S goes around in a circle, what curve does X trace out?
That’s the math question I’m solving.
To solve it, we need an obvious math fact: as a point S goes around a circle, the line going through S and any point T traces out a cone.
And another less obvious but very famous fact: when we intersect a cone with a plane P we get a curve called a ‘conic section’, which can be a circle, ellipse, parabola, hyperbola, or a line.
So, the only question is which of these curves we can actually get!
As the Sun sets, the shadow of our sundial gets arbitrarily long—so we can only get a circle or ellipse if the Sun never sets.
We only get a parabola if the Sun sets in the exact same place on the horizon that it rises—since the two ‘ends’ of a parabola go off to infinity in the same direction.
All these cases are a bit unusual. In most circumstances the curve we get will be a hyperbola or a straight line.
We get a straight line only when the Sun rises at one point on the horizon, is straight overhead at noon, and sets at the opposite point of the horizon. This would happen every day if you lived at the equator and the Earth’s axis wasn’t tilted. But in reality this situation is rare.
So, the shadow traced out by a sundial’s tip is usually a hyperbola!
You can play around with these hyperbola-shaped shadows here:
• Intellegenti Pauca, Hyperbola shadows, Geogebra.
There’s a lot more one can say about this: for example, what happens with the change of seasons? But I wanted to keep this simple!
Click on this picture for some details about a nice sundial that shows off its hyperbolae: