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The Miner's Donkey

This is a print by Joseph L. Teeters called The Miner’s Donkey. My father got it from our neighbor in 1977, who apparently got it directly from the artist in exchange for some books. Teeters was apparently then on the faculty of University of Wisconsin Eau Claire, and authored a book, Creating Tesselations.

Staffan and I went looking for any reference to this print on the internet, and we found none. So, this is perhaps the interwebs debut of this particular work. If anyone has more details about the artist, please pass them along.

A photo of the geekiest, rarest, and only item in my beer bottle collection. Yes, From the 1988 Device Research Conference, it is Device Research Brew.

All in all, I say “Bravo” to Sid the Science Kid.

Aina Explains Anatomy

So I’m teaching my son and three of his friends Physics. We gather one night every other week for two and a half hours and I talk Physics. Its a mix of explaining stuff, drawing pictures on our sketchboard, writing equations, working through problems and doing demos.

We’ve had three sessions so far and were doing some basic kinetics: velocity acceleration, force, kinetic energy. The last session was about rotational energy. The demo is watching things roll down a ramp: marbles of different weights and sizes, pieces of pipe, coins, washers and hot wheels cars. For the record, the hot wheels car beats the marble which beats the coin which beats the pipe, all because of the differing shape factors in the calculation of the moment of inertia. So far so good.

Anyway, I’ve resolved not to use calculus for any of the explanations. But my brain seems to be hard wired for thinking about this kind of physics in terms of the calculus approach, and I struggled with how to teach all those calculus ideas without the calculus itself, and how to let the kids have some access to calculation as a part of the physics learning.

Then I remembered about the Monte Carlo technique for calculating pi, and something deep inside my brain clicked: If I could teach the kids how to calculate pi by throwing darts, I could teach them about all the calculus ideas in kinematics without any calculus. Here’s how the story goes:

Lets say you put a square with a circle inscribed in it up on a dartboard. Or if you are good at darts, lots of little circles lined up in a square like this:

circles and square.jpg

Now, throw a dart at the dartboard. Stand far enough back so the dart lands randomly either inside one of the circles or not inside one of the circles. What’s the probability that the dart lands inside a circle? It is the same as the ratio of the area of a circle of diameter 1 to the area of a square of side length 1, or pi / 4.

OK, you aren’t impressed, because throwing one dart doesn’t tell us much about pi: Instead pi is telling us something about that one dart. However, throw enough darts so that 100 darts land inside the square, and I’ll bet somewhere between 75 and 82 of those also landed in one of the circles. Do it 10000 times, and between 7800 and 7900 darts will hit a circle. The more darts you throw, the closer the ratio will get to pi / 4.

OK, you still aren’t impressed, because no one would ever throw thousands of darts and the paper would get pretty ragged after all those darts. And some would land on the line and your friend Mike would say ‘inside the circle’ and you’d say “no, outside the circle” and then Chris would call ‘do-over’ and then someone would call for more cowbell and then where would you be?

But the point is that with computers there’s no need to actually throw darts — you can use the random number generator in JavaScript or AppleScript and get the same result. Here’s an apple script which calculates pi only with random numbers:


to getRandomDecimalBetween(min, max)
  set diff to max - min
  return (min + diff * (((random number from 1 to 50000) - 0.5) / 50000))
end getRandomDecimalBetween

set hitsInCircle to 0
set n to 10000
repeat n times
  set x to getRandomDecimalBetween(0, 1)
  set y to getRandomDecimalBetween(0, 1)
  if ((x * x + y * y) < 1) then -- point (x,y) must be inside the circle
    set hitsInCircle to hitsInCircle + 1
  end if
end repeat

set approximatePi to (4 * hitsInCircle) / n

and here’s a javascript version.

Essentially what the script does is choose a random point in a square on the xy plane with x and y both between 0 and 1. If x^2 + y^2 is less than 1, then the point counts as being inside the circle.

Now you are impressed, but are wondering how does this relate to physics? And the answer is that you can use the same technique to do “integrals” of some property of interest. So for example, if you want to calculate the moment of inertia of a body analytically, you do this integral (thanks wikipedia):

IIntegral.jpg

Which is ugly even for me, and I know what it means. But for a bunch of kids that don’t know calculus but can write a little script, its not hard to explain how to throw darts into a 3D space, and if you hit something, use the equations they learned the last time to calculate the kinetic energy of that little bit of the object rotating around some axis. That’s part of the homework I gave them this week. We’ll see if they come up with the right answers.

Playing with scripting of the drawing app Intaglio, I made this nice picture:

Penrose2.jpg

For the curious, here’s a script that will generate a similar tiling
property magicRatio : (1 + (5 ^ 0.5)) / 2
property invMagicRatio : 2 / (1 + (5 ^ 0.5))
property sinPiOverFive : 0.587785252
property sinTwoPiOverFive : 0.951056516
property cosTwoPiOverFive : 0.309016994
property cosPiOverFive : 0.809016994
to drawATriangle(pt1, pt2, pt3, width, fc)
tell application “Intaglio”
make new path with properties {segments:{move to, {x of pt1, y of pt1}, line to, {x of pt2, y of pt2}, line to, {x of pt3, y of pt3}, line to, {x of pt1, y of pt1}}, drawing mode:fill, stroke width:2, fill color:rgb & fc, stroke color:{rgb, 0.2, 0.2, 1, 0.4}}
end tell
end drawATriangle
to subdivideB(pt1, pt2, pt3, numSubdivisions)
set delX to (x of pt3) – (x of pt1)
set delY to (y of pt3) – (y of pt1)
set pt4 to {x:(x of pt3) – delX * invMagicRatio, y:(y of pt3) – delY * invMagicRatio}
if numSubdivisions is less than 5 then
set alpha to 0.9 – ((numSubdivisions) * 0.2)
drawATriangle(pt1, pt2, pt3, numSubdivisions, {0.3, 1, 0.4, alpha}) — change the colors here !!
end if
if (numSubdivisions > 1) then
subdivideA(pt2, pt3, pt4, numSubdivisions – 1)
end if
if (numSubdivisions > 2) then
subdivideB(pt2, pt4, pt1, numSubdivisions – 2)
end if
end subdivideB
to subdivideA(pt1, pt2, pt3, numSubdivisions)
set delX to (x of pt2) – (x of pt1)
set delY to (y of pt2) – (y of pt1)
set pt4 to {x:(x of pt2) – delX * invMagicRatio, y:(y of pt2) – delY * invMagicRatio}
if numSubdivisions is less than 5 then
set alpha to 0.9 – ((numSubdivisions) * 0.2)
drawATriangle(pt1, pt2, pt3, numSubdivisions, {0, 0.4, 1, alpha}) — change the colors here !!
end if
if (numSubdivisions > 2) then
subdivideA(pt4, pt3, pt1, numSubdivisions – 2)
end if
if (numSubdivisions > 1) then
subdivideB(pt2, pt4, pt3, numSubdivisions – 1)
end if
end subdivideA
set centerX to 280
set centerY to 350
set len to 280
— ptList is the ten outside points of the decagon, with the first one repeated at the end
set ptList to {{x:len, y:0}, {x:len * cosPiOverFive, y:len * sinPiOverFive}, {x:len * cosTwoPiOverFive, y:len * sinTwoPiOverFive}, {x:-len * cosTwoPiOverFive, y:len * sinTwoPiOverFive}, {x:-len * cosPiOverFive, y:len * sinPiOverFive}, {x:-len, y:0}, {x:-len * cosPiOverFive, y:-len * sinPiOverFive}, {x:-len * cosTwoPiOverFive, y:-len * sinTwoPiOverFive}, {x:len * cosTwoPiOverFive, y:-len * sinTwoPiOverFive}, {x:len * cosPiOverFive, y:-len * sinPiOverFive}, {x:len, y:0}}
set a to 1
repeat 5 times
set pt1 to {x:centerX, y:centerY}
set pt2 to {x:centerX + (x of item a of ptList), y:centerY + (y of item a of ptList)}
set a to a + 1
set pt3 to {x:centerX + (x of item a of ptList), y:centerY + (y of item a of ptList)}
subdivideA(pt3, pt1, pt2, 8 )
set a to a + 1
set pt2 to {x:centerX + (x of item a of ptList), y:centerY + (y of item a of ptList)}
subdivideA(pt3, pt1, pt2, 8 )
end repeat

I used to think that Creationism was a harmless, if misguided, obsession in the minds of believers whose faith is weak. A bit of time spent in Hawaii has made me recognize that Creationism is evil. Not so much because of how it makes one think of creation, but instead because of how it makes one think of destruction. On one magical snorkeling trip we saw whales and bottlenose dolphins, and then swam with an amazing assortment of fish, and then swam with spinner dolphins, and then swam with turtles. And as much as we marveled at the wonders of Creation, the dangers faced by these species are very real.

So while to me it matters very little who is right in the Creationism vs. Evolutionism debate as it applies to Creation. After all, is God any less great if one path rather the other produced our unfathomably glorious universe? But as it applies to the incredible rate at which species are disappearing from the planet, the debate matters very much to me.

The logical conclusion of Creationism is not that Man can or should do anything about the peril caused by Man’s destruction of the environment. In 1981 Ronald Reagan’s Secretary of the Interior James Watt was testifying before Congress and was asked if he agreed that natural resources should be preserved for future generations. He responded “I do not know how many future generations we can count on before the Lord returns.”   Of course, all the companies that are making money destroying the environment embrace this line of thinking, but I hope that no sensible Christian is tempted to be so blind. The line of thinking goes like this:

As surely as God planned Creation, He must also have planned the
extinctions that we are now seeing.  The disappearance of each species
is part of His plan, and it is folly for Man to think that we could
challenge this plan.  Like the natural resources that God has given us
for our sustenance, surely God intends that we use all the gifts of
Creation before He comes again.

If this is the logical conclusion of Creationism, then we must label it as the evil that it is and work against it.

Alternatively, there is the TV Evangelist line of thinking whereby we should bury our heads in the Bible. It is vapid in its own way:

As surely as God planned Creation, he also entrusted Creation to
mankind and expects Man to be its stewards.  Therefore it is of grave
concern that we we are witnessing so many extinctions in our lifetime: 
surely it must be the sign of the sinfulness of Mankind:  The
destruction that we are seeing around us is not the direct result of
Man’s actions, but God’s punishment for a world of sin.  Only by
ridding ourselves of sin and submitting ourselves in submission to God
will the destruction cease, and if God is graceful, God will re-Create
Creation in all its deserved glory.

But God does not call Christians to be ostriches. Instead, there is a very sensible and Christian line of thinking:

As surely as God planned Creation, he also entrusted Creation to
mankind and expects Man to be its stewards.  Therefore it is of grave
concern that we we are witnessing so many extinctions in our lifetime: 
surely we are called to understand how the actions of Mankind are
precipitating this crisis:  we are called to study how the disruption
of habitat imperils the species.  We are called to study how different
species respond to environmental change.  We are called to study how
genetics influence the variety of traits that a species posesses, and
how these traits change over time in response to environmental changes.
In short, we are called to study evolution.

Now that’s what I call a Christian message.

Greg Goelzhauser has had a few interesting posts recently about his experience with a ticket to a football game. Noting that he would never pay the “market” price for a good ticket ($200), he is unwilling to part with a ticket he has fallen into possession of. That is, he is unwilling to sell his ticket for $200, so he must value it more that. However, before owning the ticket, he wouldn’t even have paid $100 for a ticket, so he must value it less than $100.

For an economist, this is an example of “irrational” behavior: that something can be valued differently depending on whether or not you own it, a phenomenon called the “endowment effect”. Cash value is supposed to be independent of ownership when a sufficiently fluid market exists, as in the case of football game tickets: if someone has a ticket they value less than the market value, they can increase their perceived value by selling the ticket. On the other hand, people who value a ticket more than they value the cash in their pocket can buy a ticket and increase their perceived value.

Rather than persist in calling people irrational (though at times they certainly are), I’d prefer to look for other forces which create value in ways that can’t be arbitraged by a market: the value of community spirit, for example, or the value of a social network, or the value of one’s personal reputation. These are all forces at work in the gift economy .

The strength of traditional economics has been the ability to quantify the various economic forces which shape most of our livelihoods. Irrational behavior such as that shown in the endowment effect is a thorn in the side of economics simply because it seems to be unquantifiable. Yet I see a positive side: with enough examples such as the football game ticket, we ought to be able to sort out exactly where that extra value is created which causes the endowment effect. I don’t think people are irrational — I think that there is real value created when you own an item like the football ticket — but that value doesn’t have an associated market — it is value in the gift economy and not the exchange economy.

And with understanding of where the value resides, we can measure the value in the gift economy by looking at its interaction with the exchange economy. It may sound very silly to ask people how much a community is worth to someone. How much is your college community worth in dollars? How much is your professional circle worth in dollars? But measuring the magnitude of the endowment effect could give us a quantitative idea about the answers to those questions. In other words, if a football ticket increases in value by $100 when you own it, that must mean that there is something worth at least $100 that isn’t for sale in any store.

Its the new Honda ad. Deserving of the big-screen.

After Staffan saw it, he said “Again”. He was still saying “again” after the fourth time….

Here is why you should believe every word from the person who runs The Crystal Chamber, a web site offering crystals with supernatural healing powers.

Update (March 5, 2003)

I’ve reconsidered this. A troll is a troll, and even if he was right about the science doesn’t mean he isn’t a troll. In addition, there was some misunderstood delineation between the new age faction at the site and the mainstream pagans: I don’t know why he considered ‘uk-pagan’ an appropriate site for agitating ‘new age’ believers.

I was in error for celebrating the activity of a troll. Sorry. It’s still a funny story.

Who exactly is the “Penetrometer” aimed at?