Actually, I don't think this is true. Your tower has to taper exponentially, which means there's characteristic length scale L (probably in the dozens of kilometers) such that cross-section has to double every time you move that far down the town. The width of the tower that is 1 cm at its peak exceeds the Earth's diameter (12,700km) at its base after only 60 doublings, which happens at a height of 6,000km if L=100km. (Obviously, things get more complicated as the height becomes comparable to the distance from the Earth, but the order of magnitude should be right.)
Also, this only works if, for any given slice, the weight of the legos above is evenly distributed over the legos below. Without stronger materials to transfer this weight, the legos in the center of the bottom of the tower will fail before those on the side of the bottom.
I think it only has to get wider quadratically, not exponentially. And the weight doesn;t have to be exactly evenly distributed at the bottom, all that's necessary is that some of tghe weight of the upper cneter bricks is suppoorted by the lower outer bricks.
> I think it only has to get wider quadratically, not exponentially.
No. The total mass above a distance H from the top of the tower is
M(H) = \rho \int_0^H a(h) dh
where a(H) is the cross-sectional area of the tower a distance H from the top, and \rho is the density of the tower material. This total mass must obey
M(H) = a(H) * r / g
where r is the force per unit area that the material can support and g is the acceleration of gravity. Setting the right-hand sides of the two equations together and differentiating by H gives
r/(g \rho) (d/dH)a(h) = a(h)
which means
a(h) = exp(h (g \rho/r))
> And the weight doesn;t have to be exactly evenly distributed at the bottom, all that's necessary is that some of tghe weight of the upper cneter bricks is suppoorted by the lower outer bricks.
The distribution problem gets worse and worse as the taper continues, because more and more of the new area is further away form the center.
Also, this only works if, for any given slice, the weight of the legos above is evenly distributed over the legos below. Without stronger materials to transfer this weight, the legos in the center of the bottom of the tower will fail before those on the side of the bottom.