Fractal Dimension Calculator

1) Box-Counting Dimension

Divide the plane into square boxes of side ε. Count N(ε), the number of boxes that contain at least one point.

Dimension estimate: fit a line to ln N(ε) versus ln(1/ε). The slope is the box-counting dimension D.

Model: N(ε) ≈ C · ε^{-D} so ln N ≈ D ln(1/ε) + ln C.

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2) Higuchi Dimension

For a time series, build k-spaced subsequences and compute average curve length L(k).

Dimension estimate: fit ln L(k) versus ln k. With L(k) ∝ k^{-D}, the Higuchi dimension is D = −slope.

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3) Katz Dimension

For a discrete curve, compute total length L and maximum distance d from the first point. Then D = log10(n) / (log10(d/L) + log10(n)), where n is the number of steps.

1. Select Input Type and paste values or points.
2. Pick a Method based on your data style.
3. Adjust scales (k range or k max) if needed.
4. Press Calculate to view the dimension estimate.
5. Use Download CSV or Download PDF for reports.

1) Why Fractal Dimension Matters

Fractal dimension summarizes how a pattern fills space as you zoom in. In physics, it is used to compare rough interfaces, diffusion paths, porous materials, turbulence traces, and complex signals where ordinary Euclidean dimension is too limited.

2) Box-Counting for Spatial Patterns

Box-counting estimates scaling by overlaying grids and counting occupied boxes. In practice, stable fits often appear over a middle range of scales, not the very finest boxes where noise dominates or the coarsest boxes where saturation occurs.

3) Choosing the Scale Range

The k range controls the grid resolution. Small k gives large boxes, while larger k gives smaller boxes. A useful rule is to select a range where ln N versus ln(1/ε) looks close to linear and the reported R² is high.

4) Interpreting R² and Fit Quality

R² measures how well a straight line explains the scaling plot. Values near 1 indicate consistent power-law behavior across the chosen scales. Lower values can occur with short datasets, mixed regimes, or strong measurement noise. Try adjusting k limits to check sensitivity.

5) Higuchi Dimension for Time Series

Higuchi analyzes roughness directly in the sampled signal by measuring curve length at different step sizes. It is popular for experimental records such as vibration, geophysical traces, and physiological signals. Larger dimensions generally indicate more irregularity.

6) Katz Dimension for Quick Screening

Katz uses total path length and the farthest excursion from the start, producing a fast single-number estimate. It is useful for rapid comparisons across many recordings, but it can be less sensitive to multi-scale structure than scaling-based methods.

7) Data Requirements and Preprocessing

For reliable estimates, use long records when possible (hundreds to thousands of samples). Remove obvious outliers, consider detrending slow drifts, and keep sampling consistent. For points, include enough locations to populate multiple grid levels meaningfully.

8) Reporting and Reproducibility

When publishing results, report the method, scale parameters, and fit statistics. Export tables to keep an audit trail and enable reanalysis. Comparing methods on the same dataset can reveal whether complexity is truly multi-scale or dominated by short-scale noise.

1) Which method should I start with?

Use box-counting for spatial point patterns. For sampled signals, try Higuchi first because it captures multi-scale roughness. Katz is best for fast screening across many recordings.

2) Why does my dimension change when I adjust k?

Different k ranges emphasize different scales. Noise affects fine scales, while saturation affects coarse scales. Pick a range where the scaling plot is close to linear and the R² is strong.

3) What dimension values are “normal”?

It depends on the embedding and process. Smooth curves tend toward 1, and very rough plane-filling traces approach 2. Real signals often fall between 1.1 and 1.9 depending on dynamics and noise.

4) How many samples do I need?

More is better. Aim for at least 100 samples for quick checks, and 500+ for robust scaling fits. Sparse data can produce unstable estimates and misleading R² values.

5) Should I normalize or rescale my data?

Rescaling does not change theoretical dimension, but it can improve numerical stability. This tool normalizes point ranges for box-counting. For time series, consistent sampling and basic cleaning matter most.

6) Can I use 2D points with Higuchi or Katz?

Not directly. Higuchi and Katz here are implemented for one-dimensional sequences. If you have 2D trajectories, you can analyze each coordinate as a series or use box-counting on the set of points.

7) What does a low R² mean?

Low R² suggests weak power-law scaling across selected scales. Causes include short records, mixed regimes, strong noise, or inappropriate k limits. Try different scale ranges and compare methods for consistency.