Measuring the Philippine Coastline.

Theoretical Framework

From Classical Geometry to Fractional Dimensions

In classical Euclidean geometry, topological dimensions are strictly integers: a zero-dimensional point (D = 0), a one-dimensional line (D = 1), a two-dimensional plane (D = 2), and a three-dimensional volume (D = 3). When a 1D line is scaled by a factor of 1/s, it produces N = s¹ copies; a 2D square produces N = s² copies.

In 1918, Felix Hausdorff generalized dimension to non-integers: if scaling an object by 1/s yields N self-similar sub-units, its Hausdorff-Besicovitch Dimension is defined by N = (1/s)^D ⇒ D = log(N) / log(1/s).

The Coastline Paradox in Nature

Benoit Mandelbrot (1967) observed that natural geographical boundaries (islands, bays, and lakes) behave as statistical fractals. Unlike smooth geometric shapes (D = 1.0), measuring a real coastline with a smaller yardstick reveals previously unresolved bays, coves, and promontories—causing the total measured length to diverge towards infinity. For Laguna de Bay (D ≈ 1.134), measuring the fractal dimension provides a mathematically rigorous baseline: when human infrastructure (like road viaducts) straightens the perimeter, D drops toward 1.0, directly quantifying the loss of edge-effect microhabitats.

Algorithm Implementation

Estimator Pseudocode

The implementation remains private during the current research phase. The non-runnable pseudocode below summarizes the shared estimator workflow.

FOR each measurement scale
    measure the shoreline with box, rigid-yardstick, or bending-yardstick rules
    record the scale and resulting count
END FOR

fit the slope of log(count) against log(inverse scale)
report the slope as the estimated fractal dimension

Computational Cost

For a shoreline of n points fitted over K = 12 scales, all three estimators are linear in shoreline size in theory — and the measured scaling confirms it. Timing runs pin the process to one CPU core under the performance governor, discard a warm-up, and average three wall-clock runs.

Exponents are log-log fits of kernel time against point count on densified test rings (1k–64k points), ± standard error; raw fits in the project archive. Box counting shows mild superlinearity only under extreme densification, where its finest fitted scale tracks point spacing and the cell count grows with it — at real shoreline sizes every estimator finishes in under a millisecond.

Robustness of the Estimates

Every estimator was stress-tested against the two free choices hidden inside each method: where a walk begins (24 starting vertices, both directions) and where a grid sits (16 sub-cell placements). The parameters change where measuring starts, never how much work happens — runtime is invariant — but they do move the answer:

The honest conclusion: single-placement box-counting deltas and Taal's apparent year-to-year change are the same size as these spreads, so the paper reports grid-based deltas only alongside the grid-free yardstick walks and treats Taal as an unstable reference. Laguna's near-zero result survives every placement. The bending yardstick's hinge budget saturates by K = 8 bends per stick, and K = 1 (the conservative choice) is used for headline tables.

Review of Related Literature

Our approach builds heavily on established computational methods for self-similar boundaries. The Box Counting baseline is standard, while our Rigid Yardstick implementation follows the classic digital compass walk by Shelberg, Moellering, and Lam (1982). Our experimental Bending Yardstick sits computationally between the standard walking family and the constant-deviation variable-step (CDVS) convex-hull methods pioneered by Normant and Tricot (1991).

Locally, Philippine prior art is sparse. Alova (2025) measured the Negros Island coastline at D = 1.018 using Box Counting and coarse GADM vectors. However, there are no published archipelago-wide values, nor studies utilizing precise satellite-derived masks for Philippine lakes (like Laguna de Bay or Taal Lake). This project represents the first multi-feature algorithm-comparison fractal study tailored specifically for Philippine topographies.

Data Pipeline

Failure Modes of Existing Vision Models & Potential Contributions

Before settling on our temporal median and morphological pipeline, we evaluated several pretrained neural networks and zero-shot foundation models. Their systematic failure on tropical lacustrine environments highlighted critical gaps in current remote-sensing computer vision:

Archipelago Database

Comprehensive Nationwide Shoreline Survey

Using our native C computational engine, we processed high-resolution vectorized boundaries for 122 major Philippine coastal and inland water features (108 major islands and 14 lakes) spanning Luzon, the Visayas, and Mindanao. This represents the first archipelago-wide comparative fractal baseline for Philippine geography.

Tip: Click on any island or lake row to inspect its exact coordinate polyline and live Richardson log-log power law curve.

Fractal complexity ranking across 122 Philippine shorelines (108 Major Islands & 14 Inland Lakes). Coral: Major Islands. Emerald: Inland Lakes. Reference lines denote Euclidean boundary (D = 1.0), national mean (D = 1.1545), and the mathematical Koch Snowflake (D = 1.2618).

Findings

Multi-Temporal Composite Verification

Single-date satellite scenes in the tropics suffer from moving cumulus clouds that can fragment extracted water boundaries. Matched-date temporal compositing reduces this interference, while never-clear coverage is retained as a limitation.

Laguna de Bay, March 2026 (Sentinel-2B), with the traced boundary overlaid. Reading notes: the boundary follows the urban West Bay at left — the stretch where LLRN Phase I hugs the shore — wraps around forested Talim Island in the center, and continues into the eastern bay. The straight feature crossing the lower-left shore is the Napindan Channel crossing, whose narrow channel is removed by the 150 m rule. Faint grids in the West Bay water are fishpen structures, which read as water. Residual clouds at upper right are excluded rather than traced.

March 2026 median composite of seven clear observations. Reading notes: black holes are pixels never cloud-free during the month — mostly over the eastern ridges and a persistent cloud cluster east of Talim Island — and are treated as unknown rather than land. Sediment plumes are visible along the shallow western shore. Because every pixel is the middle of seven observations, no single passing cloud or floating vegetation mat survives into the boundary the estimators measure.

LLRN Shoreline Monitoring

Mosquito-Borne Disease Data Readiness

This is an evidence-readiness matrix, not an incidence map: it shows why a shoreline–disease correlation cannot yet be estimated without inventing precision the public records do not provide.

Sources: DOH eFOI Laguna dengue municipality request; DOH eFOI Batangas dengue release; and DOH eFOI FHSIS vector-disease response. A valid test still requires case counts, population denominators, time lags, and a direct water-stagnation measure.

Shoreline Geometry Analysis

Open shoreline arcs kept at least 100 m from unknown pixels in both periods. Cyan: 2025. Orange: 2026. March and January-March windows are reported separately. Robustness sweeps (24 start points, both walking directions, 16 grid placements) bound how much of each reported change can be produced by the measuring tools alone.

The full LLRN Phase I-adjacent corridor was clear in both January-March composites. Its three estimators changed only slightly and did not agree in direction; the corridor's own start-point and grid-placement spreads (see the robustness sweeps) are of the same order as the changes, so no construction signal is claimed.