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shminux comments on Does model theory [psychology] predict anything? (book: "How We Reason" (2009)) - Less Wrong
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Warning, spoiler ahead
A somewhat more mathematical version: assuming small number of radians n from either the pole or the equator.
For pins, the area of the (unit) sphere is pi*n^2 for the pole and 2pi*n for the equator, with the ratio n/2 for pole/equator area, which goes to zero as n goes to zero.
And as you said, for a disk "near the pole" means its normal is near the equator, so the relevant area is 2pi*n, "near the equator" means its normal is near the pole, so the result is the opposite.
An interesting aside is that disks and pins in 3D are actually Hodge duals (that's the reason torque or magnetic field can be treated as vectors sometimes).
Yes, that's how I visualized i, although it wasn't completely obvious to me why it was correct to do so. I suppose a more obvious, but unwieldy, visualization is marking two special spots (not colinear) along the perimeter of the disk and imagining how they could be animated within a sphere as the disk rotates.