VACETS Regular Technical Column

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October 3, 1996

Cubic Planet Earth

I once read a science fiction novel about our earth far in the future. Due to ecological reason, the earth was modified from its spherical shape into a cube. On such a planet, there would be no flat land any where except the center of each square surface. If you were standing at the center of one face of the cube, you would be on level ground, but if you were standing near an edge you would be on a 45-degree hill. Thus, an object placed near one edge would roll or slide back and forth along the face, even though the surface would look perfectly flat.

I suppose that the people in the future would have the technology to move huge amount of dirt from one place to the other to make the earth cubical. But, would nature allow that?

Let's imagine a cubic earth with each side of 12700 km (about the diameter of the earth, i.e., the today earth's spherical body would just fit inside a cubical earth). The spherical body would define "sea level" and 8 corners of the cube would be the peaks of 8 huge mountains. Using some simple geometry, you can show that the distance from the sphere's center to one corner of the cube is 173.2% (i.e., square root of 3) of the sphere's radius. This means that a corner of the cube, or the "peak" of the mountain, is 73.2% of a radius above sea level (the surface of the sphere). This corresponds to a 4648 km height mountain. Is it possible to have a mountain that high on earth (or any where in the universe)?

Before we get to the answer for that question, let's examine some true mountains on earth. Everyone knows the name of the highest mountain of the world. It is Mount Everest, whose height is almost 9 km above sea level and is located in the Himalayan Mountain range, exactly on the border between Nepal and Tibet. Nine of thirteen mountains that are higher than 8 km are in the Himalaya Mountain Range. However, the height of a mountain depends a good deal upon the height of its base. The Himalayan mountain peaks are by far the most majestic in the world; there is no disputing that. Nevertheless, it is also true that they sit upon the Tibetan plateau, which is the highest in the world (about 4 km). If we subtract 4 km from Mount Everest's height, we can say that its peak is only about 5 km above the land mass upon which it rests.

Is this standing to reason? Yes, it is. Suppose you had a mountain on a relatively small island. That island maybe the mountain, and the mountain wouldn't look impressive because it was standing with its base in the ocean depth. If the oceans were removed from earth's surface, then this unimpressive mountain might become a giant among the giants.

By this new standard, i.e., from base to top instead of sea level to top, are there any mountains that are higher than Mount Everest? Yes, indeed, there are. And the champion is Mauna Kea in Hawaii. This mountain is 4.2 km above see level. However, if one plumbs the ocean depths, one finds that Mauna Kea and the whole island of Hawaii stand on a land base that is over 5 km below sea level. By the new standard (base to peak), the height of Mauna Kea would be almost 10 km. It is indeed impressive.

So now we have Mount Everest as the tallest mountain on earth, almost 9 km measured from sea level to peak, and Mauna Kea as the tallest mountain, almost 10 km from base to peak. Is it a matter of chance that the highest mountains on earth are somewhat slightly less than 10 km, or could they just as well have been 20, 30, or even 100 or more kilometers high? Actually, there is a limit to the maximum height of mountains, based on the strength of materials from which mountains are formed. The rock underneath a mountain is subject to tremendous pressure from the weight of the overlying rock, which would be enough to crush or liquefy the base if the mountain was tall enough. The maximum height possible for mountains on earth is probably not much more than 10 km, as indicated by Mount Everest or Mauna Kea and by the fact that the earth's near liquid mantle is as little as about 10 km below the surface.

The maximum height of mountains depends not only on the strength of materials but also on the strength of gravity. The tallest mountain we know of in the solar system is Mars's 24-km-high volcano Olympic Mons. The fact that Olympic Mons is about 2.6 times as tall as earth earth's highest mountains correlates with Martian gravity being weaker than earth's by roughly this same factor. We don't know about the mountains on the gas giants of the solar system, but due to stronger gravity, the maximum height of those mountains must obviously be correspondingly less than those of the earth.

Now, let's get back to the possible of a cubic planet earth. As we just saw, the highest mountain on earth (Mount Everest or Mauna Kea) is slightly less than 10 km. It is no where near the 4648 km peaks of the 8 corners of the cubic earth. We also know that a mountain on earth can not get much higher than 10 km before its base is crushed or liquefied under tremendous pressure. So the idea of building a cubic earth with eight 4648 km high mountains can not be done.

Then, what is the largest cubic planet can we build? Let's assume that the tallest a mountain can have on earth is 10 km. We know that the possible height of a mountain depends on the planet's gravity and the strength of gravity on the surface of a planet is proportional to its radius, assuming a fixed density. In other words, on a planet half the size of the earth, gravity is half as strong, and mountains can grow twice as tall. These mountains can be said to be 4 times as tall as earthly mountains, when their height is expressed as a percent of the planet's radius. So for a planet that is about 0.0464 the size of the earth then its mountains can be 1/(0.0464*0.0464) = 464 times taller than those on earth when the height is expressed as a percent of the radius. This number 464 is about the same as the ratio of the 4648 km high mountains of the cubic earth to the maximum height of 10 km of the spherical earth. So the largest cubic planet that can be built will have its sides equal to 0.0464*earth's diameter = 0.0464*12700 km = 589 km. Please also note that a 130 lb. person would weight only about 6 lb. at the center of a face of that cubic planet and about 4 lb. at the edge. Wouldn't you like to live in such a planet?

Duc Ta Vo, Ph.D.

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Copyright © 1996 by VACETS and Duc Ta Vo

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