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## A mathematical puzzle (of sorts)

I happened across the following mathematical titbit, which I will set as a question to the reader. Let,

$\mu = (\sqrt{5} + 2)^{1/3}-(\sqrt{5} - 2)^{1/3}$      (1)

You are required to prove that

$\mu = 1$.

Striking, no? Obviously, sticking it into a calculator doesn’t count. I know a rather indirect way, which unfortunately involves the phrase “by inspection”. I’ll share it with you below the fold. There must be a nicer way! (more…)

## Book Review: Extreme Cosmos by Bryan Gaensler

The short version: read this book!

I’ve read quite a few astronomy books in my time, and this is one of the best. The problem with a lot of these books is that, once you’ve read one or two, they start covering the same ground. A novel example or illustration is nice, but you can read Fred Hoyle’s The Nature of the Universe from 1960 (review soon!) and get most of what we know about the lives of stars and the layout of the solar system. The most media-friendly breakthroughs have come in cosmology, which has gained more than its fair share of popular level books on dark energy, dark matter, multiverses and the like.

However, many of the major discoveries of the last few decades have been in fields like high-energy astrophysics, hypervelocity stars, supernovae, black holes, magnetars and the like. Bryan Gaensler gives an outstanding overview of these extreme objects.

A good example is his description of what it would be like to be inside a giant molecular cloud [pg 25]:

“Let’s imagine that one for these [molecular] clouds drifted through our part of the Milky Way, enveloping the Earth, Sun and the rest of the solar system. In the direction from which the cloud approached, there would be a growing inky dark patch, eventually blotting out all the starlight from half the sky. But looking in the other direction, out to free space, we wouldn’t notice any difference at all at first. The stars in that direction would seem just as bright as always.

After about 2000 years (by which point we would have penetrated around 20% of the way into the centre of the cloud), the half of the sky towards the cloud would remain totally black, but now the other half two would have started to fade. Over the centuries, the light from the various stars and constellations would have dimmed by about a factor of six – only about 150 stars would still be bright enough to be visible to the naked eye.

Wait another 2000 years, and the remaining half of the night sky would fade by a factor of 20, leaving only ten stars that we could see unaided. And if 2000 years passed once more (a total of 6000 years since our encounter with the cloud began), there would be no stars left at all visible with the unaided eye.”

This puts me in mind of a quote from Ralph Waldo Emerson: (more…)