Lol, gahd, latex is no harder then html, so if you need help figuring it out, gimme a pm, i'll put some material together if you are having trouble
Gahd, you probably should get a small brushless motor with good response and gear it down, just direct gear it and you will get very little loss, most electric cars, including the X1 are geared, its a single gear yes, and they still claim direct drive...
Things to consider:
batteries, like the crazy lithium ions used in the X1 prototype, they are biodegradable, which is cool, but also, they provide a ton of power for their size, and they should fit in the frame of your bike, no prob
you should consider weight, or rather lighten the bike as much as you can, the lighter you get it, the more efficient it gets (but hell you would know that). You can probably make a huge reduction in weight (relative ofcourse to the bike's weight) if you take off the pedals, and the rear wheel gears and change to a much more simplistic hub. Also decrease the rear tyre thickness to a road tire, and also run road tires on the front, on road (and i am assuming you are building this contraption as an alternative to scooters) you get better traction and smoother roll with a smoother tire. Another thing to think about are spokes, they are usually steel, you can switch out for some aluminum ones to once again, save on weight.
a small, but highly efficient (well as efficient as you can get it) solar charging system will extend your range on a sunny day, but do consider that you will want to plug the sucker in to charge it.
also i think simpler is better, i saw your first thought on solenoids and perma magnets, and some other designs, think about building everything into the wheel, just keeping the power source outside (whether attached or inserted into the frame is up to you (this is for better weight distribution), then you have to design 2 things, a motor-fork linkage, because a part of the motor will have to be kept in place to rotate the wheel, and a wheel, at this point you would probably be better off with a non-spoked wheel, that will fit your bike (you can make it more or less standard). with a simple hub design, you should be able to put a gear on an axle (which you will have to put in the hub, though nothing too crazy here), and a shroud, to protect both motor and the drive mechanism from the environment.
I dont think this project is undoable, infact if i were any closer, i would definitely volunteer to help with building a prototype, but as i said, i tend to think, at least in mechanics, simpler=better, well, most of the time, more efficient. But dont confuse "simpler" with "too simple", i'm the guy who takes a "too simple" idea and after making it "simpler" by adding parts, figures out better linkage for a height-adjustable wind turbine (linkage worked in my mini model, but the guy ended up getting a standard windmill, but whatever, i got the concept, and it would be cool to build :P )
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Microsoft, the leader in using innovative tactics to promote irksome experience, coupled with antiquated technology that's held together by a pyramid of makeshift afterthoughts.
Apple, the leader in using irksome tactics to promote innovative experience, coupled with an antiquated core that's enhanced by state-of-the-art afterthoughts.
Linux, the leader in not using any tactics to promote user-defined experience, coupled with state-of-the-art core enhanced by innovative afterthoughts.
