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.

Abelson, Harold, diSessa and Andera (1968) gave the first rules concerning Spirolaterals. To obtain a Spirolateral from a set of straight lines, the first of them must be one unit long and the following must be incremented one unit at each step, at the same time that they turn in a constant direction. Odds (1973) establish the variation of the rotation direction, either to the left or the right. However, he did not give a mathematical relation able to calculate open Spirolaterals. Krawczyk (2001) developed a computer program that generates Spirolaterals following the method suggested by Abelson. These are Spirolaterals obtained by enumeration without a predictive mathematical formula. Krawczyc went farther proposing Spirolaterals based in curved lines. He pointed out that there are a variety of spirolateral forms that have architectural potentiality. Following this, the architectural potentiality of Spirolaterals is the basis of this paper.

To take advantage of that potentiality a computer program was implemented to generate spatial configurations based in Spirolaterals. When a third dimension is given to the Spirolaterals they become Spirospaces. These new entities need spatial and design parameters to be useful for architectural purposes. Barrionuevo and Borsetti (2001) gave results about that work establishing the concept of Spirospaces.

The aim of this paper is to describe a work directed to improve rules and procedures concerning Spirospaces. It is expected that these procedures governed by the proposed rules can be employed as tools during the early steps in the architectural design process.

In this work some aspects concerning Spirospaces are considered. First, Spirolaterals are presented as the predecessors of Spirospaces. Second, Spirospaces are defined, together with their structural parameters. Architectural modeling is studied at the light of two special elements of the Spirospaces: Interstitial spaces and Object spaces. Next, a computer program is presented as the appropriate tool to model configurations having architectural potentiality. Finally, the results obtained running the computer program are analyzed to determine their possible use as architectural forms. Several graphic illustrations are presented showing steps going from the exploration of spatial alternatives to the selection of a specific configuration to be developed.

It is expected that the described computer program could be employed as a design aid tool. As the operation of the program generates a variety of spaces able to dwell architectural objects, it eases the search of configurations suitable to specific functions. The results obtained have the possibility of being exported to computer graphic applications able to add materials, lights and cameras.

From the detailed analysis of the results obtained, the following diverse pedagogical aspects of this work project arise: a. The use of several multiple intelligence: Howard Gardner (1985) found that a man has several distinct intelligence types among which Logical-Mathematical; Spatial; Linguistic -oriented; Musical; Intra-personal; Kinesthetic-Corporal; Interpersonal stand out. Only those types used in the task will be analyzed, making a brief description of each type. b. The architectonic-city planning aspects: architectonic-city planning interpretation of the space imagined after reading the text, with the purpose of identifying figures, shapes, volumes and colors which are expressed via an analogous space. They consist of visual, architectonic and territorial speculations without a rigorous spatial theory and it is pretended that they possess a technical precision at mathematical concept level. c. The mathematical contents: a study of the conical and square shapes present in the designs done and used in a creative manner in students’ compositions following the reading of the story chosen is carried out. An analysis of shapes is performed and mathematical problems are posed within the design context.

Traditional sketching methods have been used in task solving and the possibilities offered by the virtual tools are analyzed.

Emphasis has been put on the vertical and horizontal interchanges in the Chair, generating changes in knowledge transmission perspectives, thus allowing the sharing of contents, abilities and resources. The architectonic work imagined and created by the students will focus on these different working lines creating a harmonious and significant whole. The work is the result of multiple connections and creative proposals.

The great development of modelling achieved by the digital media and the limitations in the technical and building areas and in the existence of materials which are coherent with the resultant shapes reveal a considerable distance between the systems of ideation and simulation characteristic of the computing era and the analogous systems of production inherited from the slow industrial development. This distance has been shortened by CAD/CAM systems, which are, however, not very accessible to the architectural field. If we incorporate to the development of these divergent media the limitations which are distinctive of the material resources and procedures of the existent local technology, the aforementioned distance seems even greater.

Assuming the metaphor of living at the threshold of two ages (industrial-computing, analogical-digital, material-virtual) and the challenge of the new conceptual and operational tools in our field, we work in the mixture, with no exclusions or substitutions, proposing (by means of the development of informational complements) some alternatives of work to approach the issue under discussion from the Architecture Workshop.