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So, have been working long hours and scratching my head a lot the last three days over a tricksy little problem.* After several false starts I finally figured out the right way to approach it, did something sneaky with inventing people who don't exist and then ignoring them**, and got a mucking big ugly-looking expression.

Hey, I can cancel those bits...
...and those bits...
...and those bits too...
...and everything else, leaving zero.

I'd never realised it was possible to be pleased and disgruntled at the same time.

*Tricksy to me, anyway. I'm fairly sure there's a standard result, but I'm a reinvent-the-wheel sort of guy.
**This actually made some sort of sense at the time.
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Following on from yesterday's post, I thought I might explain a bit about the problem Russell discovered with set theory.

Russell's Paradox, with tentacles. )

1+1=2

Sep. 16th, 2007 03:14 pm
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[livejournal.com profile] ambitious_wench posted a link to a scan of part of Russell's proof that 1+1=2, and asked maths geeks to discuss why it was beautiful.

First off, I have to admit a couple of things: I'm not familiar enough with Bertrand's work to follow how the proof works, and I don't know that I'd call it beautiful. There are more than 300 pages of mathematical setup involved before he even gets to that point, and the proof isn't actually completed until eighty-something pages into the next volume of Principia Mathematica. Combined with aforementioned lack of familiarity on my part, that makes it rather too unwieldy to engage my sense of mathematical beauty. But I might be able to give at least some idea of why it's important.

And I will do it in verse. Or, failing that, a parable. )
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Unless you've been living under a rock, you've probably encountered the "men have more sexual partners than women" paradox at some stage. If you haven't, the gist of it is this: surveys that ask men and women about how many opposite-sex partners they've had in their lifetime generally find that the typical man has almost twice as many as the typical woman. (See this NY Times article for some examples.)

What is going on here? If our population is roughly 50/50 male/female, a naive look at the situation tells us that the numbers should be equal; any time a new couple forms up, both those figures should go up by the same amount, right?

There are several factors that contribute here. Some of them are well-known and get trotted out every time this sort of thing comes up, but there's a sneaky one that hardly ever gets a mention and really should.

The standard ones: sampling error, reporting error, mean vs median. )

The sneaky one. )

Puzzle!

May. 21st, 2007 02:19 am
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Via [livejournal.com profile] hasimir and various others: a self-referential multiple-choice test! Your challenge is to fill it out so that all 20 answers are correct at the same time.

BTW, with every answer you click it updates to show which are right and which are wrong. This refers to the current status of the other questions, not the final correct answers. For instance, question 3 asks how many questions have answer E; if you answer C (i.e. two 'E's), this will show up as correct when you have exactly two 'E's ticked on the test and incorrect otherwise.

And yes, it is solvable with a little bit of pen-and-paper work.

Bikes!

May. 4th, 2007 12:24 am
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I've been meaning to post about this one for a while...

When I was a teenager, I worked as an 'Explainer' at Questacon. One of the science demos I was trained to do was a bit on gyroscopic precession: if you take a bicycle wheel, spin it, and then try to tilt its axis while it's spinning, it will pull around. And we were taught (and duly taught our visitors) that Gyroscopic Precession Is What Keeps Bicycles From Falling Over. It seemed sensible enough - after all, it's much harder to balance a bike when standing still than when it's moving and the wheels are spinning. And gyroscopic effects are what stabilises a number of other things - spinning tops, for instance. Indeed, if you detach a wheel from your bike and roll it downhill on its own, gyroscopic effects due to its rotation are what keeps it upright instead of falling over on its side.

Many, many reputable physicists tell us that it's the same thing with a bicycle, or a motorbike. Straight Dope, for instance, repeats the claim. But it's hard to argue with this:


http://www.losethetrainingwheels.org/default.aspx?Lev=2&ID=34

This wonder, created by Dr. Richard Klein of UIUC, is a zero-gyroscopic bike. The two airborne wheels are in contact with the two on the ground, so they rotate at the same speed but in opposite directions, cancelling out any gyroscopic effects. Despite this, the bike is still quite easy to ride. Before Klein, English researcher David Jones had published similar findings, and his article on his fruitless attempts to build an unrideable bike is quite entertainingly written.

So why do bikes stay upright? Wikipedia has a more detailed discussion, but it comes down to forward movement. When your bike starts to fall over to the left, steering slightly to the left brings the base of the wheels back under the center of gravity, restoring stability. Most bikes will steer into the direction of the tilt even without the rider's intervention or gyroscopic effects, due to their 'trail'; you can test this for yourself by tilting a stationary bike. But as the first link above discusses, even bikes with zero trail can be ridden quite easily as long as you keep your hands on the handlebars; hands-off, you can balance with the aid of precession, but it's not nearly as easy as balancing hands-on without precession.
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Puzzle for physics geeks: Take one nonconducting icosahedron (that's a d20 for all you gamers). Replace its edges with 1-ohm resistors, connected at the vertices, so you now have a web of thirty resistors.

What is the resistance between opposite corners of the icosahedron?

(There are two ways I know of to do this one. The hard way involves a lot of simultaneous equations; the easy way takes about five seconds of mental calculation, once you know the trick.)
lederhosen: (Default)
Via Wikipedia updates: scientists convert type A/B/AB blood to O by using bacterial enzymes to eat antigens that would normally provoke an immune reaction.

(Actually, technically it should be a little better than O. Transfusing type O into an A/B/AB person is still a bit problematic because the transfused blood contains antibodies that will attack the recipient's blood cells; if I've understood this technique correctly, the conversion shouldn't add antibodies, so treated AB would be better than O :-)

And via [livejournal.com profile] cavalaxis, a very wrong way to make a very small pancake.

Gridding

Mar. 24th, 2007 05:17 pm
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Responding to problem given here.

The problem (all measurements in pixels): Given a 'canvas' size c, and an integer k (in this particular example, k= 3, 4, or 12) we want to find suitable values for the following:

t (constraint)
u (grid unit width, not counting gutters)
g (gutter width)

so that both the canvas and constraint can be divided into some multiple of k units, each of width u, separated by gutters of width g. We want to exactly cover the constraint, but we don't need to completely cover the canvas.

That translates to:
t = k*m(u+g)-g
c' = k*n(u+g)-g

for some integers m, n (greater than m), and where c' is an integer close to (and not greater than) c. There will be various constraints on t, u, and g - for instance, we'd like t to be between 360 and 500, u somewhere around 50-90, and g small.

An approach. )

EDIT: Pseudocode below. This is a bit less efficient in some parts than the implementation described above, but should be easier to follow and avoids coding in a GCD function.

Read more... )

EDIT the second: Misunderstood one aspect of the problem - the constraint doesn't have to divide by k, only the canvas. With that in mind, revised algorithm:

Read more... )
lederhosen: (Default)
This is pretty cool: guy demonstrates how to lift and position massive stone blocks singlehandedly with very simple techniques. He claims that these techniques "must have" been how Stonehenge was built; I'm not convinced of that, but they'd certainly be enough to do it.

SCIENCE!

Mar. 18th, 2007 11:38 am
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Via [livejournal.com profile] pharyngula, a gorgeous image showing how scientific disciplines connect to one another.

And while I'm here, via [livejournal.com profile] james_nicoll, how to woo publishers and agents (and make Night Travels of the Elven Vampire look like quality literature, too).
lederhosen: (Default)
Via [livejournal.com profile] mathsex, Tupper's Self-Referential Formula. It's not a deep result - all it really comes down to is that an algorithm for encoding small images can be used to encode an image of itself - but it's cute, and an illustration in how good a complex-looking formula can be at obfuscating a very simple rule.
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Via Pharyngula, the Order of the Science Scouts Of Exemplary Reputation And Above-Average Physique! I especially love the knots section.

My badges:




lederhosen: (Default)
First off, a touching Valentine's Day video for medical geeks. (Contains squicky bits.)

And the burning question of the era:

[Poll #922269]

Bonus points if you can name all those options.
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In Melbourne. We went to see Wunderkammer this morning (yay fossils! yay taxidermy! yay antique scientific/medical equipment!), dropped by [livejournal.com profile] mordwen's pool party, and then went to see the Australian Synchrotron at Clayton. We stood on the balcony looking at shiny machinery and inhaling the SCIENCE!

Migraine the other day. Boring details for my own records. )

Edit: Also, Octopus go squish!
lederhosen: (Default)
Via [livejournal.com profile] sclerotic_rings, how to make a thermic lance from tinfoil and spaghetti. The subtitles are good too.

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