Sir Arthur's magic orbit

As objects in space get further from the mass they are orbiting around, the orbit takes longer. The space shuttle and other objects in low earth orbit, can go around in about 90 minutes. The moon takes about a month to complete an orbit. Arthur C. Clarke, author of "2001, a Space Odyssey" noticed clear back in 1945 that we could take advantage of this fact to do a few special things with satellites. He noted that in an orbit at about 35,800 kilometers out a satellite would take 24 hours to go around the earth. Since the earth rotates in 24 hours, a satellite in orbit in the equatorial plane would appear to be stationary in the sky. This is the geosynchronous or Clarke orbit. If a satellite is in such an orbit a very narrowly focused telescope or antenna can be pointed at it without it leaving the field of view. You can use a lot of magnification, or antenna gain, because the satellite appears to stay put. The small carefully pointed antenna dishes used for direct-from-satellite TV work on this principle.
The orbit must be in exactly the equatorial plane or the satellite will appear to bob up and down, so the limited number of parking spots in Clarke orbit are extremely valuable and regulated by treaty. In addition the satellites aren't visible from polar areas due to the bulge of the earth getting in the way. In spite of these minor drawbacks, use of the Clarke orbit has brought worldwide wireless telephony, global data linkage for business, educational programming to poor rural areas worldwide and allowed millions of people to access programs their governments would prefer to censor, in general furthering communication and knowledge throughout the world.

This is magical enough, but recent advances in materials technology have generated new interest in an even more miraculous use that Clarke thought up for geosynchronous orbit. Take a satellite in geosynch orbit. Build towers below and above it, so that its center of mass stays at geosynch. The portion below center is traveling slower than orbital speed, so it tends to fall, pulling down. The portion above center is going faster than orbital speed and tends to fly outward. The system is stable, and is loaded in pure tension, so we can replace the towers with cables and it will remain 'tidally locked' in a straight line perpendicular to the orbit. We can even extend the bottom cable all the way to the surface of the earth, where it will have zero speed relative to the ground. If we anchor the bottom and add a little extra weight on the outer end we have a skyhook, or orbital elevator. We can grab onto the cable and pull ourselves upward to orbit for only the energy cost of doing so - pennies per pound instead of tens of thousands of dollars.!
The catch is that the cable has to be able to support its own weight. The best steel, for example, will break when you hang a vertical length of only 50 kilometers, far short of what we need. When Clarke used this idea in the science fiction novel "The Fountains of Paradise" he envisioned it as being built of diamond, the strongest material then known, and even then had to taper the cable severely - very fat at geosynch, tapering almost to a point at the ends. Constructing and setting the cable would be a nightmare, even if we could make that much diamond. Since a Mars-synchronous orbit requires 24 hours 40 minutes at an altitude of only 20,460 kilometers in the weaker Martian gravity field, it would actually be easier to build a skyhook for Mars than Earth.
We have recently invented a new form of carbon, the nanotube. This is a hollow filament only 10 atoms or so across, but capable of being made in macroscopic lengths. This material has awesomely better strength-to-weight than even diamond -so much better that a film less than a millimeter thick and tapering from 1/3 meter to only a meter wide could span earth-to-Clarke orbit and carry useful loads! There seems to be no reason we can't learn to make nanotubes in the quantities required, and we have boosters capable of carrying an entire skyhook like this to orbit, where we could simply unreel it.
There are people seriously planning on doing this in less than 20 years. The result will be an advance in space travel comparable to having replaced Columbus's ships within 100 years with a 6-lane freeway bridge across the Atlantic.
This seems a good point to introduce another idea that Sir Arthur C. Clarke thought up, known as Clarke's third law:

ANY sufficiently advanced technology is indistinguishable from magic.