Design of Strips with Geometry Shapes and Mathematical Analysis

In this research, we investigate different methods to create geometric designs for textile strips and study the geometric properties of the involved shapes. We develop three designs that contain circles, squares, and golden spiral pieces with repeating patterns and certain tangencies. One interesting part of the work is to find the tangent points and to calculate the areas of the regions to which different colors may be assigned. The main figure for Design I is a circle inscribed in a square and that for Design II is a circle inscribed in an isosceles triangle. The last design integrates golden spirals into the image. The goals of this research are to provide relationships between geometry and the considered textile designs, to examine the mathematics used to characterize the geometrical shapes, and to show how mathematics can be visualized in textile design and how it can help student learners to experience real-world applications. The main results include formulas for the areas of the involved regions in each design and where the tangent points are. In Design III, we focus on certain interesting regions bounded by pieces of circles, squares, and the golden spirals. The sequence of such areas, named as \(\{A_n\}_{n=1}^\infty \) , follows an interesting pattern. Formulas for \(A_n\) are developed using calculus ideas. The limiting situation of the ratios of two consecutive areas is provided. The last part of the thesis gives an interactive lesson plan, which involves the geometric concepts demonstrated in the textile designs, for high school students to explore real-world applications.