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
