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2026
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| Online Access: | https://doi.org/10.5281/zenodo.18450874 |
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| _version_ | 1866901375800049664 |
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| author | Cave, Scott Allen |
| author_facet | Cave, Scott Allen |
| contents | <p>This preprint introduces <strong>Euclidean Action</strong>, a fully geometric and fully discrete recursive system built from classical Euclidean invariants. Starting from a canonical equilateral triangle, the construction applies a fixed spiral similarity—uniform scaling by the golden ratio combined with a constant rotation—producing a structured sequence of similar triangles called <em>Euclidean Lifts</em>.</p> <p>At each step, standard geometric invariants (oriented area, mean edge length, mean radius, reciprocal radius, and a dimensionless aspect ratio) are extracted and assembled into a five-dimensional feature vector. These invariant measurements drive a closed recursive update in feature space through a coherence-balanced rule that suppresses uniform drift while preserving relative geometric contrasts.</p> <p>The resulting system is entirely self-contained: no physical dynamics, variational principles, or external forcing are introduced. All evolution arises from similarity geometry and invariant extraction alone. The paper analyzes the periodicity and scaling behavior of the Lift, defines the invariant feature space, proves closure of the recursion, and presents numerical examples illustrating the coupled geometric–modal evolution.</p> <p>Euclidean Action provides a minimal testbed for studying recursive geometric systems driven by invariants rather than direct deformation of figures. Potential extensions include alternative similarity rules, higher-dimensional simplices, different invariant spaces, and continuous-time limits.w</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18450874 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Euclidean Action Cave, Scott Allen <p>This preprint introduces <strong>Euclidean Action</strong>, a fully geometric and fully discrete recursive system built from classical Euclidean invariants. Starting from a canonical equilateral triangle, the construction applies a fixed spiral similarity—uniform scaling by the golden ratio combined with a constant rotation—producing a structured sequence of similar triangles called <em>Euclidean Lifts</em>.</p> <p>At each step, standard geometric invariants (oriented area, mean edge length, mean radius, reciprocal radius, and a dimensionless aspect ratio) are extracted and assembled into a five-dimensional feature vector. These invariant measurements drive a closed recursive update in feature space through a coherence-balanced rule that suppresses uniform drift while preserving relative geometric contrasts.</p> <p>The resulting system is entirely self-contained: no physical dynamics, variational principles, or external forcing are introduced. All evolution arises from similarity geometry and invariant extraction alone. The paper analyzes the periodicity and scaling behavior of the Lift, defines the invariant feature space, proves closure of the recursion, and presents numerical examples illustrating the coupled geometric–modal evolution.</p> <p>Euclidean Action provides a minimal testbed for studying recursive geometric systems driven by invariants rather than direct deformation of figures. Potential extensions include alternative similarity rules, higher-dimensional simplices, different invariant spaces, and continuous-time limits.w</p> |
| title | Euclidean Action |
| url | https://doi.org/10.5281/zenodo.18450874 |