Kotsopoulos S, Ph.D. Dissertation, Massachusetts Institute of Technology, 2005
II. Shape Computation Theory
1. Introduction
2. Computational Theory
3. Computational Design Theory
4. Shape Computation
4. 1. SHAPE CALCULUS
4. 2. SHAPE GRAMMAR
5. Constructing Design Concepts
Shape computation theory examines formal methods and tools that can be used in design.
This research focuses on shape computational methods that can be used in the studio, in designing from scratch.
The suggested design process consists of making a hypothesis (design concept), in response to a problem,
deriving its consequences, and then testing them against the available empirical standards. It is proposed that a
design concept is not arrived at by an analysis of the provided information, but it is the result of synthesis and
interpretation. It does not only express programmatic facts for the object under consideration, but also suggests a
possible new meaning for it. A design concept cannot be qualified as either true or false. The role of the design
concept is to establish a particular interrelationship among the elements that a designer identifies as crucial for
his design. Design concepts can include spatial as well as other parts: semantic, functional, etc.
In the design process one examines the consequences of one’s initial hypothesis. Deductive steps with varying
degrees of explicitness and extensiveness are used for this purpose. The general consequences of a design
hypothesis can be sketched out by rule schemata S1, S2,…Sn established from previous experience or invented.
S = {S1, S2 … Sn}
Rule schemata are general statements containing at least one free variable. They include predicates and
transformations. The formulation of a set of transformations T under which the same rule schema may apply
becomes a parameter of great importance in the development of a design.
T = {T1, T2 … Tk}
A first approach can be established by organizing rule schemata according to their general consequences. The
consequence Cj of a sequence of j < n rule schemata is what is implied by their conjunction
S1 and S2 and S3 and … and Sj ----> Cj
The above expression does not guarantee the value of the consequence Cj, which can be a matter of several
interdependent parameters. But it underlines the conditional character of the entire system: some effect Cj is
accomplished provided that some schemata S1, S2, S3,… Sj are satisfied.
Moreover, while a part of the design activity consists of formulating rule-schemata and transformations, another
part is dedicated to the specification of the particular actions. These are expressed as shape rule instances R.
A shape rule of the form R1: A1 ----> F1 determines a logical condition: if the shape A1 is found in a derivation it
can be substituted with the shape F1.
A1 ----> F1
A1
-----------------
F1
This development is applicable to both spatial and non spatial attributes of a design concept. Rules like the above
correspond to sufficient but not necessary conditions.
As a sequence of additions or subtractions would never lead by itself to the discovery of a theorem, mechanical
rule application would not lead to a design. Unless a hypothesis has been put forward, such application will lack
direction. Formal rules are not rules of discovery, leading mechanically to solutions. They only provide criteria for
checking the results of proposed actions with respect to a hypothesis.
Alternative rule instances R can be produced by substituting the free variables in a rule schema. On the base of a
sequence of rule schemata S1, S2,…Sn rules R1, R2,…Rn can be introduced as instances of the rule schemata.
The application of the rules has some final outcome G.
S1 R1
S2 R2 ==> G
: :
Sn Rn
The question of constructing a system, or a grammar, arises as soon as a number of general rule schemata, and
rules, are established. It then becomes possible to arrange them with a better sense of economy and efficiency.
Provided that the search has been carefully done, the ordering of rules does not itself create new information.
Assuming that some number n of rules is to be organized the question becomes under what ordering relation?
The final ordering of rule instances into a system is subjective and happens according to their ability to achieve
specific goals. The ordering assures that all the desired goals will be accomplished at the end.
Therefore the shape rules R may take the form:
R: { { [A1 ---> F1],…[Ai ---> Fi] } and { [G1 ---> M1],…[Gk ---> Mk] } and { [N1 ---> W1],…[Nr --->Wr] } }
The set S of elements such that { [ A1,…Ai ]…,[ F1,…Fi ] }, { [ G1,…Gk ]…[ M1,…Mk ] }, { [ N1,…Nr ]…,[ W1,…Wr ] }
are in S, is defined retrospectively.