Course Meeting Times

Lectures: 1 session / week, 1.5 hours / session


There are no formal prerequisites for this course, though it is assumed that students taking it will have a background in design.

Course Description

Shape grammars are systems of visual rules by which one shape may be transformed into another. By applying these rules recursively, a simple shape can be elaborated into a complex pattern. This course offers an in-depth introduction to shape grammars and their applications in architecture and related areas of design. More specifically, it involves manipulation of shapes in the algebras Uij, in the algebras Vij and Wij incorporating labels and weights, and in algebras formed as composites of these. Discussions center on rules and computations, shape and structure, and designs.


Grades for this seminar-style class are based on the instructor's holistic assessment of students' engagement with the material. There are no specific required assignments or exams. 

Course Outline

"Shape" = Shape: Talking about Seeing and Doing

"Calculating" = Calculating: Beyond Fancy in Imagination’s Magical Realm

For full bibliographic details on these texts, see the Primary Course Texts page. For other readings, see the Lecture Slides and Supplemental Readings page. Except where noted, all readings are by George Stiny.

Units Topics Readings

1. Overview

Shapes and symbols

  • Rules for shapes – this shape → that shape
  • Rules for symbols – description of this shape → description of that shape

The embed-fuse cycle

  • Embedding vs identity
  • Shapes fuse, symbols don’t

Shape grammars in art and design

  • Dimension i ≠ 0 vs dimension i = 0
  • Imagination : fancy :: shapes : symbols

Introduction: "Tell Me All About It" in Shape, pp. 1–59.

“Preface” and “Exhibit 2” in Calculating.

(optional) Stiny and Gips, “Shape Grammars and the Generative Specification of Painting and Sculpture.”

2. Basic Elements

i = 0 – points

i ≠ 0 – lines, planes, and solids

Content and boundaries

Points aren’t lines aren’t planes aren’t solids

Embedding vs identity

Reduction rules for fusing basic elements

  • Embedded elements
  • Overlapping elements
  • Discrete elements

Embedding basic elements


  • Parts
  • Sums, differences, and complements

“Back to Basics—Elements and Embedding,” “Counting Points and Seeing Parts,” “Shapes in Algebras and Algebras in Rows,” and “Boundaries of Shapes are Shapes” from Part II, “Seeing How It Works” in Shape, pp. 159–204.

3. Algebras of Shapes Uij

Difference of dimension i for basic elements vs dimension j for space

Generalized Boolean algebras

Table of algebras and its properties

  • Embedding
  • Transformations

“Shapes in Algebras and Algebras in Rows,” “Boundaries of Shapes are Shapes,” “Boolean Divisions,” and “Euclidean Embeddings” from Part II, “Seeing How It Works” in Shape, pp. 180–215.

“Algebras of Design.”

(optional) “Weights.”

4. Calculating with Shapes and Rules

Formulas for rule application – seeing and doing

Classifying rule applications in terms of transformations

The embed-fuse cycle

Turing machines are a special case of shape grammars

Identity + recursion (i = 0) ⊆ embedding + recursion (i ≠ 0)

Part I, “What Makes It Visual?” In Shape, pp. 61–125 and 130–158.

Paragraphs: “How Rules Work When I Calculate,” “Trying it Out,” “Spatial Relations and Rules,” “Classifying Rules with Transformations,” and “Classifying Rules with Parts” from Part II, “Seeing How It Works” in Shape, pp. 228–275.

(optional) “Turing Machines and Shape Grammars.”

5. Computer Implementation for Linear Elements

Elements and Embedding

“Classifying Rules with Transformations,” “Classifying Rules with Parts,” and “How Computers Do It” from Part II, “Seeing How It Works” in Shape, pp. 260–277.

6. Schemas for Art and Design

Parts, transformations, and boundaries

Subsets (for example, identities), inverses, copies, sums, compositions, and Boolean expressions

“What Schemas Show” from Part I, “What Makes It Visual?” in Shape, pp. 126–130.

“I Don’t Like Rules, They’re Too Rigid” from Part II, “Seeing How It Works” in Shape, pp. 277–282.

“Design is Calculating,” “Tell Me What Schema to Use,” “What the Thinking Eye Sees,” “Chinese Lattice Designs—Seeing What You Do,” “They’re Shapes Before They’re Plans,” “Getting in the Right Frame of Mind,” and “Latin and Greek, and Mathematics” from Part III, “Using It to Design” in Shape, pp. 311–354 and 368–387.

(optional) “What Rules Should I Use?”