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Mind control at its apex:

The title is a little misleading, because you expect an explanation of how a bike stays stable, but it talks about balancing the rider's weight, which is heavy compared to a gyroscopic effect. I've seen kids learn to ride, and once the bike is rolling fast enough, you can let go of the bike, and they don't freak out, and the bike stays upright with a lot less control of the rider. So the article is a switch from what it pulled you in with.
 

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n00bsauce
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"But people have built bikes with vertical head tubes and they are perfectly rideable, too. In fact, it’s quite hard to make a bike you can’t ride, and many have tried."

That paragraph is a bit misleading. To have any kind of stability you need trail build into the geometry. You don't need "rake" to create trail. If there were 0 degrees of rake and the forks were perfectly straight with the wheel mounted directly under the steering stem (no trail) I can't imagine it would be a very stable ride. It would be 100% up to the rider to make every input the bike needs to stay upright.
 

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Escorted from the White House
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This physics of bike stability video is clear and interesting. And there's more than one effect contributing to keeping the bike upright.


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From the BBC link:


It’s virtually impossible to ride along a narrow straight line just as it’s really hard to walk perfectly along a straight line, even when you’re not drunk.​

As demonstrated by Danny MacAskill doing a fence ride.
 

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Big is relative
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Garage doors keep bikes upright but only if the bike is in the big ring with the drive side crankarm level with the ground.
 
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