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If Godzilla Were Real, How Much Would He Eat?

(This post originally appeared in Huffington Post’s Entertainment section.)

The newest Godzilla is quite the beast. According to the official source, Legendary pictures, Godzilla is 355 feet tall. That’s about the height of a 40-story building! It’s also the tallest Godzilla yet (according to Wikizilla; yes, there are such sites). So, how much does such an animal need to eat to survive? Let’s do the math to find out.

To answer the question we’ll use Kleiber’s Law: an animal’s basal calorie requirement E (the least amount of calories needed to be eaten in a day to sustain life for a minimally active organism) is related to its mass M (in kilograms) by the equation

E=70M^{3/4}.

We need Godzilla’s mass. There’s no scale I know of that we could use, and I wouldn’t be comfortable asking him directly. Maybe that’s why Popular Mechanics took the easy way out and asked scientists to estimate some of Godzilla’s features. The consensus on his mass: about 160,000 tons, or about as heavy as a cruise ship!

Okay, back to the calculation. Godzilla’s mass in kilograms is 160,000,000 kg, making his daily calorie requirement roughly…

1 million calories!

Maybe this explains why he’s always eating everything in sight!

2014-05-19-gradiotowers-thumb

(And also why Hollywood — and the original creator of Godzilla — made him eat nuclear energy for food.)

Truth be told, I’m not sure how applicable Kleiber’s Law is to Godzilla. It does, however, accurately describe the relationship of mass to energy requirements for everything from cells to elephants. That’s a pretty amazing feat for just one equation. And, aside from the absurdly large appetite Godzilla has, it’s the one other thing I hope you’ll take away from this article.

Send Mom “Mathematical Flowers” for Mother’s Day!

(This post originally appeared in Huffington Post’s Science section, and later on Princeton Press’ blog.)

Did you know that you can create beautiful flowers with an equation? Read on to find out how.

Before I tell you what that equation is, let me convince you first. The picture below shows a particular graph of this equation (on the left) along with an actual flower (on the right).

Picture1-1

 

Pretty neat, huh? But wait, there’s more! Let’s change just one number in this equation and see what happens.

Picture2-2

 

A fourth petal! (And yes, I’ve also changed the color.) Thus far there is one clear takeaway: These “mathematical flowers” are strikingly similar to the real thing. Let me give you one last example:

Picture-3

 

So, what’s the equation that can seemingly duplicate many of the pretty flowers we see around us, and what is the number that controls how many petals are generated? The equation is

r=cos(ntheta),

and by changing n we control the number of petals. But to generate recognizable flowers we need to choose n to be a natural number (1,2,3,…). In that case the equation above produces the “mathematical flowers” in the three pictures you saw. For example, in the first picture n=3, in the second picture n=2 (not 4), and in the third picture n=5. When n is not a natural number, you get curves that don’t look anything like flowers. For example, here’s the peanut-like curve you get for n=1/6:

peanut

I produced all of these graphs on my computer, but you don’t have to have special software to do it yourself. You can now share the joy of receiving “mathematical flowers” with the moms in your life by using the widget I wrote that produces several of these virtual flowers. Here it is (it’s free to use).

I’m sure mom will be as surprised as you were to see just how realistic these mathematical flowers look. And if your mom is a math enthusiast, or would just like to understand how the r-equation produces those flowers, read on and you’ll be able to explain it to her.

First, have a look at the diagram below.

PolarCart

Here I’m showing you two different ways to plot a point, in this case (a,b). In the method we’re most familiar with you move a units to the right from the origin (the point where the two axes intersect) and b units up from there. That’s called graphing in Cartesian coordinates. But you can also graph the same point in polar coordinates. To do so, you first move r units away from the origin, and then rotate counter-clockwise by the angle in the picture (the Greek letter “theta”).

One quick way to generate lots of points is to graph an equation, like y=x2. This rule tells us that the y-value is the square of the x-value. For example, when x=1 we gety=1, and when x=2 we get y=4. Therefore, the points (1,1) and (2,4) (in Cartesian coordinates) are on the graph of y=x2.

Our “mathematical flowers” are just the graphs of the cosine equation above, but where each point is plotted in the polar coordinate system. For example, when the angle theta is zero we get r=1, meaning that the point (1,0) is on the graph of every mathematical flower, regardless of what n is. As theta ranges from 0 to 360 degrees, the graphs generate the flower-like curves I showed you in the first three pictures. (I took the extra steps of shading in the graphs and putting them on a nice background.)

Now that you know how those “mathematical flowers” were generated, let me leave you with an interesting thought. Instead of creating curves with an equation and then comparing the results to real flowers, we could reverse this reasoning and come to a much more interesting possibility: Maybe flowers follow mathematical laws as they grow! For many flowers, and indeed many other examples of natural phenomena, this is indeed true. This surprising fact is just another example of hidden mathematics all around us.

Happy Mother’s Day!

These Two Numbers Make Spring Possible (and the other seasons too)

(This post originally appeared on Princeton Press’ blog.)

March 20th. Don’t recognize that date? You should, it’s the official start of spring! I won’t blame you for not knowing, because after the unusually cold winter we’ve had it’s easy to forget that higher temperatures are coming. But why March 20th, and not the 21st or the 19th? And while we’re at it, why are there even seasons at all? Read on to find out the answers.

The answer has to do with 2 numbers. Don’t worry, they’re simple numbers (not like pi [1]). Stick around and I’ll show you some neat graphs to help you understand where they come from, and hopefully entertain you in the process too.

The first star of this show is the number 92 million. No, it’s not the current Powerball jackpot; it’s also not the number of times a teenager texts per day. To appreciate its significance, have a look at our first chart:

f1
Figure 1: The average surface temperature (on the vertical axis) of the planets in our solar system sorted by their distance from our sun (the horizontal axis). From left to right: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto (technically, Pluto lost its planet status in 2006).

 

That first planet on the left is Mercury. It’s about 36 million miles away from the sun and has an average surface temperature of 333o. (Bring LOTS of sunscreen.) Fourth down the line is the red planet, Mars. At a distance of about 141 million miles from the sun, Mars’ average temperature is -85o. (Bring LOTS of hot chocolate.) We could keep going, but the general trend is clear: planets farther away from the sun have lower average temperatures [2].

If neither 333o nor -85o sound inviting, I’ve got just the place for you: Earth! At a cool 59o this planet is … drumroll please … 92 million miles from the sun.

We actually got lucky here. You see, it turns out that a planet’s temperature T is related to its distance r from the sun by the formula , where k is a number that depends on certain properties of the planet. I’ve graphed this curve in Figure 1. Notice that all the planets (except for the pesky Venus) closely follow the curve. But there’s more here than meets the eye. Specifically, the formula T=k/sqrt{r} predicts that a 1% change in distance will result in a 0.5% change in temperature [3]. For example, were Earth just 3% closer to the sun—about 89 million miles away instead of 92 million—the average temperature would be about 1.5o higher. To put that in perspective, note that at the end of the last ice age average temperatures were only 5o to 9o cooler than today [4].

So our distance from the sun gets us more reasonable temperatures than Mercury and Mars have, but where do the seasons come from? That’s where our second number comes in: 23.4.

Imagine yourself in a park sitting in front of a bonfire. You’re standing close enough to feel the heat but not close enough to feel the burn. Now lean in. Your head is now hotter than your toes; this tilthas produced a temperature difference between your “northern hemisphere” and your “southern hemisphere.” This “tilt effect” is exactly what happens as Earth orbits the sun. More specifically, our planet is tilted about 23.4o from its vertical axis (Figure 2).

f2
Figure 2. Earth is tilted about 23.4o from the plane of orbit with the sun (called the ecliptic plane).

 

Because of its tilt, as the Earth orbits the sun sometimes the Northern Hemisphere tilts toward the sun—roughly March-September—and other times it tilts away from the sun—roughly September-March (Figure 3) [5].

f3
Figure 3. Earth’s tilt points toward the sun between mid-March and mid-September, and points away from it the remaining months of the year. The four marked dates describe how this “tilt effect” changes the number of daylight hours throughout the year. Assuming you live in the Northern Hemisphere, days are longest during the summer solstice (shorter nights) and shortest during the winter solstice (longer nights). During the equinoxes, daytime and nighttime are about the same length.

 

Now that you know how two numbers—92 million and 23.4—explain the seasons, let’s get back to spring in particular. As Figure 3 shows, there are two days each year when Earth’s tilt neither points toward nor away from the sun. Those two days, called the equinoxes, divide the warmer months from the colder ones. And that’s exactly what happened on March 20th: we passed the spring equinox.

Before you go, I have a little confession to make. It’s not entirely true that just two numbers explain the seasons. Distance to the sun and Earth’s tilt are arguably the most important factors, but other factors—like our atmosphere—are also important. But that would’ve made the title a lot longer. And anyway, I would’ve ended up explaining those factors using more numbers. The takeaway: math is powerful, and the more you learn the better you’ll understand just about anything [6].

 


 

[1] The ratio of a circle’s circumference to its diameter, pi is a never-ending, never repeating number. It is approximately 3.14.

[2] Venus is the exception. Its thick atmosphere prevents the planet from cooling.

[3] Here’s the explanation for the mathematically inclined. In calculus, changes in a function are described by the function’s derivative; the derivative of T is T'(r)=-kr^{-3/2}/2. This tells us that for a small change dr in r the temperature change dT is (-kr^{-3/2}/2)dr. Relative changes are ratios of small changes in a quantity to its original value. Thus, the relative change in temperature, dT/T, is

frac{dT}{T}=left(-frac{kr^{-3/2}dr}{2}right)left(frac{r^{1/2}}{k}right),

which is minus 0.5 times the relative change in distance, dr/r. The minus sign says that the temperature decreases as r increases, confirming the results of Figure 1.

[5] Just like in our thought experiment, the Southern Hemisphere’s seasons are swapped with our own; when one is cold the other is warm and vice versa.