Tuesday, February 26, 2008

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.


L!N said...

I think one of the most fundamental flaws in architectural education (and common practice) is that representation is accepted as a "standard" for things are done, and orthographics are accepted with unwavering zealous. Such, I suppose, is the business aspect of a profession.

To characterize projections as but Euclidean geometry is not entertained. Additionally, the fundamentals of any curriculum does not sufficiently offer the students even the opportunity to investigate theories behind common practice.

I think all options would be clear, if all individuals were offered sufficient background in mathematics.

thefretbreaker said...

all right angles are congruent? that can't be right! wait...my understanding of congruency is that the shapes would be identical when superimposed. A 45 degree right angle triangle and a 30-60 right angle triangle would not be congruent, right?

My only concern on experimenting with new methods of representation is that they tend to turn (ha! alteration at its worst) into an art of representation, and they shy away from the initial essence of the idea, the creation of the space. I think that throughout time, the methods of representations derived out of their needs and priorities. Architects in the arab world didn't care about cosmic order as much as Renaissance architects did, they cared more about orientation and articulation between private and public spaces; information not necessarily conveyed in drawings. The artisan work comes after the fact, due to a need to employ local craftsmen.
I do agree that one needs to question and rethink the design process, which representation happens to be a major ingredient in. However, I think that this rethinking process needs to come out of a need to address a certain outlook on modern society and not just a change of representation for representation sake. I would say the ShoP architects are the best example of architects who understand the technological shifts that are happening in the world today, and having shifted their representational methods in order to better communicate with frabricators, manufacturers...etc.

L!N said...



"right angle" != "right TRIangle"


Triangles with angles adding to a sum of 180 degrees are so two-D!!

Luc Wilson said...

This is something that is intuitive and seems obvious but if you are starting with nothing you have to assume something as true to be able to create a system. But you can an image the incredible realm of possibilities for creating new systems of geometry if you assume something as basic as “all right angle are congruent” is never true.

Zak said...

The LTL work you're referring to is some of the most compelling, and successful architectural representation I have ever seen. However, they still rely on orthographic drawings to get their buildings constructed. I wonder if our whole profession could be challanged to merge more with the construction industry and the highly skilled craftsman that were so highly saught after during times such as the Gothic era. It we could all express with clarity our intentions as well as a firm such as LTL, could we simply present a series of perspectives, or *gasp* even an animation to a builder and have them simply construct a building from there, leaving them to come up with the details? This would be an interesting step, however, it sort of gives more credit to some of these design-build firms such as Shop Architects.