Mathematics has profoundly impacted the discipline of origami, facilitating the emergence of sophisticated design methodologies. Origami constitutes the art of meticulously folding paper into various sculptures without the necessity of cutting the material or employing adhesives for assembly. This traditional Japanese craft of paper folding unveils complex geometric patterns manifested through its crease lines. We highlight the dynamic relationship between artistic ingenuity and mathematical principles in the realm of advanced origami design, as well as the role of origami as a practical pedagogical instrument for the instruction of geometric concepts, delving into its mathematical intricacies. In this review, we will examine the process by which the construction of a representational origami figure is accomplished through the application of mathematical algorithms. Furthermore, it investigates the techniques that have broadened the creative horizons within the field of origami and its different applications.