Abstract
We study positive-root constructions based on the Pandrosion rectangle. A decentered moving arc gives a globally monotone fifth-order correction. A nested-radical support uses two successive mean proportionals and has exact cubic local order nine with coefficient 7/648. Its unmodified scalar family converges from every positive start for all real p > 2; degree doubling and inversion extend it to every p ≠ 0. These convergence results and exact cubic order are checked in Lean; general local order at least nine has a separate analytic proof. A complete integer-degree inventory bounds a nested-radical step by 10p + 59 traces. We compare rectangle supports of orders one, two, three, four, five and nine at a common starting state. A separate power-free subdivision construction encloses real-degree roots with an explicit width and drawing bound. Figures detail the first nested-radical step for the cube root of two. Historical priority, minimum drawing cost and runtime superiority remain unestablished.
Overview
This preprint studies positive-root constructions based on the Pandrosion rectangle. A decentered-arc support provides a fifth-order correction that is globally monotone for real degrees p > 2. A support with two nested radicals achieves exact local order nine in the cubic case, with relative coefficient 7/648.
The unmodified scalar family converges from every positive starting point for p > 2. An adaptation using degree doubling and inversion extends the result to nonzero real degrees, still for positive inputs. A separate subdivision construction encloses real-degree roots without evaluating a residual power.
Constructions
Two successive mean proportionals, intersections and length transfers make the geometric iteration explicit.
Drawing inventory
For integer degrees p ≥ 3, the protocol uses at most 10p + 59 traces per nonterminal step: 89 at degree three, under the preparation exclusions specified in the paper.
Scope of the proofs
Section 8 describes 104 audited statements for the nested-radical results, covering the cubic case, polynomial certificates, real-parameter analysis and the extension of degrees. A separate retained audit of seven declarations concerns subdivision.
The general local order of at least nine and the convergence of the decentered arc have written analytic proofs. The drawing inventories are not a cost semantics formalized in Lean. The results certify neither a physical drawing nor every rounding error of an implementation.
Related resources
- Project repository and geometric constructions
- Interactive construction: real-degree subdivision
- Zenodo archive, DOI and complete bibliography
Citation
Ivan Besevic (2026). Geometric Root Calculators: A ninth-order cubic construction and real-degree root solvers. Preprint, version 27-NR, October 18, 2026, 26 pages. DOI: 10.5281/zenodo.23449823.
The Zenodo record specifies an MIT license. The supplements described in the manuscript are separate from the PDF available on Zenodo.
The construction, step by step.
This scene follows the cubic case in Section 4.2 of the manuscript. It illustrates the paper’s nested-radical construction, which is distinct from the generic engine of solver 449.
Two square roots.
One cubic correction.
Construct an approximation to ∛2 with two mean proportionals, then read the result in the Pandrosion rectangle.
Diagrams and explanations are accessible without 3D. Three.js scenes are hosted on this website, without a visualization CDN. Three.js license.