Geometry

2506 Submissions

[3] ai.viXra.org:2506.0102 [pdf] submitted on 2025-06-23 20:43:13

The Known Circle: A Physical Thought Experiment Challenging Symbolic Incompleteness

Authors: Christopher R. Parks
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)

The Known Circle proposes that a perfect physical circle, formed and measured in an idealized system, can yield an exact, finite area through proportional mass — without invoking p as an infinite symbolic constant. By treating p as an emergent product of measurable systems, the paper challenges the assumption that circular area must remain symbolically incomplete. This is not a numerical approximation, but a conceptual challenge to the structure of mathematical limits andtheir dependence on symbolic infinity.
Category: Geometry

[2] ai.viXra.org:2506.0064 [pdf] replaced on 2025-06-20 06:48:42

Two-Sided Symmetry and Holonic Maps: From Koestler’s Holarchy to Intuitionist Geometry and Archetypal Resonance

Authors: Stephen P. Smith
Comments: 13 Pages.

This paper explores Arthur Koestler’s concept of Janus-faced holons within a dynamic holarchy, integrating insights from CPT symmetry, intuitionist mathematics, Michael Schneider’s generative geometry, and Karl Friston’s free energy principle. It critiques static holonic diagrams and proposes a more resonant, fractal, and bilaterally symmetrical mapping rooted in archetypal forms. Drawing from sacred geometry, musical structures, and biological patterns, the essay argues that reality unfolds through intuitive construction and pre-existing mathematical orders. Holonic development, like nature itself, reflects a deep, two-sided balance—unifying form, transformation, and perception in a cosmological vision where cognition participates in creation.
Category: Geometry

[1] ai.viXra.org:2506.0016 [pdf] submitted on 2025-06-04 22:14:30

Observer-Centric Internal Spherical Geometry

Authors: Larry Whitaker
Comments: 9 Pages. (Note by ai.viXra.org Admin: Please cite listed sceintific references)

We present a practical mathematical framework for interior spherical geometry that en-ables complete sphere characterization and three-dimensional positioning from an observer-centric perspective. The methodology addresses fundamental limitations in traditionalspherical geometry by eliminating dependence on external reference points and establish-ing the observer as the coordinate system origin. Our approach integrates perpendicularchord measurement techniques with computational algorithms to provide accurate interiorpositioning for practical applications. The framework introduces measurement protocols forgeometric calculations, establishes cone-based coordinate systems with scalable precision,and addresses measurement uncertainty propagation. Applications include navigation systems for enclosed environments, geometric modeling of spherical structures, and scientificinstrumentation requiring interior perspective calculations.
Category: Geometry