Wednesday, December 26, 2007

Solution to the World's Hardest Easy Geometry Problem

A few weeks ago, someone posted this problem to a project mailing list at Google. For a few days afterward, every office I walked into had a white-board covered with elaborate color-coded triangles. By my rough estimate, various proofs that x = 20 probably cost Google on the order of a quarter of a million dollars in lost productivity over the next few days. It would obviously be Evil to allow this scourge to spread, so here's my solution.

First, we need some auxiliary lines and points. Draw a line through D parallel to AB, and let F be the intersection with BC. Now draw line AF. (It's just a mirror image of BD.) Let G be the intersection between AF and BD. Finally, draw line CG. It's easy to see that the line CG bisects the angle ACB, so ACG and GCB are both 10 degrees.

It's helpful to keep the endgame in mind while working toward it. Just from the fact that the angles of a triangle add up to 180 degrees, We know that angle AEB is 30 degrees. If we knew angle DEF, we could easily compute x: DEF - AEB. So, how to find angle DEF? First we need to establish three more-or-less unrelated equations between segment lengths:
  1. DF = FG. Angles ABG and BAG are both 60 degrees, so ABG is an equilateral triangle. Since angle DGF = AGB, angle DGF is also 60 degrees. By symmetry, angles FDG and DFG are equal, so they must be 60 degrees, too, and DFG must be equilateral. So the sides are equal.
  2. CF = AF. Since angles FCA and FAC are both 20 degrees, ACF is isosceles, which means that CF = AF.
  3. CE = AG. This is the trickiest one. It hinges on the fact that triangles ACG and ACE have equal sides. That's a little surprising, but true: both have angles 10, 20, and 150, and they have a common side AC. So they're congruent and have a common side, so they're equal. That means in particular that CE = AG.
Now subtract equation 3 from equation 2:
  • CF - CE = AF - AG
  • EF = FG
The rest is easy. By equation 1, FG = DF, so EF = DF, which means that triangle DEF is isosceles with DE as the base. We know that angle DFE is 80 degrees, so both DEF and EDF are 50 degrees. Finally, x = DEF - AEF = 50 - 30 = 20 degrees.

Sunday, December 23, 2007

Yet another solution to Fermi's Paradox

If intelligent civilizations are common in the universe, where are they? Even assuming technology only a little better than Earth's, it would take less than 100 million years to populate the galaxy. Why haven't they come to visit? There have been a lot of ideas over the years. Last night I thought of yet another solution, and maybe an idea for a SF galactic civilization that doesn't require a warp drive or worm holes.

Suppose that our current understanding of physics is basically correct, and there isn't some exotic trick that allows us to exceed the speed of light. And suppose you want to design a galactic society, where it's practical for humans to range over the Milky Way and make occasional visits to Andromeda and other nearby galaxies, all within reasonable durations of subjective time. How do you do it?

First, it isn't impossible to travel galactic distances even in an unextended human lifetime. You just have to accelerate to relativistic speeds. This idea is a staple of science fiction; a great example is Joe Haldeman's The Forever War. The problem with traveling at relativistic speeds is the famous twin paradox: the traveler is fine, but when she returns home, she finds that her friends are all long dead, or in extreme cases, that the Sun has gone nova.

However, there's a way to avoid the twin paradox: have everyone travel at relativistic speed so they stay synchronized. Here's how it works. Once every 10 years (subjective time) the entire human race, which is spread all over the Milky Way, climbs into space ships, which take off and accelerate at 2 g long enough to reach nearly the speed of light (roughly a year). Then they all decelerate to their destinations, hang out for a year, and do it over again. Since they're all doing it simultaneously, they stay synchronized. They spend approximately equal subjective time traveling and staying put. Assuming that we manage to extend human lifetime to (say) a few thousand years, humans could range over the entire galaxy without breaking any physical laws or losing track of their friends and relatives.

Take this a step further. After some time, humans encounter evidence of another, larger and more advanced, galactic civilization doing the same thing. They figure out from available evidence the next synchronization point of the advanced civilization, and arrange to synchronize with it. They find that they are just the latest of thousands of civilizations to join. Possibly they find that there is an ongoing debate as to whether to join an even more far-ranging civilization that synchronizes even more infrequently and spans multiple galactic clusters. Imagine thousands of individual civilizations stuttering into the future, touching down only every few thousand years and leaving scarcely a trace,. Occasionally, tributary cultures join larger streams, populating ever more vast spans of time, but also populating time ever more sparsely.

The effect of all this is that the universe is densely populated with a huge, galaxy-spanning civilizations --- but only for brief, rare intervals. And all of human civilization has fallen into a gap in their timeline.

Saturday, December 15, 2007

Best Science Blogs

What's the best science blog? According to the 2007 Weblog Awards, it's a tie between Phil Plait's Bad Astronomy and Steve McIntyre's Climate Audit. Over 50,000 votes were cast. When the voting officially closed, Climate Audit held a narrow lead, and Steve McIntyre's partisans began to celebrate. However, votes continued inexplicably to pour in, and eventually Phil Plait's Bad Astronomy surged into the lead. Well, actually it was quite explicable: there was a lot of cheating going on. Rather than spend days trying to sort things out, which was probably impossible anyway, Steve and Phil agreed to call it a draw.

So, voting aside, which blog is actually better? I've been reading both for quite a while. Bad Astronomy is pretty light fare. In his Intro, Phil writes
The Bad Astronomy web pages are devoted to airing out myths and misconceptions in astronomy and related topics.
He spends a lot of time on people who believe that the moon landing was a hoax, creationism, and astrology. I guess someone needs to point out this nonsense, but I find it a bit tedious after a while. Still, it's a quick read, and he intersperses it with interesting pictures of wildlife in his back yard, silly videos, etc. There's really nothing wrong with Bad Astronomy, but how it came to be a contender for Best Science Blog I have no idea.

On the other hand, Climate Audit is an amazing blog. It isn't a blog about science so much as actual science brought to you in real time. The mission of Steve's blog is to audit climate science: dig up the data, try to replicate the computations, look for mistakes, look for bad assumptions. A typical thread will start with Steve posting a few paragraphs about his attempts to replicate a calculation, and asking for help. What follows is astonishing: dozens of experts in various disciplines chime in with their ideas. Many of the big names in climate science drop in from time to time. The result is fascinating and often surprising.

I'll warn you that the going is often pretty heavy: I have a B.S. in math and a PhD in Computer Science, and some of it is not easy reading for me. But watching scientific theories being hypothesized, tested, critiqued, and modified in real time is surprisingly addictive, at least for me.

I don't know if Climate Audit is the best science blog on the web, but it's far more interesting than Bad Astronomy.