jwgh | Annoying science fiction device detected

I started reading John Brunner's Meeting at Infinity. Early on in the book we encounter that hoary old bit of science fiction nonsense, the inconstant π:

Pi, it seemed, was invariant. However, certain deductions from curved-space mathematics indicated conditions under with it would assume values different from the familiar 3.1416. It would remain an irrational number of course. But the physical conditions for altering its value could be described.

So the guy creates a machine that creates an area where the value of π is different, and this turns out to be a way to reach alternative universes whose history differs from Earth's proportionately to the difference between its value of π and ours.

I think that this sort of thing comes from a misunderstanding of noneuclidean geometry (or, I suppose, a desire to annoy mathematicians). There are geometries where if you measure the circumference of a circle and divide it by the diameter you'll get a number other than 3.14159265... This doesn't change the value of the number π, though, and it has nothing to do with π's irrationality. It makes about as much sense to posit a universe where 2 has a different value.

Anyway!

Flat | Top-Level Comments Only

In _Eon_ by an author I can't recall (Greg Bear? Dunno) some of the characters enter The Way, an apparently infinite passage that extends into another dimension, or something, from inside an Earth-orbiting asteroid. At one point, I recall that the characters encountered a sort of dimple in their path, and, entering it, found that their instruments were reporting a different value for pi. Okay, we get it, as if an infinited tunnel extending out of a finite asteroid isn't odd enough, something's deforming space or something. Here's my question:

I don't know about the unanticipated transdimension journeys you go on, but what're the odds anyone bothered to invent, much less pack along, a pi-o-meter? I'm thinking we've got the tricorder syndrome.

The exploration party paused to get its bearings, all feeling strangely uneasy, though none would admit it. The navigator pulled out his pocket compass, looked at it, and gasped.

"What is it, Joe?" asked the captain.

"The compass ... look at the dial!" came the response.

The captain examined the compass. "I don't see anything --" he began, then suddenly stopped. The circular dial's circumference was quite obviously almost four times its diameter.

"Dr. Haller! This compass now has over 400 degrees of bearing!" (Each degree divided up into minutes and seconds of unusual angular width, of course.)

I can't think of one offhand, but it'd be better if they could come up with a reason why certain human biological or neurologic functions wouldn't work or would work unpredictably different in curved space, so the red-shirt that happened to stumble into the curved space would collapse and die dramatically.

Or less dramatically, but more scientifically interesting, is a region where the flora and fauna have a different golden ratio, or different fractal patterns that don't exist where pi (etc.) is different. "I've just learned the most amazing things about your samples, Dr. Haller!" Naturally, those creatures would mysteriously die as soon as they were removed from their 22/7ths environment, or at least begin to do so. And something about the 22/7ths universe could save all mankind, ...

Um, anyone else feel a Scientifiction Playhouse episode coming on?

Some of this might actually be in the Brunner book I'm reading. I'll let you know!

Ideally the Scientifiction Playhouse would make a reference to the π=24/7 troll, and maybe the old Douglas Hofstadter chestnut 'If π equalled 3, this sentence w

uld l

k like this." [The 'o's are supposed to be replaced with hexagons, because the ratio of a hexagon's perimeter to its diameter is 3.] Or maybe not.

I'm not sure how the anomaly is encountered, though. Maybe by miners tunneling deep into the Earth, affected by the increased gravitational field there?

My first instinct was to go with the traditional unexplored remote island, but yeah, a mine would make more sense. And you're right abotu the 22/7, 24/7 thingie, and I never made it through any Hofstadler (please do not revoke my geek credentials, I'm sure I have a copy of G.E.B. around here somewhere), but I can work that in, I think.

What I was thinking was that maybe the minecarts (or whatever objects with wheels you would have on a remote desert island) would suddenly stop moving, because their wheels suddenly turned hexagonal due to the changing value of pi.

Unfortunately UNSENET isn't really a good medium for typing sentences with hexagons instead of os. I was shocked that there doesn't appear to be a Unicode hexagon!

ISAGN!

Hmmmm, for some reason changing the golden section makes my brain hurt much faster than changing pi. It must have to do with the way the golden section shows up even when all you're talking about is adding whole numbers (as in Fibonacci series). I guess it's easier for me to deal with the thought of impossible twists lurking in a big, squishy, multidimensional space than in a few poor little discrete integers.

This also reminds me of a crackpot paper posted down in the physics labs about Planck's constant varying with wavelength that I remember reading, and realizing after poking through the guy's equations that there was a much simpler way to state his premise that he somehow never got around to saying: Photons have rest mass. That would certainly be a tidy explanation for why the sky is dark, wouldn't it

Well, the thing is that you can define π in ways that don't have anything (at least on the surface) to do with geometry. For instance, the normal (Gaussian) distribution is:

P(x)dx = (1/srt(2π)) e^(-(z^2)/2)) dz

and you could use that define π if you really wanted to.

(Now that I think of it, I think that the book about the guy who finds out he's God also has him change the normal distribution at one point, with the result that people start suffocating when all of the air in a room suddenly collects in one corner, etc.)

Alternatively (if somewhat artificially) you could define pi as follows:

π = 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ...

You can also take the route of defining the sine function from looking at e^x for complex x, then finding its period (2π), which again has nothing obvious to do with geometry. (If some alien civilization did this, I imagine that some mathematician might have been surprised to discover that half the period of the sine function turns out to be exactly the ratio of a circle's circumference to its diameter!)

It's unclear to me whether these things are also supposed to change when people talk about changing π. I would expect not and that what's being talked about is the ratio of a circle's circumference to its diameter. Things like Brunner's mentioning that even if π has a different value it has to be irrational make it clear to me, though, that in at least some cases there's some confusion about what's what.

I'm interested in reading that Egan story.

I vaguely remember another novel where a character gradually comes to realize that he's God. At one point he makes π be a rational number (there's a comment to the effect that it's sort of a relief for God to stop having to come up with more and more digits) and scientists and/or mathematicians are able to figure out that that's happened somehow -- maybe computers calculating π suddenly started coming up with this different number? (I think it would have been neat if e^πi suddenly stopped equaling -1 or something).

At the end of the book the Universe is destroyed and God creates a new one where planets follow square orbits.

Does this ring a bell for anyone?

Also, in a curved space the ratio of circumference to diameter depends on the size of the circle. In the limit of a small circle it converges to pi.

One case I can think of where that doesn't happen is in a space with a conical singularity, like a cosmic string. Then it just depends on whether the singularity is inside the circle.

A gee-whiz science book I read ages ago had a bit about how a fly landing in a circle the size of Trafalgar square or something exerts enough gravity to change the 30-somethingth decimal place of pi.

I'm also reminded of Greg Egan's story "Luminous", about some mathematicians who discover a 'defect' in mathematics, away out in the realm of very very large numbers, and find ways to manipulate the border between standard and nonstandard mathematics, and of course soon discover that there are concerned parties on the other side...

Annoying science fiction device detected

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Unicode Hexagon

different golden ratio

Re: different golden ratio

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um, spoiler warning