McDonald’s Geometric Equations: A Theoretical Framework for Shape Transformations and Addition

ABSTRACT This paper introduces a novel geometric framework termed "McDonald's Transforms," which systematizes the expansion and contraction of polygons through iterative spatial relationships. We establish a specific correlation between the diagonal of a square and the side lengths of its successive transformations, demonstrating that the diagonal of a square constructed from the diagonal of a parent square is exactly twice the side length of the original. We extend this logic to general area transformations for squares and triangles, defining a "transformation order" that dictates scaling through power functions. Additionally, we propose a theory of "Shape Addition," modeling the enclosure of shapes within grids as an arithmetic sequence. This framework provides a simplified approach to calculating geometric scaling and grid expansion without complex trigonometric derivations. Keywords + MSC Classification Keywords: Geometry, Shape Transformation, McDonald’s Transforms, Geometric Scaling, Arithmetic Progressions. MSC Codes (2020): Primary 51M04 (Elementary problems in Euclidean geometry); Secondary 97G40 (Plane and solid geometry). INTRODUCTION Motivation: Standard geometric scaling often relies on separate applications of the Pythagorean theorem or area ratios. This work seeks to unify these into a singular transformational theory. Background: The relationship between a square's side and its diagonal (s\sqrt{2}) is well known; however, McDonald’s theory simplifies the iterative process of using these diagonals as bases for subsequent shapes. Contribution: This paper introduces the "Transformation Order" (X), a constant that defines how different shapes (squares vs. triangles) scale relative to their parent shapes during expansion and contraction. Preliminaries / Definitions * Parent Shape (A): The initial geometric figure before transformation. * nth Transform (n): The number of times a transformation is applied, where n > 0 denotes expansion and n < 0 denotes contraction. * Transformation Order (X): The ratio of the area of the first transformed shape to the area of the parent shape. * Conventional vs. Unconventional: Transformations using integers versus those using fractions or decimals. Main Results Theorem 1: McDonald’s Theory of Squares If a square B is constructed using the diagonal of square A as its side length, then the diagonal of square B (L) is exactly twice the length of the side of square A. Theorem 2: General Transformation Formula The area (A'), perimeter (P'), or volume (V') of a transformed shape is determined by the formula: A' = X^n \cdot A P' = X^n \cdot P V' = X^n \cdot V Where X=2 for squares and X=4 for triangles. Theorem 3: Theory of Shape Addition The total number of shapes (y) resulting from enclosing a central shape with others of equal size follows the arithmetic sequence: y = a + xn For squares, this is specifically defined as y = 1 + 8n. Examples / Applications * Square Expansion: Given a parent square with side 2cm (Area = 4cm²), the 4th transform (n=4) yields an area of 2^4 \cdot 4 = 64\text{cm}^2. * Triangle Contraction: For a triangle with Area 96cm², a contraction of n = -2 returns the area to the parent state of 6cm² (4^{-2} \cdot 96 = 6). * Grid Growth: Applying shape addition to a square (n=2) results in 17 total squares (1 + 8(2) = 17). Discussion / Conclusion McDonald's Geometric Equations provide a streamlined method for calculating multidimensional scaling. While this document focuses on squares and triangles, the concept of "Transformation Order" can be theoretically extended to any regular polygon or 3D object. Future research should aim to define the constant X for n-sided polygons and explore the implications of "Unconventional Transformations" (fractional n) in fractal geometry. References * Ejike, M. C. (2026). McDonald's Geometric Equations (McDonald's Transforms). Private Circulation.