Overview
Floating-point arithmetic introduces systematic approximation errors that obscure fundamental mathematical relationships in geometric calculations, producing results like 2.999999999999999 instead of the exact integer 3. These compounding and confounding precision losses stymie a full-featured implementation of Buckminster Fuller's Synergetics framework, which requires symbolic operations on all-integer accounting based upon ratios geometrically based upon high-frequency shapes. Here we present Symergetics (Symbolic Synergetics), an open source Python package which provides exact rational arithmetic methods framed within the vectorial geometry of the Synergetics framework. The package implements a Quadray coordinate system for tetrahedral geometry within the Isotropic Vector Matrix (IVM) lattice, exact volume calculations for Platonic solids using IVM units (tetrahedron = 1, octahedron = 4, cube = 3, cuboctahedron = 20), and pattern analysis algorithms for Scheherazade numbers (1001^n) and primorial sequences using exact arithmetic. The computational implementation achieves high test coverage with rigorous validation, demonstrating 100% precision preservation across 953 test cases. Applications include active inference modeling, crystallographic analysis, materials science, and computational geometry. Complete implementation details are available in the core modules and computation modules . The package is distributed under Apache 2.0 license at the Symergetics repository: https://github.com/docxology/symergetics . Towards a symbolic and Synergetic future, together we go!
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Methods and contributions
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Findings and contributions
- The package reports 100% precision preservation across 953 test cases.
- Exact arithmetic returns 11/12 for 3/4 + 1/6 rather than the float 0.9166666666666666, with no approximation errors detected in any test case.
- The paper reports that its Platonic-solid volume relationships in IVM units were independently verified, confirming Fuller's original Synergetics calculations.
- Quadray-to-Cartesian conversions are reported to round-trip exactly.
- Primorials are computed exactly, e.g. 10# = 6,469,693,230.
Methods
- SymergeticsNumber exact-rational wrapper over Python fractions.Fraction — Arithmetic is done with a Fraction-based class that keeps exact fractional representations and auto-simplifies via GCD.
- Quadray (four-axis tetrahedral) coordinates in the IVM lattice — Points are represented as (a, b, c, d) along four tetrahedral axes, normalized by subtracting the minimum coordinate.
- IVM-unit volume calculation for Platonic solids — The package computes polyhedron volumes in IVM units with the tetrahedron as the unit volume.
- Pattern analysis of Scheherazade numbers (1001^n), primorials, palindromes — Exact-arithmetic routines analyze powers of 1001, primorial sequences, and multi-base palindromes.
- Test suite of 953 test functions across 32 files — Validation is via a pytest suite covering arithmetic, coordinate transforms, geometry and integration cases.
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Citation
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