Reading the Comics, July 21, 2018: Infinite Hotels Edition


Ryan North’s Dinosaur Comics for the 18th is based on Hilbert’s Hotel. This is a construct very familiar to eager young mathematicians. It’s an almost unavoidable pop-mathematics introduction to infinitely large sets. It’s a great introduction because the model is so mundane as to be easily imagined. But you can imagine experiments with intuition-challenging results. T-Rex describes one of the classic examples in the third through fifth panels.

The strip made me wonder about the origins of Hilbert’s Hotel. Everyone doing pop mathematics uses the example, but who created it? And the startling result is, David Hilbert, kind of. My reference here is Helge Kragh’s paper The True (?) Story of Hilbert’s Infinite Hotel. Apparently in a 1924-25 lecture series in Göttingen, Hilbert encouraged people to think of a hotel with infinitely many rooms. He apparently did not use it for so many examples as pop mathematicians would. He just used the question of how to accommodate a single new guest after the infinitely many rooms were first filled. And then went to imagine an infinite dance party. I don’t remember ever seeing the dance party in the wild; perhaps it’s a casualty of modern rave culture.

T-Rex: 'David Hilbert was a mathematician and hotelier who was born in 1892. He built an infinite hotel, you guys! THE INFINITE HOTEL: A TRUE STORY. So Hilbert built this infinite hotel that was infinitely big and had infinitely many rooms; I believe this was a matter of some investment. But build it he did, and soon after a bus with infinity people in it showed up, with each of them wanting a room! Lucky for Hilbert he had his infinite hotel, so each guest got a room, and the hotel was filled up to capacity. Nice! But just then another friggin' bus showed up, and it ALSO had infinity people in it!' Utahraptor: 'Nobody builds for TWO infinite buses showing up right after the other!' T-Rex: 'Turns out they do! He just told every guest already there to move into the room that was double their current room number. So the guest in room 3 moved into room 6, and so on! Thus, only the even-numbered rooms were occupied, and everyone on the new bus could have an odd-numbered room!' Utahraptor: 'Amazing!' T-Rex: 'Yep! Anyway! It's my understanding he died an infinitely rich man infinity years later.'
Ryan North’s Dinosaur Comics for the 18th of July, 2018. The strip likely ran sometime before on North’s own web site; I don’t know when.

Hilbert’s Hotel seems to have next seen print in George Gamow’s One, Two Three … Infinity. Gamow summoned the hotel back from the realms of forgotten pop mathematics with a casual, jokey tone that fooled Kragh into thinking he’d invented the model and whimsically credited Hilbert with it. (Gamow was prone to this sort of lighthearted touch.) He came back to it in The Creation Of The Universe, less to make readers consider the modern understanding of infinitely large sets than to argue for a universe having infinitely many things in it.

And then it disappeared again, except for cameo appearances trying to argue that the steady-state universe would be more bizarre than what we actually see. The philosopher Pamela Huby seems to have made Hilbert’s Hotel a thing to talk about again, as part of a debate about whether a universe could be infinite in extent. William Lane Craig furthered using the hotel, as part of the theological debate about whether there could be an infinite temporal regress of events. Rudy Rucker and Eli Maor wrote descriptions of the idea in the 1980s, with vague ideas about whether Hilbert actually had anything to do with the place. And since then it’s stayed, a famous fictional hotel.

David Hilbert was born in 1862; T-Rex misspoke.

Teacher: 'Sluggo --- describe an octagon.' Sluggo: 'A figure with eight sides and eight angles.' Teacher: 'Correct. Now, Nancy --- describe a sphere'. (She blows a bubble-gum bubble.)
Ernie Bushmiller’s Nancy Classics for the 20th of July, 2018. Originally run, it looks to me, like the 18th of October, 1953.

Ernie Bushmiller’s Nancy Classics for the 20th gets me out of my Olivia Jaimes rut. We could probably get a good discussion going about whether giving an example of a sphere is an adequate description of a sphere. Granted that a bubble-gum bubble won’t be perfectly spherical; neither will any example that exists in reality. We always trust that we can generalize to an ideal example of this thing.

I did get to wondering, in Sluggo’s description of the octagon, why the specification of eight sides and eight angles. I suspect it’s meant to avoid calling an octagon something that, say, crosses over itself, thus having more angles than sides. Not sure, though. It might be a phrasing intended to make sure one remembers that there are sides and there are angles and the polygon can be interesting for both sets of component parts.

Literal Figures: a Venn diagram of two circles, their disjoint segments labelled 'Different' and their common area labelled 'Same'. A graph, 'Height of Rectangles', a bar chart with several rectangles. A graph, 'Line Usage': a dashed line labelled Dashed; a jagged line labelled Jagged; a curvy line labelled Curvy. A map: 'Global Dot Concentration', with dots put on a map of the world.
John Atkinson’s Wrong Hands for the 20th of July, 2018. So this spoils a couple good ideas for my humor blog’s Statistics Saturdays now that you know I’ve seen this somewhere.

John Atkinson’s Wrong Hands for the 20th is the Venn Diagram joke for the week. The half-week anyway. Also a bunch of other graph jokes for the week. Nice compilation of things. I love the paradoxical labelling of the sections of the Venn Diagram.

Ziggy: 'I wish I'd paid more attention in math class! I can't even count the number of times I've had trouble with math!'
Tom II Wilson’s Ziggy for the 20th of July, 2018. Tom Wilson’s still credited with the comic strip, though he died in 2011. I don’t know whether this indicates the comic is in reruns or what.

Tom II Wilson’s Ziggy for the 20th is a plaintive cry for help from a despairing soul. Who’s adding up four- and five-digit numbers by hand for some reason. Ziggy’s got his projects, I guess is what’s going on here.

Cop: 'You were travelling at 70 miles per hour. How much later would you have arrived if you were only going 60?' Eno: 'No fair --- I hate word problems!'
Glenn McCoy and Gary McCoy’s The Duplex for the 21st of July, 2018. So the strip is named The Duplex because it’s supposed to be about two families in the same, uh, duplex: this guy with his dog, and a woman with her cat. I was reading the strip for years before I understood that. (The woman doesn’t show up nearly so often, or at least it feels like that.)

Glenn McCoy and Gary McCoy’s The Duplex for the 21st is set up as an I-hate-word-problems joke. The cop does ask something people would generally like to know, though: how much longer would it take, going 60 miles per hour rather than 70? It turns out it’s easy to estimate what a small change in speed does to arrival time. Roughly speaking, reducing the speed one percent increases the travel time one percent. Similarly, increasing speed one percent decreases travel time one percent. Going about five percent slower should make the travel time a little more than five percent longer. Going from 70 to 60 miles per hour reduces the speed about fifteen percent. So travel time is going to be a bit more than 15 percent longer. If it was going to be an hour to get there, now it’ll be an hour and ten minutes. Roughly. The quality of this approximation gets worse the bigger the change is. Cutting the speed 50 percent increases the travel time rather more than 50 percent. But for small changes, we have it easier.

There are a couple ways to look at this. One is as an infinite series. Suppose you’re travelling a distance ‘d’, and had been doing it at the speed ‘v’, but now you have to decelerate by a small amount, ‘s’. Then this is something true about your travel time ‘t’, and I ask you to take my word for it because it has been a very long week and I haven’t the strength to argue the proposition:

t = \frac{d}{v - s} = \frac{d}{v}\left(1 + \left(\frac{s}{v}\right) + \left(\frac{s}{v}\right)^2 + \left(\frac{s}{v}\right)^3 + \left(\frac{s}{v}\right)^4 + \left(\frac{s}{v}\right)^5 + \cdots \right)

‘d’ divided by ‘v’ is how long your travel took at the original speed. And, now, \left(\frac{s}{v}\right) — the fraction of how much you’ve changed your speed — is, by assumption, small. The speed only changed a little bit. So \left(\frac{s}{v}\right)^2 is tiny. And \left(\frac{s}{v}\right)^3 is impossibly tiny. And \left(\frac{s}{v}\right)^4 is ridiculously tiny. You make an error in dropping these \left(\frac{s}{v}\right) squared and cubed and forth-power and higher terms. But you don’t make much of one, not if s is small enough compared to v. And that means your estimate of the new travel time is:

\frac{d}{v} \left(1 + \frac{s}{v}\right)

Or, that is, if you reduce the speed by (say) five percent of what you started with, you increase the travel time by five percent. Varying one important quantity by a small amount we know as “perturbations”. Working out the approximate change in one quantity based on a perturbation is a key part of a lot of calculus, and a lot of mathematical modeling. It can feel illicit; after a lifetime of learning how mathematics is precise and exact, it’s hard to deliberately throw away stuff you know is not zero. It gets you to good places, though, and fast.

Wellington: 'First our teacher says 25 plus 25 equals 50. Then she says 30 and 20 equals 50. Then she says 10 and 40 equals 50. Finally she says 15 and 35 equals 50. Shouldn't we have a teacher who can make up her mind?'
Morrie Turner’s Wee Pals rerun for the 21st of July, 2018. Originally ran the 22nd of July, 2013.

Morrie Turner’s Wee Pals for the 21st shows Wellington having trouble with partitions. We can divide any counting number up into the sum of other counting numbers in, usually, many ways. I can kind of see his point; there is something strange that we can express a single idea in so many different-looking ways. I’m not sure how to get Wellington where he needs to be. I suspect that some examples with dimes, quarters, and nickels would help.

And this is marginal but the “Soul Circle” personal profile for the 20th of July — rerun from the 20th of July, 2013 — was about Dr Cecil T Draper, a mathematics professor.


You can get to this and more Reading the Comics posts at this link. Other essays mentioning Dinosaur Comics are at this link. Essays that describe Nancy, vintage and modern, are at this link. Wrong Hands gets discussed in essays on this link. Other Ziggy-based essays are at this link. The Duplex will get mentioned in essays at this link if any other examples of the strip get tagged here. And other Wee Pals strips get reviewed at this link.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

4 thoughts on “Reading the Comics, July 21, 2018: Infinite Hotels Edition”

  1. This is, in fact, my favorite context for rational functions. a) helps learners get some of the illogic of speeding, b) combines with different speed limits to make reasonably complex functions. If you speed x miles/hour over the speed limit…

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    1. That’s a good context, yeah. I had at one point thoughts to make a more specific post about this kind of variations problem, but never had the chance to put it together. I’m glad a comic strip gave me the push to actually write it; it’s a great deal easier doing this sort of thing when something else tells me what to write.

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