Matemática viva: alguns padrões matemáticos e geométricos da natureza

Abstract

This study aims to investigate the mathematical patterns found in nature, with emphasis on fractals, spirals, and numerical sequences, analyzing their formation rules and potential applications in mathematics education. It starts from the contemporary conception that mathematics is essentially the science of patterns, redefining the role of the mathematician as an observer and interpreter of regularities present in physical, biological, and even behavioral

phenomena. The research adopts a quantitative approach, using bibliographic and exploratory- descriptive methodology, based on works by classical and contemporary authors, as well as a

variety of educational materials. Both physical and digital collections were consulted, along with technological tools such as artificial intelligence, to search for illustrative images and construct geometric models. The analysis focuses on recurring structures in nature, such as the symmetry of snowflakes, fractals found in tree branches, river basins, and biological systems, as well as logarithmic spirals observed in shells, galaxies, hurricanes, and flowers. The Fibonacci sequence is addressed as a fundamental numerical pattern, widely found in natural phenomena, and when related to spirals, it reveals the harmony embedded in the golden ratio. These elements are discussed not only for their visual or aesthetic appeal but also for their functionality in spatial organization and the efficient growth of living organisms. The study proposes a reflection on the application of these patterns in mathematics teaching, in alignment with the guidelines of the Brazilian National Common Curricular Base (BNCC), emphasizing their importance for developing geometric thinking and promoting a more meaningful, contextualized, and investigative learning process. It concludes that mathematical patterns not only describe natural reality but are part of its very essence, highlighting mathematics as a universal language that integrates science, art, and nature.