In Urban Design there is a very useful plan called the Figure Ground Drawing. It's usually black and white and is at the scale of roads and buildings - the city. The buildings are black (figure) and the background or space between is white (ground). This type of drawing shows density to open space ratios as a graphic abstraction. I like this drawing because it represents the physicality of an urban situation from an external perspective. Much like walking around the spaces of a city - the buildings become objects - their interiors separated into another world.
The Euclidean notion of the line has length but zero thickness. This fits well with the Figure Ground Drawing as it could be said the boundary between the black and white is a line without any thickness. It also fits well with our experience of the city as a collection of objects with space between. And it could be stretched further to say an object (a building) has interior space and exterior space - the boundary between the two being a zero thickness Euclidean line.
One of the motivations of Modernist Architecture was to breakdown the idea of internal and external space. Ultimately to express Architecture as the subject rather than the object. Consciousness as the subject. No separation - all as one. Below is a Modernist Figure Ground Drawing.
Space takes a step forward and the figure (object) is transformed into a line. Ironically Euclid's line is manifested by giving it real thickness - the wall. In Classical Architecture the wall doesn't exist - there is only the zero thickness boundary between object and space. And this is the paradox I felt on my first visit to Mies' Barcelona Pavilion. The very act of eliminating the boundary between the inside and out concretised the impenetrable nature of form. Transparency is rendered opaque by removing the illusion of the internal.
My design for the National Wildflower Centre took the impenetrable Miesian wall as figure and found space within its depth. The internal is sculpted out of the very fabric of the illusory line. A return to the Classical. An object with an interior. The zero depth line re-emerges as the boundary around the (Miesian) wall.
It seems there is no way out. The interior is just an illusion.