Telling Time on a Torus | Infinite Series

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? Then stick around.

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Previous Episode:
How to Divide by Zero

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.

Written and Hosted by Gabe Perez-Giz
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow and Meah Denee Barrington

Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon.

Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!

And thanks to Nicholas Rose, Jason Hise, Thomas Scheer, Marting Sergio H. Faester, CSS, and Mauricio Pacheco who are supporting us at the Lemma level!

[pbsinfiniteseries]

*Whoops! Square at **6:20** is NOT the Klein bottle!* (which would require not criss-crossing the "gluing" of points at the left and right). The construction I showed is the (real) projective plane. My bad, everyone. I'm a topological dum-dum. Thanks to everyone who pointed out my error.

[xatnu]

At 6:15 they stated this identification gives the Klein bottle, but in truth it gives an even weirder surface called the "real projective plane". This plane is kind of like "half" of a klein bottle, in the sense that if you glue two of them together by identifying any pont of one to a point of the other, you obtain an object equivalent to a klein bottle.

[gareththompson]

Damn I thought I had clicked on a PBS Space Time video for a minute. Welcome back to the spotlight Gabe. Matt has been doing a great job with the show, but I've missed you.

[ragha]

So, swap hour and minute hands to give a second helical curve. Intersection of both curves are valid solutions..

[earthbjornnahkaimurrao]

now i know why this channel is called infinite series: because every video tells you to pause and watch 2 more videos before watching this one. As such you must always watch an infinite number of videos before ever watching any of the videos. Quite the paradox!

[marshallgatten]

Pacman can only cross the edges at 90 degrees, whereas every line on your helix crosses at an angle. For this reason, a far more apt analogy would be Asteroids. And the animation would have the added benefit of showing the ship shooting in a direction other than it's heading so that the "bullet" crosses at one angle while the ship crosses at another while the "paper" remains unchanged. Other than that ridiculously nitpicky thing, I loved this video. Great job!

[sebastianelytron]

This is some seriously complex stuff. Go easy on our brains, Gabe. With this episode you *torus* a new one!

[rikschaaf]

You made it quite easy. Just overlap the square with its mirror around the diagonal and mark the points that overlap. The special case is all the overlapping points on the diagonal. Those are all the points that have the minute and hour hand in the same position.

[NiloRiver]

Touching the esoteric without fantasizing so much. Thank you!

[slam_down]

Welcome back Gabe, please please please do a spacetime episode or cameo <3

[gregoryfenn]

For me, this video was fantastic and an intuitive intro to "pacmanifying" a 2d plane/sqaure. It's a really cool way to model otherwise hard-to-imagine shapes. I wouldn't have seen that coming from the title of the video... so I would argue the title should have been more to do with the pacmanification.

[groethendieck]

error at 6:25, it is actually the projective plane and not the klein bottle...

[pierrecurie]

8:18
-1 and 4 are in the same equivalence class mod 5 (ie 4-(-1) = 5). Neither of them are in the same equivalence class as 1. It shouldn't be too surprising that 1 and -1 aren't the same.

I think it's more an issue of convention whether to use 0, 1, 2, 3, 4 or -2, -1, 0, 1, 2. There are a few situations where it's more convenient to use -1 than n-1.

[lierdakil]

A fun way to look at the challenge is to figure out what exactly does "switching the hands" mean in the mathematical sense.

First, let me introduce a function fraction(x), denoted {x}=fraction(x), which returns fractional part of a number. Then, if we call hour hand position in terms of full revolution H (that is, H=0 at 00:00:00, and H=1 at 12:00:00), then minute hand position in terms of full revolution M = {12 H}.

So, switching the hands is just switching M with H, so that H = {12*M}. But for that position to be also valid, M = {12*H} must still be respected. Hence,
H={12*{12*H}}.
let us now substitute H=x/12, 0 ≤ x < 12
x/12 = {12*{x}}
and represent x as a sum of its whole part 0 ≤ h < 12 and fractional part 0 ≤ r < 1:
(h+r)/12 = {12*r}
repeating the procedure for r, we arrive at
12*h+k = 143*f
where 12*r = k + f, 0 ≤ k < 12, 0 ≤ f < 1, k is whole.

Left side is whole, so right side also has to be whole. It can only be whole if f = m/143 where m is whole:
12*h+k = m

m smaller then 143: since 0 ≤ f < 1, 0 ≤ m/143 < 1, so 0 ≤ m < 143.

The rest is trivial.
H = x/12 = (h+r)/12 = (h+(k+f)/12)/12 = (12*h+k+f)/144 = (m + m/143)/144 for m = {0, 1, ..., 142} -- one just needs to convert this to

Some formulae:

Let us define floor(x) as a number x rounded to integer towards negative infinity, denoted ⌊x⌋=floor(x). For whole numbers, ⌊x⌋ = x - {x}

Then

hours = ⌊H*12⌋
minutes = ⌊{H*12}*60⌋
seconds = ⌊{{H*12}*60}*60⌋
fractions = {{{H*12}*60}*60}

[skoockum]

"Rock out with ...your clock out"

[RadicalCaveman]

Thanks to you, I learned a new mathematical term: "pacmanified."

[MasterNeiXD]

Great video. The equivalences just blew my mind.

[TheCimbrianBull]

Mindblown! All of this math stuff makes my head spin. That torus shape gave me a craving for donuts! :-)

[jeremyags]

Good to see you here Gabe! Love your Spacetime episodes

[eofirdavid]

Very good video. Starting with an interesting problem and then showing how mathematical objects arise from it naturally (and the pacman example is great). This problem shows naturally why and where you can use quotient spaces, and probably this video should have come before the one on quotients.

Regarding the division algorithm remark in the end - you don't toss away the negative numbers because they are not positive. You just choose a representative from each class mod n, and luckily in the integers we can choose all the representative to form a nice set (i.e. 0, 1, ..., n-1). There are also problems where it is better to use the set -n/2, ..., n/2, and not to mention other rings where there is no notion of "positivity" at all.