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Neither have I.

Argh. For no reason that I can discern, I can no longer connect to the internet through my laptop's wireless network. Pretty frustrating.

I just read "A Sound of Thunder".

This is the Bradbury short story where people pay to go back in time to shoot dinosaurs. One guy freaks out and accidentally steps on a butterfly. When they return to their normal time, everything is subtly different, and a dictator type just won the presidential election.

I dunno. This story seems kind of dumb and smart at the same time.

As far as being smart, it was fairly original and profound to suggest the way the trivial in complex situations can have unpredictable results.

As for being dumb, it just seems stupid that the way these people who have time travel would decide to make money off it, by having them go back in time and hunt dinosaurs. At one point, a character says something like they're going back before Moses.

Well, wouldn't it have been pretty interesting to have gone back in time and find out the true story of Moses? Or to see the Titanic sink? Or to see Julius Caesar command a battle? Or to see the construction of stone hedge? Or the evolution of man?

The professionals in the story were careful to avoid a time paradox, and it was only by the screw up of a client that one occurred. But by the ridiculous lengths they had to go through, it should have been apparent one was inevitable.

What about for my scenarios? These guys don't have to see Moses, or the first man, first hand. They could be over a mountain with a telescope. They could hover above the Titanic in a helicopter. If they're still worried about being seen, they could launch a satellite with a high resolution camera.

It was obviously done for the sake of making a point, but it was sort of funny that killing the butterfly resulted in huge and unpredictable changes to the future ... except that it just resulted in one thing in their present going the opposite way it could have.

If you would imagine something eventually big happening in our past, say, Russia joining the axis powers, who subsequently win World War 2, you probably wouldn't then be able to have, say, Kerry win the 2004 Presidential election.

Once the changes start becoming big, then future events really should become not only different, but downright unpredictable. To suggest otherwise almost contradicts the apparent point of the story.

The other way of looking at is is that by their present, it just so happens that by pure coincidence, the changes have finally snowballed into something big.

I found and read a translation of Euler's original paper on the sums of reciprocal powers.

Some of the middle of it was hard to follow directly, because of the absence of modern notation, but this is what I gather it amounts to.

Most descriptions I've seen of how he solved the problem involved factoring the polynomial sin(x) or cos(x) (or sin(pi * x) / (pi * x)). In the paper, it's more like he's dealing with

-sqrt(2) * sin(pi * (x-1)/4)

This allows him to not only sum reciprocal even powers, but also gives

pi / 4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11....

Euler says that this should confirm that result for anyone who doubted it! I guess in the days before computers, the exactness of that result could have been more mysterious.

Alternatively, it also could have confirmed to him that this method works.

It also gives the other alternating sums of other odd powers, like

1/1^3 - 1/3^3 + 1/5^3 - 1/7^3 + ... = pi^3 / 32


Nowadays, another way to do problems like this is with Fourier series.