🧵 Mapping Combinatorial Search Spaces to Continuous Search Spaces via Circle Transforms

Anonymous at Tue, 11 Feb 2025 18:08:39 UTC No. 16582771

Framework (TSP inspired):

Key Invariance Principle: The theory centers on a transform that maps geometric configurations through force circles, which encode geometric constraints and reveal pathways between valid configurations. Crucially, at any chosen radius, the original figure can be exactly reconstructed, this is because the transform circle’s circumference is set equal to the Euclidean perimeter, ensuring that the arcs between candidate midpoints maintain their true lengths.

The fundamental elements are:

1. Configuration Mapping Mechanism
- Each point in a geometric configuration is associated with a "force circle"
- The force circle is centered on its point with radius equal to the point's distance from the configuration's centroid
- These force circles remain invariant through transformations

2. Transform Circle Properties
- Points are mapped onto a circle with radius R = perimeter/(2π)
- Angular positions preserve relative distances from the centroid
- Arc lengths between adjacent points maintain proportional relationships from the original configuration

3. Geometric Preservation Properties
- Valid configurations emerge where force circle intersections create midpoints
- Total perimeter is preserved through arc length conservation
- When scaling between configurations (e.g., square with C=4 to C=4.828), proportional relationships are maintained (see attached)

4. Configuration Discovery Mechanism
- As the transform circle's radius changes, points spread or contract while maintaining their geometric relationships
- Force circle intersections create paths that midpoints follow
- When a midpoint lies within multiple force circles, it can split to reveal alternative valid configurations
- Each split point represents a distinct valid geometric configuration

Attached visualization demonstrating a square edge length one, transformed with 2 edge 2 diagonals through radius scaling.