h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

One that comes to mind occasionally, goes (I think) something like:

Ingredients:
- A number of rods, eg pins or needles or matchsticks (all of the same length)
- A hard, flat surface with parallel lines marked on it, where the distance between lines is the length of the rods.

Procedure:
Drop the rods onto the surface from a moderate height.

Results:
The ratio of rods lying between lines -to- rods touching or crossing lines approximates to pi.
The more rods you drop, the closer to pi that ratio becomes.


I am still greatly intrigued by the memory - but is it so? One rod at a time? Clear up before the next drop? ...

This is known as Buffon's Needle.

I'm currently doing a large project examining the practicalities of it. It's probable that the scouts of certain rock ants use it to select appropriate nesting sites when the colony is migrating.

There's the one about rivers - that the ratio of their length source-to-mouth to their length bends included approximates to pi.

Bouncy: more! tell me more about the ants!

King Bomba: Yes, heard that one, too, though so many variables makes it seem

I don't believe the one about the rivers. A young river rises in the mountains and flows more or less straight to the sea. But as the river ages, it starts to meander, which increases the length so the ratio goes up. Perhaps it is that it approaches pi as the river ages.

What it really boils down to, I think, is seeing a pattern and wanting it to be pi. When the *average* of the ratios of the rivers (and of those that were chosen by Hans-Henrik Stolum, who noticed it...) turned out to be a slight bit above 3. Therefore "Bingo, it's pi!" It's all a bit fluffy and vague, really.

"I don't believe the one about the rivers. A young river rises in the mountains and flows more or less straight to the sea. But as the river ages, it starts to meander, which increases the length so the ratio goes up."

Surely that's operating on the assumption that the river will be flowing across a perfectly flat terrain. Rivers don't work like that.

B

Rivers: School geography taught me about minor irregularities of flow leading to meanders, bends, hairpins then floods cutting across necks, making ox-bow lakes...
That's on flatter land, of course...

Rivers are pretty well all old rivers by now, so could (arguably) be more or less settled into an 'average'(!) pattern within their flood plains. Thus it's conceivable...

However, as you're suggesting, it takes a bit of faith in pi. Also surveys & resurveys after floods...


Does anyone *know*?

I have a feeling that rivers may well be close to being fractals so you would be able to find almost any value for total length : length from point A to B ratio.

More about the ants?

Well, the original method was by dropping needles randomly onto lined paper. If the needles are all the same length and the lines are evenly spaced, then you can get a formula for how often they could intersect. This involves pi, so you can do the experiment to get how often they actually intersect, plug in the results and rearrange the formula to find it.

Tnere is a paper somewhere which extended upon this work by working out that you can generalise is to: if you randomly place a set of lines of known length in a fixed 2D area, the number of intersections should be inversely proportional to the area.

A guy in the 1960s devised a cunning method to estimate how long the roots are in a soil sample by couting the intersections in a cross section of known area.

Then in 2000 another guy studying albipennis ants noticed that they seemed to select nest sites (flat areas in rock) based on how big they were, even though the sites were dark and very large and the ant should have no way of knowing. He thought that their pheremone trail could be approximating the random set of lines above, so they would just have to 'count' how often they recrossed it. Then he and some other people did some experiments to demonstrate this was very likely the case.

There's been some other bits of follow-up work involving mobile robots with pens and other such fun stuff.

And now I'm making a computer model of it.

Thinking about it, I recall another, completely separate, method for finding pi. Take an n-sided regular polygon. Work out a formula for the area or perimeter. Let n tend to infinity: the formula for area or perimeter tends toward the equivalent formula for a circle. Solve for pi.

My point, MuB, was that rivers change length, getting longer as they age. So they can only have a ratio of pi at one particular age, unless they asymptote at the pi ratio.

But they also get shorter at the same rate - ox-bow lake theory and all that guff.

B

Thanks, Bouncy. When you said
< seemed to select nest sites (flat areas in rock) based on how big they were>
I jumped to (soldier-, or safari- ?) ants who build bridges of themselves. However, I googled & found
http://beheco.oxfordjournals.org/cgi/content/abstract/16/2/488
Not completely clear, but I ain't trying for the complete paper (I'd stand no chance!).

Rivers: length due (directly) to age seems (to me) to be a minor consideration compared with the river finding a suitable route to the lowlands (which length is likely to be moderately static over long times?).
Intuitively then (my intuition!), it's the route through their own 'deposit beds' that's potentially subject to most change. Again intuitively, I can see that length being maintained, approximately, by essentially random meanderings & re-meanderings.

I suppose that if river routes in general are 3 or so times the crow-flight distance, then it's people that will tend to converge on pi, no matter what the river does...?

Build a pyramid and some crackpot will find pi.

Build a pi and they'll come... It had to happen:

http://www.artmusicdance.com/vaspi/highlights.htm "proof of the existence of God"

and, only there could it happen:
http://www.inwit.com/inwit/writings/indianapilaw.html : "In 1897 a bill was proposed in the Indiana Legislature that would have legally established the value of pi. This value would then have been copyrighted and used in state math textbooks. Other states would have to pay to use this value."

Build a piramid the ruins of common sense.

The bill was proposed in Indiana by a crackpot mathematician; note that it was rejected.

All that reminds me rather of the Natural Law party who stood for election the UK general elections some years ago.

They were (are?) followers of the Maharishi Mahesh Yogi and had a full double page advert for their science of 'natural law' in the broadsheets of the time where they equated a 'decadic superstring' equation with the Maharishis Vedic laws of god and earth (or whatever).

As well as doing strange mathematics they also attemted 'yogic flying' which invloved bouncing up and down on crash mats whilst doing the lotus position.

Much the same thing and utterly hilarious

Rivers are categorized depending on their length to curve ratio. I don´t know why any of those should be near pi, rather than just differing values.

Strangely, there was an article in today's paper about crackpots in Russia building pyramids to focus life-giving energy. No mention of pi in the article, though.