Syllogisms seem to work by relating things first in a more general way and then in a more specific way. The relationship of the general and specific supports the argument by either agreeing or denying the relationship.

Probably everything Barr says can probably be construed to somehow relate to geometry but before going to far into it, I found the discussion starting on page 5 about how something can still be alike even when its been taken apart and reassembled so long as the order is still the same. Not sure why it bothers me, perhaps its because I think of it in terms of geometric operations of stretching and for for cutting and reattaching there isn't such a neat geometric operation.

Again, probably everything can eventually be construed to relate geometry to architecture (at least the parts of it we like) but for the sake of discussion I think one area where it doesn't quite relate is when architects design an effect or environmental change without necessarily designing how it can happen.
If for example: if an architect may only be concerned with the temperature of the room insomuch that it is habitable by people, they wouldn't care about the geometric arrangement of the room, or mechanical systems (geometrically designed to perform) necessary to produce a specific temperature, just that the effect is achieved.