Spherion: Geometry & Algebra
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What is Spherion?
Spherion Geometry
Spherion (𝕊) is a new geometric approach and conceptual computational framework that does not fit neatly into any existing mathematical or physical category. However, we can accurately situate it in terms of disciplines, operations, and the type of mathematical object it most closely resembles. The primary operation involves recursively defining and overlapping subdivisions of spherical space into zones of increasing resolution, along with probabilistic associations among the zones that involve uncertainty. On this site, I will elaborate further on this topic, presenting ideas regarding Spherion to both professionals and laypeople. Spherion was created and developed by Favst McFrey (Fausto Machado Freire).
Spherion Algebra
Spherion Algebra (𝕊ₐ) is the formal language behind Spherion geometry, with rules that depart from Euclidean arithmetic. Instead of numbers, 𝕊ₐ manipulates nested zone codes that subdivide a sphere and overlap like 3-D Venn diagrams. Its core moves—fan-out, collapse, weigh—recursively split or fuse zones while attaching a probability. The resulting idempotent lattice treats union as addition, intersection as multiplication, and propagates uncertainty by simple coefficients. On this site I’ll unpack these ideas for experts and newcomers, showing how 𝕊ₐ can replace π, square roots, and decimals with discrete rotation-equivariant tokens. Spherion Algebra was created by Favst McFrey (Fausto Machado Freire).
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“Spherion Algebra” introduces a new formal language for reasoning about space, uncertainty, and computation on the sphere. Instead of using coordinates or decimal numbers, it encodes location as a sequence of nested zones—discrete tokens that divide the sphere into hierarchically arranged regions.
This algebra defines operations such as fan-out, collapse, and weigh, which expand, merge, or assign probabilistic weights to zones.
These operations create an idempotent lattice where:
• Union functions like addition,
• Intersection functions like multiplication,
• and uncertainty naturally propagates across levels.
Spherion Algebra offers a rigorous theoretical base and computational framework applicable to geospatial, probabilistic systems, and uncertainty modeling. Its combination of recursive subdivisions—defined by complex zone codes—and probabilistic links provides deep insights into spherical geometry.
Instead of using π, square roots, or continuous functions, Spherion Algebra constructs spatial models with finite symbolic precision. The system maintains rotational symmetry and handles ambiguity naturally. Based on geometric intuition but designed for modern computation, Spherion Algebra offers a new toolkit for spatial databases, probabilistic inference, and AI in curved space.
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