Telling Time on a Torus

[MUSIC PLAYING] What shape do you most associate with a standard analog clock?

Your reflex answer might be a circle.

But a more natural answer is actually a torus.

Surprised?

Than stick around.

I'll explain what I mean and pose a clock puzzle challenge problem that adopting this viewpoint might help you to solve.

[MUSIC PLAYING] Some configurations of a clock, like the hour hand at 3 with the minute hand at 12-- represent valid times of day.

If the hands sweep around continuously at their usual steady rates, this configuration will actually happen every 12 hours, at precisely 3 o'clock.

But other configurations are invalid.

Like the hour hand at 3 with the minute hand anywhere other than 12, since the hour had moves slightly off the 3 as soon as the minute hand leaves the 12.

Fine-- now, take a valid configuration but swap the positions of the hands.

Is that new configuration also valid?

Well, if the hands overlap, then yes, since swapping wouldn't change the reading on the clock.

But if the hands don't overlap, you certainly won't get a valid time in general.

Just look at 3 o'clock.

Remember, swapping the hands here won't give you 12:15 or any other legitimate time for that matter.

So are there any other solutions?

And how would you find them?

This is an old problem that can be tackled in lots of different ways.

But there is a visual approach that I especially like, because it illustrates how the quotienting concept that Tai-Danae introduced last time can give you unexpected problem-solving power when applied to shapes.

So without giving away the answer or even the whole method, I'm going to take a detour into quotient shapes to give you just enough ammunition to attack the problem this way if you choose to.

Of course, you're free to tackle the problem however you like.

And later in the episode, I'll circle back to refine the question and tell you how to submit answers.

Before we get started though, you might want to pause me and get some important background from two of Tai-Danae's earlier episodes-- her intro to quotienting and her episode about topology and open sets.

So go check those out.

And then rendezvous back here with me, to rock out with your clock out.

Let me start by quite literally dissecting the clock.

Meaning, let's detach the two hands and mount each one on its own clock face.

Now, orient those clock faces at 90 degrees to each other.

And connect them like this, so that the entire minute hand gets dragged by the hour hand along a wider circle.

If you play around with this contraption, you'll notice that, no matter how you position the two hands, the tip of the minute hand always falls on the surface of this torus.

That's why a torus is a natural backdrop for analyzing two-hand clocks, because each point on the torus corresponds to exactly one possible configuration of the two hands and vice versa.

OK, now initialize the clock to midnight.

And let the hands run at their usual steady rates.

As time passes, the tip of the minute hand traces out a helical curve that winds 12 times around the torus before closing on itself and starting over.

Apparently, the points traced out by that curve should correspond to all the valid clock configurations that actually occur during each 12-hour cycle.

Now, a helix wrapped around a torus isn't exactly the easiest thing to work with.

But as it turns out, this coil can be re-imagined as just a bunch of simple straight lines on a square.

And the key to that simplification is the concept of quotient shapes or, to be more precise, quotient spaces.

Here's the idea.

If you had a square sheet of a stretchable material, like foam rubber, you could build a torus out of it in two stages.

First, you could fold it and glue these two sides together to make a cylinder.

And then you could fold the cylinder in the other direction and glue the open circular edges together to make a torus.

But there's a way to achieve the same effect without folding or gluing anything or even visually imagining any folding or gluing.

Here's the alternative procedure.

Step 1-- why don't we just declare each pair of points that are directly across from each other on opposite edges of the square to be equivalent?

If you think about it, that's tantamount to defining an equivalence relation on the square that also treats all four corners as equivalent-- that's got to be true by transitivity-- and that also makes each interior point equivalent only to itself.

Now, step 2-- treat each equivalence class or bucket as if it were a single point.

In other words, quotient out the square by this equivalence relation.

Remember, treating equivalence classes themselves as your basic objects, rather than the things inside those classes, is essentially what quotienting is.

To get a visual sense of what this is, it's like we've turned the square into a Pac-Man game, where hitting any wall brings you back in through the opposite wall, without changing your overall heading.

Now, my claim is that Pac-Manifying the square by quotienting out this particular equivalence relation, gives you something topologically equivalent to the torus.

In other words, the quotienting effectively gives you folding and gluing without any of the 3D visual overhead.

Technically, to formally establish this correspondence, I'd still need to show two things-- one, that the Pac-Man square, equipped with its new concept of identical points, inherits a notion of open sets or neighborhoods from the original square; and two, that there exists a one-to-one mapping between the Pac-Manified square and an actual torus that maps neighbors in one space to neighbors in the other space.

Demonstrating all that rigorously would take us too far off track today.

But both facts turn out to be true.

And at least with the cylinder, you should be able to convince yourself that this all works out and that a helical barber-pole curve on the cylinder would, in fact, correspond to a line on a square that has Pac-Man edges at the left and right.

Now, this quotienting procedure turns out to be very general.

Just change the equivalence relation and you can build all sorts of funky shapes out of a square.

For instance, say you equate opposite points on the left and right edges, like before, but now equate all the points on the top edge just with one another.

You get a cone.

Or equate diagonally opposite points on just the top and bottom edges and you get a Mobius strip.

If you throw in diagonally opposite points on the left and right edges on top of that, you get a Klein bottle.

This list goes on and on.

And you can use base shapes other than a square.

If you've never seen this before, I encourage you to play around with the options, because it's pretty fun.

But let's get back to the clock puzzle.

If you buy my arguments so far, then the points on the Pac-Manified square, just like points on the torus, should represent all possible configurations of the clock.

And the winding helix, which captured just the valid configurations that actually occur during times of day would correspond to a line that climbs up the square by repeatedly exiting through one side and re-entering on the opposite side.

So just call the bottom edge the minute hand axis and the left edge the hour hand axis, set a convenient scale of your choice for each axis, and you should be able to tackle the valid swaps problem with just some high school math and a little patience.

I will leave you with one hint.

Think about what geometric operation on the square, swapping the clock hands, would correspond to.