R' Micha Berger wrote:
> Because one doesn't see the sekhakh, the tzeil is disconnected
> from the sekhakh. If the connection is psychological (need I
> revive the "ta'am and taste" thread?) rather than physical,
> geometry is irrelevant.

This fits nicely with the way I'm looking at it. The sechach needs to 
be above, and visible so that the connection is made.

But as regards the idea that the shade I'm sitting in has to come 
from the sechach rather than from the walls or from nearby 
buildings...

Some people (such as myself) ask about sitting by the southern wasll 
of the sukkah while in the northern hemisphere. Others (such as R' 
Chaim Manaster) take this further by asking about the sun's 
inclination in arctic regions.

The problem with this approach is that it does not go far enough. We 
have to take it to the logical conclusion. We need a case where the 
sun's inclination is such that the whole concept of "the shade of the 
sechach" has no basis in reality. Like at night!!!

It is not possible to build a sukkah in such a way that a portion of 
it is shaded by the sechach at *night*. There is no way to sit in a 
sukkah at night and be shaded by the sechach. Yet all shitos agree 
that it is possible to perform the mitzvah of living in the sukkah at 
night. Therefore, the mitzvah does not require one to actually be 
shaded by the sechach.

My conclusion is that when the halacha talks about "sitting in the 
shade of the s'chach", this is an idiomatic phrase which actually 
means, "sitting in a place which WOULD be in the shade of the s'chach 
IF the sun were directly overhead."

Akiva Miller