Geometry and Mechanics, Inescapable Tools of Architectural and Structural Design

• segunda época • año 9 • núm. 18 • México • unam • diciembre 2018 • 16-27 Abstract The aim of this paper is to describe some teaching procedures that have been applied in the education on structures for architecture students at the National Autonomous University of Mexico (unam). Even though most of the architecture curricula at Mexican higher education institutions include the subjects of mechanics and geometry, they are taught separately and are not applied to architectural concepts and structural design. In order to mitigate these undesirable teaching conditions, a group of professors has attempted to simultaneously incorporate mechanics and geometry into the conception and design of structures. In response to the aforementioned circumstances, 'Mechametry' was created as a new subject in the unam architecture curriculum. Mechametry has been offered for 10 years to undergraduate students, as well as to students in the Design of Lightweight Structures specialization course. Nowadays, many former students are working for companies in Mexico and around the world and are directly involved in the design and construction of shells and other spatial structures, primarily tensile surface structures. Some selected projects by these students will be shown and described in this paper, as will projects that have received awards in international contests for architecture students.


Introduction
In current attempts by structural designers to produce sustainable architecture and engineering, lightweight structures become relevant, as they meet environmental criteria, such as maximum efficiency with minimum materials and making better use of natural resources.
The conception, design and construction of lightweight structures should therefore be carried out by architects and engineers with the greatest possible knowledge of the handling of the form, through deep knowledge of geometry, and the best possible understanding of structural behavior, through deep knowledge of mechanics. Certainly, the ability of each structural designer to conceive and design a new structure is measured through their personal ability, a skill with which they were born and which is augmented by the training they have acquired in universities or other higher education institutions. An architect or an engineer with a deep knowledge of geometry and mechanics can therefore properly conceive and design a better sustainable structure, which must not lack that characteristic that was defended countless times • segunda época • año 9 • núm. 18 • México • unam • diciembre 2018 • 16-27 by the architect Felix Candela, that great builder of reinforced concrete shells in Mexico: beauty.
Since the last century, we can add the names of many other renowned designers and builders in addition to Felix Candela: Vladimir Schuchow, Antonio Gaudí, Eduardo Torroja, Pier Luigi Nervi and Frei Otto are some of the most outstanding individuals that represent a common conception, design and construction of sustainable architecture and engineering. It is curious that, in the 60's and 70's, the term sustainability was not used; however, all of the efforts of these builders were aimed at conceiving and designing sustainable architecture and engineering. All of them, architects or engineers, can be considered to be the earliest defenders of what we now identify as sustainable architecture or engineering.

Mechanics and geometry
After many years of teaching separate courses on structural geometry and mechanics to undergraduate architecture students, the author of this paper came to the conclusion that, if the contents of both subjects were taught simultaneously, explaining the close relationship between form and structural behavior, students would obtain a clearer and more objective view of how structures work and behave and consequently would be able to conceive, design and construct better structures. Students would thus understand that a form resulting solely from the designer's formal whims can become beautiful and shocking, even if it is totally unsustainable. The author is convinced that future architects and engineers must acquire, at the university level, the sufficient and necessary knowledge to practice professionally in a responsible fashion and with the conviction to propose and produce sustainable architecture and engineering.
Based on the above, the author proposed the creation of a single subject in the architecture curriculum called "mechametry" -mechanics and geometry -where students simultaneously learn and exercise these two fields of knowledge and come to understand how they complement each other. There was strong resistance on the part of the university administration to the acceptance of this term, but over time, and given the program's results, they have officially accepted the term mechametry.
Several architects and engineers who took the mechametry course offered to undergraduate students in the Faculty of Architecture, as well as to graduate students taking the Design of Lightweight Structures specialization course, are currently working in Mexico and around the world as freelancers or for agencies dedicated to the design and construction of lightweight structures, mainly tensile surface structures. Many of them have received national and international awards.

Didactic methodology
Below is a description of the general sequence of one semester of the elective course on mechametry. Each course consists of 16 two-hour sessions, giving a total of 32 hours per course. It is important to note that an average of 45 students enroll in the course each semester. With more than 300,000 students, the unam is one of the largest universities in the world and the main campus in the south of Mexico City is called University City.
Session 1: The professor starts out by inviting the students to reflect on the importance and the close relationship that exists between the shape and the mechanical behavior of a structure, presenting and discussing examples of past and contemporary buildings in which this relationship is emphasized. The professor also shows examples of finished buildings where form was conceived without considering the importance of adequate and sustainable structural behavior. Session 2: Mechametry students have already studied statics and strength of materials. However, some fundamental concepts -force, stress, tension, inertia-are repeated again with a focus on the conception and design of a lightweight structure. Students also simultaneously conduct analytical exercises. Session 3: The students have already studied descriptive and analytic geometry in previous semesters. Here we discuss the relationship between form and structural behavior for conical sections and catenaries and their application to the conception and design of a lightweight structure. Once again, the students conduct analytical exercises. Sessions 4: During the 10 subsequent sessions, students form teams of no more than five students each, with each team developing three lightweight structures. They can apply any lightweight structural system: tensile sur-face structures, grid shells, tensegrity systems, pneumatic structures, structures with nodes and bars, etc. Each team presents its project with a physical model and evaluates its successes and failures while looking for solutions that could improve its structural design. The students consider all the observations made during their evaluation and then, they redesign an improved version of their project.
Graphic: unam -Structures Laboratory archive. and builds a physical model that will be placed on the grounds of the Faculty of Architecture. In this case, the model has to be large enough to demonstrate the knowledge they have acquired during the 16 sessions of the course.
At the end of the course, the students are evaluated on attendance, participation, individual and group work.

International recognition
Many architects and engineers who took the mechametry course as undergraduate architecture students or as graduate students in the Design of Lightweight Structures specialization course are currently working, in Mexico and abroad, as freelancers or employees of companies dedicated to the design and construction of lightweight structures, primarily tensile surface structures. Many of them have also won national and international awards for their work. The recognitions and awards given to former students of the mechametry course that has been taught by the author over the past 15 years are listed below. Many of these students have also collaborated on the ongoing research projects of the unam Faculty of Architecture's Lightweight Structures Laboratory under the direction of the author.

Award-winning projects by former students
Over 20 former students of the mechametry course are currently working as architects or engineers, either as freelance professionals or at companies dedicated to the design and construction of lightweight structures. Some of these projects are listed below: mechanical behavior of a structure and its form. Students carry out a series of exercises in which they must conceive and design lightweight structures, both individually and in groups, constructing physical models that are discussed in front of the class and in which successes, failures and possible solutions are discussed. Students then present a final project that incorporates all of the observations made in their partial evaluations, thus designing an adequate and logical lightweight structure in terms of its form and its structural behavior. Finally, students have submitted their projects to international student competitions, many of them receiving prizes that demonstrate the effectiveness of the teaching strategies shown herein. The author is confident that this is a good methodology for teaching structures to architecture and civil engineering students, but this strategy does not aim to be the only or the best way to achieve these objectives.