The paper is divided into four parts. Part I identifies fundamental theoretical problems, contrasts the application of computation to architecture and to music, and draws upon several different areas for insight into the nature of making; Part II reviews particular architectural implications of these considerations, introduces the concept of computational composition in architecture, and presents a brief overview of important precedents; Part III proposes new goals for computer-aided architectural design and presents a framework for computational composition; finally, Part IV presents recent work directly related to the ideas presented in the previous parts and leads to the Conclusion. The appendices contain a pseudo-Prolog expression of Alvar Aalto's architectural language and notes on features of the PADL-2 solid modeler that are architecturally interesting.

Mathematics and especially geometry have found increasing application in the computer-based design environment of our day. The computer has become the central tool in the modern design environment, replacing the brush, the paints, the pens and pencils of the artist. However, if the artist does not master the internal working of this new tool thoroughly, he can neither develop nor express his creativity. If the designer merely learns how to use a computer-based tool, he risks producing designs that appear to be created by a computer. From this perspective, many design schools have included computer courses, which teach not only the use of application programs but also programming to modify and create computer-based tools.

In the current academic educational structure, different techniques are used to show the interrelationship of design and programming to students. One of the best examples in this area is an application program that attempts to teach the programming logic to design students in a simple way. One of the earliest examples of such programs is the Topdown Programming Shell developed by Mitchell, Liggett and Tan in 1988 . The Topdown system is an educational CAD tool for architectural applications, where students program in Pascal to create architectural objects. Different examples of such educational programs have appeared since then. A recent fine example of these is the book and program called “Design by Number” by John Maeda . In that book, students are led to learn programming by coding in a simple programming language to create various graphical primitives.

However, visual programming is based largely on geometry and one cannot master the use of computer-based tools without a through understanding of the mathematical principles involved. Therefore, in a model for design education, computer-based application and creativity classes should be supported by "mathematics for design" courses. The definition of such a course and its application in the multimedia design program is the subject of this article.

Our results fall in three major categories: 1) pedagogical discoveries about learning to design with a computer, which is greater than the sum of learning to design and learning about computers; 2) design exercises based on the Macintosh environment, exploiting the unique graphic qualities of the machine while simultaneously developing the ideas and drawing skills needed in the preliminary stages of design; 3) descriptions of the studio environment, including hardware, software, workstation layouts, security solutions, and other practical information that might be useful to others who are contemplating a similar project.

The "Calor" problem requires the move of a famous residence to a hot arid climate. The residence must then be redesigned in the original architect's style so the building becomes as energy efficient as possible in its new arid environment. The students are required to use as design criteria a new building program, the design philosophy of the original architect, and appropriate passive energy techniques that will reduce the thermal stress on the building. The building's energy response is measured by using an envelope energy analysis program called "Calor".

Much of the learning comes from imposing a new set of restraints on a famous piece of architecture and asking the student to redesign it. The students not only need to learn and use a different design philosophy, but also develop new skills to communicate their ideas on the computer. Both Macintosh and IBM computers are used with software ranging from Microsoft Works, Superpaint, AutoCAD, MegaCAD, Dr Halo, to Calor.