Alberto Perez-Gomez and a brief history of geometry

Alberto Perez-Gomez and Louise Pelletier, in Architectural Representation and the Perspective Hinge, put in to question architects unwavering faith in the conventional set of projections (plan, section, elevation) that have been used to represent the idea of a building. He points out that they are symbols for a building not the actual building. “For architects it is important to remember that a symbol is neither a contrivance nor an invention—nor is it necessarily a representation of absolute truths or transcendental theological values.” It is clear that the conventional set of architectural projections is neither arbitrary nor a given.

This idea of an (un)stable foundation of conventional architectural representation can be put in relation when compared to the foundations of Euclidean Geometry. In Elements, thirteen books on geometry, Euclid defines an axiomatic system in which a finite set of axioms are take as true (without proof) and all theorems are proved from the initial set of axioms. We cannot prove anything with out accepting something as true or given. Euclid’s five axioms are:
1. Any two points can be joined by a straight line.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. Parallel postulate. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. (essentially, given a line M and a point P not on that line there exists only one line that goes through P and never intersects M.)

He also included 23 definitions such as point, line, and surface in addition to 5 “common notions,” which included, "things which equal the same thing are equal to one another,” and “if equals are added to equals then the sums are equal.” These are concepts that we have to accept as true if we are going to prove anything. Accepting these basic ideas and definitions as true, Euclid proved the theorems that appear in Elements and formed the basis for geometry for the next two thousand years.

Understand Euclidean geometry allows us to deconstruct conventional architectural representation. In architecture we have accepted projection and associated definitions as givens and proved our whole system of representation based on these assumptions. So what happens if we don’t accept the stated axioms as givens? In the early 19th century mathematicians began to question the parallel postulate. Could it be derived in terms of the other four axioms (and hence wasn’t an axiom but a theorem)? Were there theorems that had been derived that didn’t rely on the parallel postulate? It turns out that the first 28 theorems he proved are not derived based on the parallel postulate. This questioning of the fundamental axioms of geometry has led to new geometries. For instance, if we take the parallel postulate as false we get two new axioms, either 'there exists an infinite number of lines through P parallel to M' or 'there exists no lines through P parallel to M.' The former results in hyperbolic geometry and the latter in elliptic geometry.

So what happens if we don’t accept the basic foundation of conventional architectural representation as true? What if we think of them as false? What other systems of representation can be derived?

The most important idea to remember is that treating an underlying principal as false doesn’t necessarily give us the opposite of it. We get systems that are derived from various rule sets. For instance, Gothic architecture “operating though well-established traditions and geometric rules that could be applied directly on site,” was derived from an entirely different system of architectural representation than today. Conversely, contemporary architects are redefining rules within the conventional rule set. This still produces new systems of representation, but ones based on some of the same rules, similar to the Euclidean/Hyperbolic/Elliptic geometry relationship.

Lewis Tsurumaki Lewis has taken the generally accepted rule of ‘conventional architectural representations (plan, section, axon) need to remain separate to convey information’ and negated it, “conventional architectural representations need to be combined (ex. orthographically-projected-plan-sectioned-isometrics) to convey information.’ They have defined a new system based on the traditional conventions of representation and the redefinition of one of those rules. Additionally, SHoP architects have redefined architectural representation based on efficiency resulting in axonometric construction drawings and a general abandonment of orthographically projected architectural representation.

Now we are faced with the challenge of inventing new systems and conventions of architectural representation that are informative and appropriate to contemporary techniques and practices. Do we negate the traditional conventions and start over? Or do we return to older systems, such as the gothic tradition of building? The appropriate approach to rethinking the traditional system of architectural representation is to redefine specific axioms of convention that allow us to construct new systems out the fragments of tradition.

Lynn, go back to math class

In Animate Form Lynn co-opts math terms to define new terms for discussing animation and digital process. He is generally right, but not in precise ways, and sometimes wrong. For instance, he defines a type of animate form as “topologically entities” and discusses the implications of “topological form.” There is no topological form. Topology is field of geometry that is concerned with determining when things are the same. Determining when two things are exactly the same is nearly impossible and not practical to study. In topology two things are considered the same when there exists a homeomorphic map between them. Simply (and not rigorously), this means that two things are the same when you can deform one into the other with out adding or taking away any holes. Lynn takes the term topology from this idea, using it to describe surface manipulation. Not quite right.

Lynn further describes the temporal component of topology as, “the immanent curvatures that result from the combinatorial logic of differential equations.” This makes absolutely no sense. Simply, combinatorics is a type of pure math concerned with counting. A differential equation is an equation or an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders. (wikipedia) One is a type of equation in calculus and one is a branch of pure math. While there are relationships between the two, I’m not convinced Lynn knows what he is actually saying. He is taking combinatorial to be synonymous with combination. I think he really means the immanent curvatures that result from the combination of differential equations.

Lynn needs to either describe (correctly) why and how he is using math terms or not use them at all.