Is it better to build a space elevator from GEO down to the surface of the Earth? Having just finished Arthur C. Clarke's "The Fountains of Paradise", Clarke seems to make a distinction between starting construction of a space elevator from geosynchronous orbit and working on our way down until we hit a point on Earth and constructing starting from Earth and going up.
My question is, is there a difference between the two? Theoretically speaking, is starting from GEO and coming down different or even easier than starting from Earth and just building a really tall building and going up? It seems to me that starting from the surface and going up might be easier. We don't even have to restrict ourselves to the equator. Is there some fundamental difference or a deep physical reason why one works better than the other?
Building a really really tall building/space elevator (36,000 km up) from the surface seems to work just as well because the top of the building/elevator will be "in sync" with its base on the surface, no? Effectively behaving like geosynchronous even away from the equator at higher latitudes? Its not like the top will be "moving" at different speeds than its base hence the elevator breaks. I thought only free-bodies such as satellites will have different speeds depending on their height in orbit because they are constantly "falling". If something is connected to the surface via a rigid solid body then it will move at the same rotational speed as its base point on the surface of the Earth.
Thanks you.
 A: Building up from the ground is essentially impossible. One reason is stability, as John Rennie points out in his answer, but a far more fundamental reason* has to do with compressive strength. When a space elevator is completed, the cable is under a lot of tension. However, if you build up from the ground then while you're building it it's under compression most of the time, because there's nothing "pulling" it from the top. Materials with the tensile strength required not to snap when the elevator is complete are just about in the range of plausibility, but compressive strength is much harder, and it's quite likely that no material with sufficient compressive strength to build a tower thousands of kilometers tall will ever exist.
Building from GEO downwards solves this problem by having the cable under tension the whole time, so you just don't need to worry about compressive strength. I suppose in theory you could build upwards and keep it under tension by pulling the top upwards with a massive rocket the whole time, but the energy requirements for that would be ridiculous. Building from the top down is a lot more plausible.
Building upwards away from the equator won't work, because the centrifugal force is directed in a direction parallel to the plane of the equator (perpendicular to the Earth's axis of rotation), so although a space elevator under tension could exist away from the equator, it would point away from the ground at an angle, not vertically upwards. If you tried to build upwards (assuming you have a material with the impossible compressive strength required) then once it got tall enough the centrifugal force would start pulling it sideways and it would fall over. If you did want to build an elevator far from the equator, the best way would probably be to build it at the equator at first (from geosynchronous orbit down) and then slowly move the base until it's where you want it.
* I say compressive strength is "more fundamental" because, if you did have a material with high enough compressive strength, the stability problem could conceivably be solved with guy ropes, or by attaching rockets to the structure to keep it stabilised. However, the compressive strength issue can only be solved by having a material strong enough or by having the cable in tension.
A: The trouble with building columns is that they are fundamentally unstable i.e. if you perturb them away from the vertical the perturbation grows and the column falls over. This problem is worse the higher the column relative to it's base, so for very high columnar structures you need extra butressing to keep the column stable. By contrast perturb a suspended cable away from the vertical and it will simply fall back to vertical again. For this reason if we ever manage to build a space elevator I suspect it will be built from orbit down towards the ground.
