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Okay, I was too interested and had to do a quick calculation...
GPE down the slope:
Rider is k above the minimum position on the bike (k is one crank length - cranks are parallel with the ground)
Bike and rider at minimum position on the slope
GPE_start = m_rider * g * kGPE up the slope:
Rider at minimum position on the bike (cranks perpendicular to the ground, riders weight on lower pedal)
Bike and rider distance d up the slope
GPE_end = (m_rider + m_bike) * g * d sin(theta)d = k / [(1 + m_bike/m_rider) * sin(theta)]
d is (metric) gear inches / 4 - we only considered a quarter revolution of the cranks
For k = 0.175m, m_bike = 10kg, m_rider = 75kg, theta = 20deg, we get d = 0.451m or 17.8 inches
So that's 71.2 gear inches, which is bang on a 43/16 on a 25mm, 700c tyre, a fairly typical fixie gear ratio! How satisfying!
To cycle up a 45 degree slope with the same bike weight, cranks, etc., you only need a 34/26!
Of course I'm ignoring drivetrain losses, loss of traction, what the upper foot is doing, the use of your muscles, etc.
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frankenbike
So, my guess (and I've been drinking) is that the limit is when the vertical component of the propulsion (rider weight on the pedals causes bike to move forwards and up a slope) is counter balanced by the gain in gravitational potential energy from moving up the slope.
Lower gear = less forward motion (and therefore a smaller vertical component).
ICBA with the maths right now, will see if being sober in the morning helps.