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December 3, 2015

42

Posted by John Baez

In The Hitchhiker’s Guide to the Galaxy, the number 42 was revealed to be the Answer to the Ultimate Question of Life, the Universe, and Everything. But we never learned what the question was!

That’s what I’ll explain this Saturday in Montreal.

My talk is part of the Canadian Mathematical Society winter meeting. But it’s open to the public, and I’ll keep the math simple and fun. So, come along and bring your kids!

It’s from 6 to 7 pm on December 5th at the Hyatt Regency Montreal, at 1255 Jeanne-Mance. It’s in “Rooms Soprano A & B”, but I won’t be singing — they must have confused me with my cousin Joan.

If you can’t come to the talk, you can still see the slides here:

If you don’t already know the joke, watch the movie clip first!

Earlier that day I’ll be at Prakash Panangaden’s session on Logic, Category Theory and Computation, and I’ll give a talk there at 10 am, right after André Joyal’s hour-long talk on “Simplicial Tribes for Homotopy Type Theory”. His abstract says “A tribe is a categorical model of homotopy type theory. We would like to show that the category of tribes has the structure of a fibration category, but path objects are missing. To correct this, we introduce the notion of simplicial tribes.”

This session is in Pavillion Kennedy of UQAM on the third floor in room PK-3605. My talk will be on categories in control theory, and the slides are here:

Posted at December 3, 2015 4:24 AM UTC

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15 Comments & 0 Trackbacks

Re: 42

Here is something that could be related, 42^2 + 56^2 = 70^2 ?

Posted by: Mark Thomas on December 7, 2015 9:00 PM | Permalink | Reply to this

Re: 42

At the end of the second book, and also the TV show, Arthur Dent and Ford Prefect are stranded on prehistoric Earth, and they realize the Ultimate Question is inside Arthur’s head. Anyway, they determine that the “Ultimate Question” is “5 x 6”. The joke is that it doesn’t match the answer.

Posted by: Jeffery Winkler on December 8, 2015 7:08 PM | Permalink | Reply to this

Re: 42

Can’t recall where this is from, but if you remember the C preprocessor,

#include <stdio.h>
#define SIX 1 + 5
#define NINE 8 + 1

int main() {
  printf("Six times Nine is %d\n", SIX * NINE) ;
  return 0 ; 
}
Posted by: Jesse C. McKeown on December 8, 2015 8:19 PM | Permalink | Reply to this

Re: 42

Jeffrey wrote:

Anyway, they determine that the “Ultimate Question” is “5 x 6”.

Elsewhere I read:

At the end of the radio series, the television series and the novel The Restaurant at the End of the Universe, Arthur Dent, having escaped the Earth’s destruction, potentially has some of the computational matrix in his brain. He attempts to discover The Ultimate Question by extracting it from his brainwave patterns, as abusively suggested by Ford Prefect, when a Scrabble-playing caveman spells out forty two. Arthur pulls random letters from a bag, but only gets the sentence “What do you get if you multiply six by nine?”

“Six by nine. Forty two.”

“That’s it. That’s all there is.”

“I always thought something was fundamentally wrong with the universe”.

Six times nine is, of course, fifty-four. The answer is deliberately wrong. The program on the “Earth computer” should have run correctly, but the unexpected arrival of the Golgafrinchans on prehistoric Earth caused input errors into the system—computing (because of the garbage in, garbage out rule) the wrong question—the question in Arthur’s subconscious being invalid all along.

Quoting Fit the Seventh of the radio series, on Christmas Eve, 1978:

Narrator: There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable.

There is another theory, which states that this has already happened.

Some readers who were trying to find a deeper meaning in the passage soon noticed that 6 × 9 is actually 42 in base 13 (as 4 × 13 + 2 = 54, i.e. 54 in decimal is equal to 42 expressed in base 13). When confronted with this, the author claimed that it was a mere coincidence, famously stating that “I may be a sorry case, but I don’t write jokes in base 13.”

Posted by: John Baez on December 8, 2015 8:34 PM | Permalink | Reply to this

Re: 42

Yeah, you’re right! I was trying to remember something I read over 30 years ago!

Posted by: Jeffery Winkler on December 9, 2015 7:01 PM | Permalink | Reply to this

Re: 42

My talk went pretty well; the room was packed, standing room only, and a lot of people said they liked it — even my wife! My main mistake was not giving any explanation of why the positive integer solutions of

1a+1b+1c=12 \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{2}

give ways for 3 regular polygons to meet snugly at a vertex. I didn’t want to get into this so near the start, but apparently some audience members were left scratching their heads for minutes afterwards.

After the talk there were questions. Since the announcer had mentioned some other talks on ‘my favorite numbers’, André Joyal asked me what’s the smallest number I consider uninteresting.

That’s a reference to an old joke. After explaining the joke, I decided to take it seriously, and guessed 14.

Afterwards I regretted this. 14 is a Catalan number, the number of faces of the 3d associahedron. As someone reminded me, it’s also the dimension of the group G 2G_2. And someone else pointed out the Kuratowski’s closure-complement problem: 14 is the maximum number of different subsets of a topological space that can be generated by repeatedly taking closures and/or complements.

(This must be something like the free lattice of some sort on one generator. What sort, exactly?)

So I switched to 15, but then someone pointed out Conway–Schneeberger Fifteen Theorem: if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers.

After this, we agreed that 13 would be a good choice for the first mathematically uninteresting number — despite its cultural significance.

True, there are 13 Archimedean solids. But does this make the number 13 interesting?

Posted by: John Baez on December 9, 2015 7:28 PM | Permalink | Reply to this

Re: 42

As the Wikipedia article suggests, the 14 of the Kuratowski closure-complement problem doesn’t have much to do with topology; it has more to do with the Moore closure operators.

That is to say: if T:P(X)→P(X)T: P(X) \to P(X) is any closure operator (order-preserving map on a power set P(X)P(X) such that S≤T(S)S \leq T(S) and TT(S)=T(S)T T(S) = T(S) for all S∈P(X)S \in P(X)), then there are at most 14 distinct sets that can be generated by starting with an SS and applying TT and complementation ¬\neg. One can show formally that T¬T¬T¬T=T¬TT \neg T \neg T \neg T = T \neg T, so that the monoid of operations generated by TT and ¬\neg contains no elements other than

1,¬,T,¬T,T¬,¬T¬,T¬T,1, \qquad \neg, \qquad T, \qquad \neg T, \qquad T \neg, \qquad \neg T \neg, \qquad T \neg T,

\,

¬T¬T,T¬T¬,¬T¬T¬,T¬T¬T,¬T¬T¬T,T¬T¬T¬,¬T¬T¬T¬\neg T \neg T, \qquad T \neg T \neg, \qquad \neg T \neg T \neg, \qquad T \neg T \neg T, \qquad \neg T \neg T \neg T, \qquad T \neg T \neg T \neg, \qquad \neg T \neg T \neg T \neg

of which there are 14.

More interesting from an actual topological perspective is that generically there are 13 (ha!) subsets that can be created by starting with a subset AA and closing up under closures, interiors, and unions. The nLab mentions a few more details on this and gives references.

Posted by: Todd Trimble on December 10, 2015 5:46 AM | Permalink | Reply to this

Re: 42

Thanks! The nLab material, I imagine written largely by you, is very helpful.

If the 13 operations you mention could be put in some nice bijective correspondence with the 13 Archimedean solids, then I’d say the number 13 was interesting. But I don’t sense any magic of this sort in the air.

Posted by: John Baez on December 10, 2015 7:17 AM | Permalink | Reply to this

Re: 42

My initial vote for a non-interesting number is 10. The usual reason for it to be at all worth mentioning is that most of us have that many fingers. I’m sure there must be a better reason to like ten on it’s own merits, and I can’t think of that reason mostly because of all the base-ten-related facts that crowd in. What am I forgetting?

Posted by: stefan on December 11, 2015 9:52 PM | Permalink | Reply to this

Re: 42

Superstring theory lives in 10 dimensions, and while it gains much of its strength from the number 8, the magic of the number 8 rubs off onto the number 10, basically because the ambient 10-dimensional spacetime should be understood as the 2 dimensions of the string worldsheet plus the 8 transverse dimensions.

In particular, 10-dimensional Minkowski spacetime is the highest-dimensional one that can be identified with the space of 2×22 \times 2 self-adjoint matrices valued in a normed division algebra. In 10 dimensions this normed division algebra is the octonions, 𝕆\mathbb{O}! The space of 2×22 \times 2-self adjoint octonionic matrices is a Jordan algebra, but as a mere vector space it’s isomorphic to 𝕆⊕ℝ 2\mathbb{O} \oplus \mathbb{R}^2, and that exhibits the 10 = 8 + 2 splitting.

Only in dimensions 8n+28n+2 do Minkowski spacetimes admit Majorana-Weyl spinors: that is, spin-1/2 particles that spin only clockwise and are their own antiparticle.

As a spinoff of all these facts, the space of Majorana-Weyl spinors in 10d Minkowski spacetime can be identified with 𝕆 2\mathbb{O}^2.

As a further spinoff, there is a mystical isomorphism between the double cover of the Lorentz group in 10 dimensions and SL(2,𝕆)SL(2,\mathbb{O}). It’s ‘mystical’ because it takes some work to define this group, 𝕆\mathbb{O} being nonassociative.

As a further spinoff, we get superstring theory.

And as yet another spinoff, we get the 10-dimensional lattice E 10\mathrm{E}_{10}, and a nice octonionic description of this lattice. This is an even unimodular Lorentzian lattice; such things exist only in dimensions of the form 8n+28n + 2. It’s easy to describe them directly, but in 10 dimensions there’s also a nice description in terms of the Cayley integral octonions. This helps us understand the discrete subgroup of the Lorentz group that preserves this lattice: the so-called E 10\mathrm{E}_{10} Weyl group.

So, personally, I think the number 10 is fascinating.

Posted by: John Baez on December 11, 2015 10:12 PM | Permalink | Reply to this

Re: 42

1010 is also the number of ways to partition 66 objects into two sets of 33, and therefore enters into various exceptional behaviours of S 6S_6 and A 6A_6. For instance, the partitions correspond to the 1010 pairs of mutually polar anisotropic lines in the symplectic projective 33-space over 𝔽 2\mathbb{F}_2 (via S 6≃PSp(4,2)S_6\simeq PSp(4,2)), and the number of points in the projective line over 𝔽 9\mathbb{F}_9 (via A 6≃PSL(2,9)A_6\simeq PSL(2,9)). The first isomorphism is ultimately related to the Weyl group of E 8E_8 and the second to the Weyl group of E 6E_6. So 1010 is an interesting number in the context of finite Lie theory too.

Posted by: Tim Silverman on December 12, 2015 12:23 PM | Permalink | Reply to this

Re: 42

Fascinating indeed! That’s exactly what I was hoping for! It was lurking in my mind that 10 played a role in superstring theory, but not looking it up I was stuck recalling either 10+1 or 26 as the more important dimensions…and the fact that 10 = dimension of 2x2 self-adjoint matrices over the octonions will definitely stick in my memory better.

So I wonder if there is anything in the study of base ten representation of integers that informs or is informed by this algebraic perspective? For instance, could we use any facts about the normed division algebras to attack the open questions of multiplicative persistence?

In base ten it is unknown if there are numbers for which it takes an arbitrarily large number of steps of [multiply all the digits together] in order to reach a single digit. Here’s the OEIS sequence showing what is known.

Of course the physical applications are more interesting! But that’s what I was getting at–I’ve always been a little bit bothered by the attention given to base 10 just because of historical significance. Thanks for the great description of its inherent importance!

Posted by: stefan on December 12, 2015 3:36 PM | Permalink | Reply to this

Re: 42

I don’t know any way that interesting facts about the number 10 are related to properties of base 10.

People have remarked that the 10 dimensions of superstring theory and the 26 dimensions of bosonic string theory are reminiscent of the 10 digits and 26 letters of the alphabet — but this is nothing more than a very nice joke.

The Pythagoreans tried hard to explain why 10 was great, apparently neglecting the fact that we happen to have 10 fingers. Aetius wrote:

Ten is the very nature of number. All Greeks and all barbarians alike count up to ten, and having reached ten revert again to the unity. And again, Pythagoras maintains, the power of the number 10 lies in the number 4, the tetrad. This is the reason: If one starts at the unit (1) and adds the successive number up to 4, one will make up the number 10 (1 + 2 + 3 + 4 = 10).

The Pythagoreans were fond of the tetractys, namely this thing:

If you click the link, you’ll see they associated the rows of the tetractys to the 0-simplex, the 1-simplex, the 2-simplex and the 3-simplex. So, to them, the 3 dimensions of space were connected to the 10 dots of the tetractys.

Posted by: John Baez on December 12, 2015 5:14 PM | Permalink | Reply to this

Re: 42

Nice. Triangular numbers are rather fundamental in several ways, and I hadn’t known this bit of their history. I suppose studying base-10 properties of numbers can also be justified as a starter step in studying base-p properties, too.

Posted by: stefan on December 13, 2015 3:24 PM | Permalink | Reply to this

Re: 42

Note that j(τ) = 1/q + 744 + 196884 * q + 21493760 * q^2 + 864299970 * q^3 + … From Wolfram Alpha: Euler’s constant = .5772156649….

log(744) / log(pi) = 5.77607095 approx.

(mass top quark)/(mass electron) = 340901 approx. 196884/340901 = .57754010695… approx.

Posted by: David Brown on December 11, 2015 8:20 PM | Permalink | Reply to this

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