The Wonderful, and Wonderfully Misleading, Powers of Ten

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I was lucky enough to study in my first year of college under the designer/architect Charles Eames, inventor of the molded plywood chair (among many other things). He showed us the first draft of a movie he called “Powers of Ten” which zoomed in and out in a sequence of images depicting what you would see if every ten seconds you multiplied the distance by a factor of ten. The movie moved outward to galactic scales, then inward to atomic particle scales, covering 40 powers of ten. On the trip out, the first version had a neat clock that showed you what percent of the speed of light you were moving. This was dropped from the version referenced.

Eames remade the movie in 1977, advised by a number of scientists including Philip Morrison, a physicist who also narrated the film. You can find it at http://www.gearthblog.com/blog/archives/2012/02/the_powers_of_ten.html

Philip Morrison is one of my heroes. He worked on the Manhattan Project and could read a substantial book in a day. He wrote the book reviews during the golden years of Scientific American when Martin Gardner was writing “Mathematical Games” and the magazine featured beautiful pointillist pencil drawings.

The link takes you to the Google Earth blog, unsurprisingly, since Google Earth uses the same technique to zoom you in to your chosen site (except it “only” zooms by 7 powers of ten, i.e. ten million times). Another term for powers of ten is order of magnitude.

Powers of ten is a way of thinking about exponential growth. The concept is a two-edged sword. One edge is an indispensable tool for understanding nature; the other is among the most misleading concepts in modern life. “Powers of Ten” allows us in just a few minutes to visualize the smallest thing we have direct evidence for (quarks) and the largest (super-clusters of galaxies), over 40 powers of ten.

Thinking in terms of powers of ten is routine for mathematicians, scientists and statisticians. But in our daily lives things grow by adding up. We think and act linearly, 1, 2, 3, 4, 5, 6, 7, 8, and so on. Thinking exponentially you get 1, 2, 4, 8, 16, 32, 64, 128 and so on. Already in just eight steps, powers of two has outdistanced our thinking by a factor of eight. Using powers of ten, you get 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000, 10,000,000. Eight steps and you go from an individual to nearly the population of New York City.

This huge disparity in modes of thinking can be seen more clearly using examples.

There is a famous Chinese story about a poor man who was granted a wish from the Emperor. He asked that he be given one grain of rice, but added the stipulation that on each successive day, he would receive twice what he had received on the previous day (powers of 2, or doubling). In a month he owned just over a billion grains (230) and it wasn’t long before he owned all the rice in China.

Another often-cited example is lilies on a pond. Let’s say your pond is 750 feet in diameter, which comes out to 10 acres – nice big pond. Let’s assume a lily takes up 1 square foot, and that the number of lilies doubles every year.

You start with one lily, hardly noticeable. After sixteen years, you have just the lily patch you were looking for, covering about about 3/4 of an acre. The next year, however, the lilies cover one-and a half acres and you begin to worry. The next year they cover 3 acres, the year after that 6 acres, and before the end of the next year the entire pond is covered. So it took 16 years to get where you wanted, then less then 3 years to obliterate the pond. This illustration from the wonderful YouTube channel XKCD illustrates the point humorously.

exponential_curve

Some wonderful examples that translate a huge power of ten (67 orders of magnitude) can be found illustrated at https://www.youtube.com/watch?v=ObiqJzfyACM on one of my favorite YouTube science channels, “Vsauce” narrated by Michael Stevens. The whole 20 minute episode is worth watching, but the examples illustrating a huge power of ten starts at minute 14.

Finally, to illustrate the futility of manned space flight, consider this. It is hard to get a grasp on the sobering fact that the universe is almost empty of matter as we experience it. Of the three components of the mass of our universe, ordinary matter accounts for maybe 2-3 percent, the rest being mysterious “dark matter” and the even more ubiquitous and mysterious “dark energy.”

Our solar system, which compared with outer space is as crowded as a subway platform, is terrifyingly empty. I made a conceptual model of our solar system at a scale of 1 billion to one, set in a place familiar to many. Here is an illustration showing the model:

Manhattan small

 

Imagine that the sun is a ball about four and one-half feet in diameter, on a pedestal in the Grand Army Plaza in front of the Plaza Hotel at 59th Street and 5th Avenue in Manhattan.

The orbit of Mercury, the inmost planet, crosses 5th Avenue a block away, at 60th Street, and is about half the diameter of your little fingernail. Venus at 61th Street and Earth near 62nd street are about the size of your middle fingernail. Our moon is half the size of a pencil eraser and is 15” from the earth.

Mars, another block away, is a little bigger than Mercury. Then there is a gap where the asteroid belt occurs, with the largest asteroid being virtually invisible, the size of a fine grain of sand. Outside the gap is the first of the “giant” gaseous planets, Jupiter, which in this model is the breathtaking size of a large grapefruit, with an orbit that crosses 5th Avenue around 70th Street. Another 7 blocks gets us to softball-sized Saturn. At 95th Street we encounter the orbit of Uranus, which like Neptune at 115th Street is 2” in diameter.

In this model, the orbit of Neptune, now the outermost planet, swings out over the Hudson and into Union City, NJ, then back across the river and down Houston Street, across Brooklyn and Queens passing through the Sunnyside Railroad Yards before crossing the East River back into East Harlem. This is plotted on another Google Earth image:

Manhattan large

Now that you have a sense of the sizes of these bodies and their orbits, try to imagine away everything else: the earth below, New York City, the sky above, and replace it with black emptiness. Just a 4-1/2 foot ball of fire surrounded by tiny spheres slowly orbiting in one plane, the outermost of which is almost 3 miles away and the size of a golf ball. And this is a high density of matter by the standards of the universe.

Within this disk of orbiting motes 5 ½ miles in diameter we humans have traveled the distance from your elbow to your fingertips. Mars is 150 feet away, while the nearest star at this scale is the distance of a trip around the earth, 25,000 miles. We talk a good game about fleeing into the galaxy when the going gets too tough here on earth, but the facts are starkly clear: we will forever need to cling to the nurturing surface of our tiny speck in the vast emptiness that surrounds us.

It pays to keep our nest in good shape, as it is all the home we will ever have.

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