Current architecture is characterized by specifying new typologies but not experienced until now in the engineering and architecture disciplines. Architectural free forms with non-euclidean geometries are being developed, which answer to logical analysis and parametric design, and which are tending to the mimesis of behaviour with the nature. Innovative research is developing advanced constructions of complex geometries and free forms by architects like Yushang

Architecture is totally linked to the knowledge and study of geometry. The re-entrant hexatruss structure is part of the auxetic structures. These structures, characterized by its negative Poisson's ratio, change their geometric configuration from a line to a surface and from a surface to a volume or spatial framework. This project is based on establishing those re-entrant hexatruss geometries to be able to build new spaces in architecture, studying analytically geometric and structural properties of auxetic materials and structures that can be a part of the architecture construction.

If a lengthened structure gets fat in all directions and the same structure gets narrower when it is pressed, it is an auxetic structure. This happens because these kinds of structures have a negative Poisson's ratio (NPR). In this paper a comparative study of unity patterns of CAD auxetic geometries is presented. A Computer Aided Design library of auxetic geometries will be developed to apply them to architecture. The geometric behavior of the eighteen auxetic separate patterns is tested from the developed library in order to develop a systematic comparison, analyzing properties of these forms, such as their maximum achievable area reductions in relation with the total length of bars of the structure, in order to obtain a growth factor.

Auxetic materials have several properties very useful to be applied to architecture structures. This paper is aimed to test structurally a specific auxetic structure model. This hypothesis will be check: if auxetic materials have innovative properties in nanoscale then they will also have these properties in macroscale. But there are some differences for these different scales. In the nanoscale auxetic structures have rigid knots with flexible bars, but in the scale of architecture they will have articulated knots and a cable that stabilizes the set.A unity of the hexagonal re-entrant structure will be tested in order to obtain their structural characteristics. The application of this structure and their behavior in architecture are not yet known, that’s why this auxetic model will become an experimental model to establish a structural evaluation of one of the most innovative auxetic geometries, to apply to the construction of new architectures. The results of research and investigation will become apparent by their structural evaluation, through the utilization of manual calculus.The re-entrant hexagonal geometry provides a strong foundation for research of application of new structural systems on the production of architecture, while identifying transformations that new geometries and their application techniques, will contribute to the development and divulgation of new spatial and typological solutions. That is the reason to claim a detailed analysis to advance on the design and construction of new architectures.

"Push-and-pull" efficient structures have been inconceivable between XVIII centuries. It is because of the incapacity of obtain an efficient behaviour of tensioned material. Since XVIII centuries, architecture began to experiment new processes of adaptation defining new typologies in the engineering and architecture that had not been experienced until that moment. Tensegrities and tensioned structures were developed from the study of tension and geometry. Since then it has been emerged studies of this type by researchers who have continued or have been interested in these works.Tensegrities are bar and cable structures that work only in compression or tension efforts. Bars and cables are balanced, but in appearance the growth is disorderly. Most of deployable structures are based on tensegrity systems.This paper presents a synthesis of the art state of the set of technical solutions to the design of architectures based on tensegrities and tensioned structures that will be applied to the building construction. The research of adequate materials to tension efforts will be crucial in this study.This study provides a strong documental foundation for research on the significance of new structural and constructive systems in the production of the current architecture based on tensegrities and tensioned structures.