Advanced International Journal for Research

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Definition of Perfect Shape and Their Values in n-Dimensional World

Author(s) Chung Seop Lee, Lauren Choi, Sang Hyeon Kim
Country United States
Abstract This paper introduces the concept of the perfect shape, defined as an ????-dimensional figure whose enclosed measure and boundary measure are connected through a differential relationship. Specifically, a shape is perfect when the derivative of its ????-dimensional quantity with respect to its defining size parameter equals its (n−1)-dimensional quantity. Through geometric and differential analysis, the paper demonstrates that this condition holds exclusively for hyperspheres—figures whose boundary points are equidistant from a central origin. Circles and spheres thus serve as lower-dimensional instances of this universal form. Building upon this foundation, the paper proposes a conjecture linking the measures of perfect shapes across consecutive dimensions. Observing that differentiation and integration connect adjacent dimensions, it is hypothesized that higher-dimensional expressions of the hypersphere can also be derived by multiplying the measure of the n-dimensional figure by 2^n, reflecting a recursive geometric pattern among perfect shapes. This framework offers a new perspective on dimensional growth and suggests that higher-dimensional geometry may be constructed through the calculus of adjacent dimensions. Although this study presents a theoretical conjecture, future research will aim to validate the proposed 2^n-scaling relationship through topological data analysis and AI-based modeling, providing a computational approach to bridging analytic and geometric views of higher-dimensional space.
Keywords Geomety, Topology, Perfect Shapes
Field Mathematics
Published In Volume 6, Issue 6, November-December 2025
Published On 2025-12-27
DOI https://doi.org/10.63363/aijfr.2025.v06i06.2733
Short DOI https://doi.org/hbg692
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E-ISSN 3048-7641
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