Tau not Pi

This one’s for the nerds. It’s hard to argue against the elegance of using tau instead of pi. Natalie Wolchover explains, while Kevin Houston makes the argument in video:

Malthusian Growth

I think I may fairly make two postulata.

First, That food is necessary to the existence of man.

Secondly, That the passion between the sexes is necessary and will remain nearly in its present state.

— Malthus (1798): An Essay on the Principle of Population via The Concise Encyclopedia of Economics

“Malthusian” is often used as a derogatory term to refer to alarmist predictions that we’re going to run out of some natural resource. I’m afraid I’ve used the term this way myself, however, according to Lauren Landsburg at the Concise Encyclopedia of Economics, Malthus is being unfairly maligned. He wasn’t actually predicting catastrophe but wondering why the catastrophes don’t usually happen.

What Thomas Malthus did, in 1798, was point out that while populations grow at a geometric rate – the U.S. population, he noticed, doubled every 25 years – but resources, like food, only increase at an arithmetic rate. As a result, any naturally growing population will eventually run out of resources.

malthus-base — The red line shows geometric growth. No matter how much you start off with, the red line will always end up crossing the blue line.

The linear equation has the form:

$y = m t + b$

where y is the quantity produced, t is time (the independent variable), and m and b are constants. This should not look to unfamiliar to students who’ve had algebra.

The geometric equation is a little more complicated:

$y = a^{gt} + c$

here a, g and c are constants. g is the most important, because it’s the growth rate – the higher g is the faster the curve will rise. You can play around with the coefficients and graph in this Excel spreadsheet .

At any rate, after the curves intersect, the needs of the population exceeds how much it can produce; this is the point of Malthusian catastrophe.

malthus-cat — The intersection point is where the needs of the population exceeds the production.

The observation is, indeed, so stark that it is still easy to lose sight of Malthus’s actual conclusion: that because humans have not all starved, economic choices must be at work, and it is the job of an economist to study those choices.

— Landsburg (2008): Thomas Robert Malthus from The Concise Encyclopedia of Economics.

A Nefarious Application of Math

Cynical, but, if you consider the current “kinetic military action” in Libya, way to close to reality. Indeed, this highlights the question: When does it become too easy to go to war?

Jonathan Schell sums it up in the Guardian:

American planes are taking off, they are entering Libyan air space, they are locating targets, they are dropping bombs, and the bombs are killing and injuring people and destroying things. It is war. Some say it is a good war and some say it is a bad war, but surely it is a war.

Nonetheless, the Obama administration insists it is not a war. Why?

…, the balance of forces is so lopsided in favour of the United States that no Americans are dying or are threatened with dying. War is only war, it seems, when Americans are dying, when we die. When only they, the Libyans, die, it is something else …

— Schell (2011): Libya: it’s not a war if Americans can’t get hurt

What’s Needed for a Nation’s Peace

The Fund for Peace has been doing a lot of thinking about what it takes for a country to be considered peaceful, and what it takes for a state to fail. For the last seven years they’ve been putting together maps of the world with an index of how stable different countries are.

fsi-finland — Finland - the most sustainable state, at least according to the Failed State Index.

While it’s pretty in-depth and makes for rather sobering reading, it’s worth taking a look at the criteria they’ve come up with to determine a country’s stability. It may be useful to include some of this information in the cycle where we focus on peace.

Their criteria for instability include:

Demographic pressure (such as having too many young adults, as we’ve seen in Egypt)
Amount of refugees and internally displaced peoples (refugees are people who’ve crossed international borders). Both leaving or entering refugees can undermine stability.
Historical Injustice – communities can have an understandably hard time forgetting the past, just look at Isreal/Palestine.
Brain Drain – when countries start to fail, the first to leave are the ones who can afford to. Yet these intellectuals and professionals, with their college degree are vital for creating a stable and prosperous country.
Inequality – especially when driven by active discrimination (wealth inequality is something to watch out for).
Economic decline – pushes trade into the black market and increases criminality and corruption.
Illegitimacy of the state – if people don’t believe the people in government have everyone in the country’s best interests at heart, and are only looking out for themselves and their friends, then there’s probably going to be trouble.
Public Services go kaput – It’s a really bad sign when the government can meet people’s basic needs – like picking up the garbage.
The Rule of Law goes kaput – when you’re ruled by the caprice of men, and your rights under the law are not respected, you may begin to consider and agitate for other options for government.
Personal Armies – forces that are tied to individual leaders, like private militias or super-secret police for example, are very damaging to a country’s cohesion.
Fighting elites – healthy countries need robust arguments in their political class – think checks and balances – but it can go too far and lead to things like extreme nationalism and ethnic cleansing.
Invasion – both overt invasion and covert meddling in the affairs of a country are unhealthy for that state’s stability.

It’s also very nice that you can download their index data as a MS Excel spreadsheet, which you can let students analyze to answer their own research questions. For example, I was wondering what was the difference between the best, the worst and the USA, so I plotted this graph.

fsi-best-worst-usa — Comparing the best (Finland), worst (Somalia) and the USA using the Fund for Peace's Failed State Indicies.

The USA is much closer to Finland than Somalia, thank goodness, but should probably watch out for that Uneven Development (wealth inequality).

I think something like this would make a good experiential exercise for the science of geography.

Solving Quadratics

Solving quadratic equations requires finding the factors, which is not nearly as easy as multiplying out the factors to get the unfactored equation.

Instead, you have to do a bit of trial and error, to figure out which pairs of numbers multiply to give the constant in the equation and then add together to give the coefficient of the x term.

factoring — Factoring quadratic equations.

It gets easier with practice. Or you could use the quadratic formula, where if the equation were:

$a x^2 + b x + c = 0$

The solutions would be found with:

$x=\frac{-b \pm \sqrt {b^2-4ac}}{2a}$

So in the equation used in the diagram:
$x^2 + 7 x + 10 = 0$

you get:

a = 1

b = 7

c = 10

Putting these values into the quadratic equation gives:

$x=\frac{-7 \pm \sqrt {7^2-4(1)(10)}}{2(1)}$

which simplifies to:

$x=\frac{-7 \pm \sqrt {49-40}}{2}$

$x=\frac{-7 \pm \sqrt {9}}{2}$

$x=\frac{-7 \pm 3}{2}$

With the whole plus-or-minus thing ( $\pm$ ), this last equation gives two solutions:

(1): $x=\frac{-7 + 3}{2} = -5$

and,

(2): $x=\frac{-7 - 3}{2} = -2$

Now, you may have noticed that the solutions are negative, but when the equation is factored in the illustration, the result is:

$(x + 5) (x + 2) = 0$

The difference is that, although we’ve factored the equation, we have not solved it. When I say solve the equation, I mean find the values of x that would result in the left hand side of the equation being equal to the right, which is zero. Since multiplying anything by zero will give you zero, and the two factors multiply each other, the left-hand-side of the equation will equal zero when either one of the two factors equals zero.

So:

(x + 5) = 0
x = – 5

and:

(x + 2) = 0
x = -2

Finally, we can plot the line:

$y = x^2 + 7 x + 10$

using the Graphing Calculator Pro app, or this somewhat crude Parabola-Line Excel Graphing Worksheet, to show that the line crosses the x axis at -5 and -2.:

Illustrating the Multiplication of Quadratic Factors

all-terms-color — Each term needs to multiply the other two terms in the opposite parenthesis, so start with the reds, then do the two mixed colors, then the blues, and finally, combine the mixed colors.

I’ve been playing around with ways of showing how to multiply out quadratic factors like the one above. I’m still not perfectly happy with these animations but they’re the best I’ve come up with so far. A smoother, Flash or svg animation might work better though.

In this second version the terms being multiplied are highlighted. I like how the highlighting gives some more stability to the animation, but I’m always leery of putting too much color or bells and whistles because they tend to complicate the picture. In this case at least, I think all the colors have meaning and are useful.

all-terms-color-hi — Multiplication of quadratic factors.

It’s 10 PM and the Moat is Empty

My students and I had a great chance to use the our recent geometry work when we figured out how long it would take to drain the new moat in front of the school.

It’s not really a moat, it’s going to be a flower bed that will soak up some of the runoff that tries to seep into the school’s doors every time a spring or fall thunderstorm sweeps through.

The hole was dug on Thursday evening and filled with rainwater with water, half a meter deep, by Friday morning’s rain. At least we know now that the new beds are in the right place to attract runoff.

But to fill the trenches with gravel, sand and soil, we needed to drain the water. With a small electrical pump it seemed like it would take forever; except that we could do the math.

The pump emptied water through a long hose that runs around the back of the building where the topography is lower. I sent two students with a pitcher and a timer (an iPod Touch actually) to get the flow rate.

They came back with a time of 18.9 seconds to fill 4 liters. I sent them back to take another measurement, and had them average the to numbers to get the more reliable value of 18.65 seconds.

Then one of the students got out the meter-stick and measured the depth of the moat at a few locations. The measurements ranged from 46 cm to about 36 cm and we guesstimated that we could model the moat as having two parts, both sloping. After measuring the length (~6 m) and width (2 m), we went inside to do the math.

moat-volume — Rough sketch of the volume of water in the moat.

With the help of two of my students who tend to take the advanced math option every cycle, we calculated the volume of water (in cm³) and the flow rate of the pumped water (0.2145 cm³/s). Then we could work out the time it would take to drain the water, which turned out to be a pretty large number of seconds. We converted to minutes and then hours. The final result was about 7 hours, which would mean that the pump would need to run until 10 pm.

And it did.

Tsunami Geometry: Calculating the Height of a Tsunami using Basic Geometry

Since we’re working on geometry this cycle, I thought it would be an interesting exercise to think about how we could use geometry to think about how the strength of tsunamis decreases with distance from the source.

Of course, we’ll have to do this using some intense simplification so we can actually apply the tools we have available. The first is to approximate the tsunami as a circular wall of water centered on the epicenter of the earthquake.

tsunami-geometry — Simplified tsunami geometry.

This lets us figure out the volume of the wave pretty easily because we know that the volume of a cylinder is:

(1) $! V_c = \pi r^2 h$

The size of the circular water wall we approximate from the reports from Japan. The maximum height of the wave at landfall was somewhere in the range of 14 m along the northern Japanese coast, which was about 80 km from the epicenter. Just as a wild guess, I’m assuming that the “effective” width of the wave is 1 km.

Typically, in deep water, a tsunami can have a wavelength greater than 500 km (Nelson, 2010; note that our width is half the wavelength), but a wave height of only 1 m (USSRTF). When they reach the shallow water the wave height increases. The Japanese tsunami’s maximum height was reportedly about 14 m.

At any rate, we can figure out the volume of our wall of water by calculating the volume of a cylinder with the middle cut out of it. The radius of our inner cylinder is 80 km, and the radius of the outer cylinder is 80 km plus the width of the wave, which we say here is 1 km.

Calculating the volume of the wave

However, for the sake of algebra, we’ll call the radius of the inner cylinder, r_i and the width of the wave as w. Therefore the inner cylinder has a volume of:

(2) $! V_i = \pi r_i^2 h$

So the radius of the outer cylinder is the radius of the inner cylinder plus the width of the wave:

(3) $! r_o = r_i + w$

which means that the volume of the outer cylinder is:

(4) $! V_o = \pi (r_i + w)^2 h$

So now we can figure out the volume of the wave, which is the volume of the outer cylinder minus the volume of the inner cylinder:

(5) $! V_w = V_o - V_i$

(6) $! V_w = \pi (r_i + w)^2 h - \pi r_i^2 h$

Now to simplify, let’s expand the first term on the right side of the equation:

(7) $! V_w = \pi (r_i ^2 + 2 r_i w + w^2) h - \pi r_i^2 h$

Now let’s collect terms:

(8) $! V_w = \pi h \left( (r_i ^2 + 2 r_i w + w^2) - r_i^2 \right)$

Take away the inner parentheses:

(9) $! V_w = \pi h (r_i ^2 + 2 r_i w + w^2 - r_i^2)$

and subtract similar terms to get the equation:

(10) $! V_w = \pi h ( 2 r_i w + w^2 )$

Volume of the wave

Now we can just plug in our estimates of width and height to get the volume of water in the wave. We’re going to assume, later on, that the volume of water does not change as the wave propagates across the ocean.

(11) $! V_w = \pi (14) ( 2 r_i (1000) + (1000)^2 )$

rearrange so the coefficients are in front of the variables:

(12) $! V_w = 14 \pi ( 2000 r_i + 1000000 )$

So, at 80 km, the volume of water in our wave is:

(13) $! V_w = 14 \pi ( 2000 (80000) + 1000000 )$

(14) $! V_w = 7081149841 m^3$

Height of the Tsunami

Okay, now we want to know what the height of the tsunami will be at any distance from the epicenter of the earthquake. We’re assuming that the volume of water in the wave remains the same, and that the width of the wave also remains the same. The radius and circumference will certainly change, however.

We take equation (10) and rearrange it to solve for h by first dividing by rearranging all the terms on the right hand side so h is at the end of the equation (this is mostly for clarity):

(15) $! V_w = \pi ( 2 r_i w + w^2 ) h$

Now we can divide by all the other terms on the right hand side to isolate h:

(16) $! \frac{V_w}{\pi ( 2 r_i w + w^2 )} = \frac{\pi ( 2 r_i w + w^2 ) h}{\pi ( 2 r_i w + w^2 )}$

so:

(17) $! \frac{V_w}{\pi ( 2 r_i w + w^2 )} = h$

which when reversed looks like:

(18) $! h = \frac{V_w}{\pi ( 2 r_i w + w^2 )}$

This is our most general equation. We can use it for any width, or radius of wave that we want, which is great. Anyone else who wants to calculate wave heights for other situations would probably start with this equation (and equation (15)).

Double checking our algebra

So we can now figure out the height of the wave at any radius from the epicenter of the earthquake. To double check our algebra, however, let’s plug in the volume we calculated, and the numbers we started off with, and see if we get the same height (14 m).

First, we’ll use all our initial approximations so we get an equation with only two variables: height (h) and radial distance (r_i). Remember our initial conditions:

w = 1000 m
r_i = 80,000 m
h_i = 14 m

we used these numbers in equation (10) to calculate the volume of water in the wave:

V_w = 7081149841 m³

Now using these same numbers in equation (18) we get:

(19) $! h = \frac{7081149841}{\pi ( 2 (r_i) (1000) + (1000)^2 )}$

which simplifies to:

(20) $! h = \frac{7081149841}{ 2000 r_i \pi + 1000000 \pi }$

So, to double-check we try the radius of 80 km (80,000 m) and we get:

h = 14 m

Aha! it works.

Across the Pacific

Now, what about Hawaii? Well it’s about 6000 km away from the earthquake, so taking that as our radius (in meters of course), in equation (20) we get:

(21) $! h = \frac{7,081,149,841}{ 2,000 (6,000,000) \pi + 1,000,000 \pi }$

which is:

h = 0.19 m

This is just 19 cm!

All the way across the Pacific, Lima, Peru, is approximately 9,000 km away, which, using equation (20) gives:

h = 0.13 m

So now I’m curious about just how fast the 14 meters drops off to less than 20 cm. So I bring up Excel and put together a spreadsheet of tsunami height at different distances. Plot on a graph we get:

So the height of the tsunami drops off relatively fast. Within 1000 km of the earthquake the height has dropped by 90%.

How good is this model

This is all very nice, a cute little exercise in algebra, but is it useful? Does it come anywhere close to reality? We can check by comparing it to actual measurements; the same ones used by NOAA to compare to their model (see here).

tsunami-model-data-honolulu — The red line is the tsunami's water height predicted by the NOAA computer models for Honolulu, Hawaii, while the black line is the actual water height, measured at a tidal gauge. Other comparisons can be found here.

680_20110311-TsunamiWaveHeight — Tsunami wave heights in the Pacific, as modeled by NOAA. Notice how the force of the tsunami is focused across the center of the Pacific.

The graph shows a maximum height of about 60 cm, which is about three times larger than our model. NOAA’s estimate is within 20% of the actual maximum heights, but they’ve spent a bit more time on this problem, so they should be a little better than us. You can find all the gruesome details on NOAA’s Center for Tsunami Research site’s Tsunami Forecasting page.

Notes

1. The maximum height of a tsunami depends on how much up-and-down motion was caused by the earthquake. ScienceDaily reports on a 2007 article that tried to figure out if you could predict the size of a tsunami based on the type of earthquake that caused it.

2. Using buoys in the area, NOAA was able to detect and warn about the Japanese earthquake in about 9 minutes. How do they know where to place the buoys? Plate tectonics.

buoys-warning — The locations of the buoys in NOAA's tsunami warning system.

Update

The equations starting with (7) did not have the 2 on the r_iw term. That has been corrected. Note that the numerical calculations were correct so they have not changed. – Thanks to Spencer and Claude for helping me catch that.