# Partially Auxetic Structure of Rotating Rectangles
## 5 Conclusions  
In summary, a modified structure of rotating rectangles was designed and produced for which a metamaterial mechanical phase transition was determined. This modified structure behaves stably with large changes in dimensions.  
Modified structures of rotating rectangles, built from rectangular unit cells with selected a/b and x parameters, show the occurrence of a metamaterial phase transition. It consists of changing Poisson's ratio from positive to negative values while continuously changing the dimensions of the structure. This means that, for example, when the structure is stretched, the elongation of the structure in both orthogonal directions initially increases. Then, with a further continuous increase in elongation in one direction, the elongation in the direction perpendicular to it decreases.  
For low values of the parameter x (position of the axis of rotation), this effect occurs for a/b \(\geq\) 2 (ratio of the length of the sides of the rectangular unit cell). However, larger values of this parameter (x \(>\) 0.1) can also appear for less elongated rectangular unit cells.  
Measurements on real physical models confirmed the calculated theoretical values and measured values of linear dimension changes for areas of auxetic behavior of structures. In this respect, the work sets the direction for the mass production of auxetic structures for applications requiring large negative Poisson's ratios and small elongations in the direction perpendicular to the force causing the deformation of the structure.  
An additional but no less important feature of such a modified structure of rotating rectangles (or squares) is the independence of Poisson's ratio from the elastic properties of the unit cells. Poisson's ratio only depends on the change in linear dimensions of the modified structure and is, therefore, related to a new class of mechanical metamaterials. This example of auxetic structures has a Poisson's ratio that is not related to material properties but only results from movement of the structure. Poisson's ratio here is, therefore, purely geometrical.  
The proposed structures can be used to build a structure that, given a certain change in Poisson's ratio, will exhibit the expected auxetic behavior with the required dimensional changes. This type of structure may be needed in applications where a large negative Poisson's ratio is required, and the change in dimensions in one direction is small.  
The article presents the design and development of a modified metamaterial structure that can be used in robotics.  
Such modified structures of rotating rectangles composed of numerous unit cells made of rigid materials can also exhibit a predetermined degree of deformation, e.g., tensile expansion. For this reason, they can also be used as metamaterials for the study of buckling and fracture mechanics. This is particularly important because only real physical structures allow us to understand their macroscopic mechanical properties. Only the bulk behavior of a metamaterial can indicate its auxetics and stability, which is the starting point for potential and expected applications.  
[PERSON] came up with the idea presented here, including the model and geometric relationships, prepared the first draft of the manuscript, and constructed physical models used as proof. [PERSON] arranged the editing and proofreading of the work. All authors have read and agreed to the published version of the manuscript.  
The present paper was financed partly by the Polish Ministry of Education and Science within statutory activity.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
Special thanks go to [PERSON] for his help in implementing unit cells.  
The authors declare no conflicts of interest. The study is part of a strategic area of interest covering the analysis and assembly of auxetic systems.