About

Fractional Brownian surfaces model natural terrain, cloud density fields, and material roughness. The underlying stochastic process depends on the Hurst exponent H, which controls spatial correlation: values near 1.0 produce smooth, rolling landscapes, while values near 0.0 yield jagged, high-frequency noise. This tool implements the Diamond-Square algorithm on a grid of size (2ⁿ + 1)², applying random midpoint displacement at each recursion level. The displacement magnitude decays by a factor of 2^−H per subdivision, directly linking the Hurst parameter to the spectral density falloff.

Incorrect roughness calibration produces unrealistic terrain. A surface intended for hydraulic erosion simulation requires H ≈ 0.7 - 0.8, matching empirical measurements of real mountain ranges. This generator exports both a colorized 3D preview and a raw grayscale heightmap suitable for import into game engines or GIS software. Note: the Diamond-Square algorithm introduces slight directional artifacts along grid axes. For production terrain, consider post-processing with erosion filters.

Formulas

The Diamond-Square algorithm generates a fractal heightmap on a square grid of side length N = 2ⁿ + 1. At each recursion level k, two steps alternate:

Diamond step: For each square of side s, the center point receives the average of its four corners plus a random displacement:

h_center = h₁ + h₂ + h₃ + h₄4 + R ⋅ δ_k

Square step: For each diamond, the midpoint of each edge receives the average of its diamond neighbors plus displacement.

The displacement scale decays exponentially:

δ_k = δ₀ ⋅ 2^{−H ⋅ k}

Where R ∈ [−1, 1] is a uniform random variable, δ₀ is the initial displacement amplitude, and H is the Hurst exponent (0 < H ≤ 1). Higher H produces smoother surfaces because high-frequency components are suppressed more aggressively. The resulting surface has a spectral density proportional to 1f^{2H + 2}, characteristic of fractional Brownian motion in two dimensions.

Reference Data

Terrain Type	Hurst Exponent H	Character	Real-World Analog
Jagged Peaks	0.10 - 0.25	Extremely rough, sharp spikes	Karst limestone, coral reef
Rocky Mountains	0.25 - 0.40	High frequency, steep gradients	Alpine peaks, volcanic ridges
Eroded Hills	0.40 - 0.55	Moderate roughness, natural look	Appalachian hills, weathered cliffs
Rolling Plains	0.55 - 0.70	Gentle undulations	Great Plains, Scottish Highlands
Smooth Dunes	0.70 - 0.85	Low frequency, broad features	Saharan dunes, glacial moraines
Near-Flat Plateau	0.85 - 1.00	Very smooth, minimal variation	Salt flats, frozen lakes
Cloud Density	0.60 - 0.75	Soft, billowy gradients	Cumulus cloud fields
Ocean Floor	0.45 - 0.65	Mid-ocean ridges + abyssal plains	Atlantic seabed
Martian Terrain	0.50 - 0.60	Cratered, moderately rough	Mars Valles Marineris
Lunar Surface	0.30 - 0.50	Impact craters, regolith	Moon highland terrain
Glacier Surface	0.65 - 0.80	Crevassed but broadly smooth	Antarctic ice sheet
Fracture Network	0.15 - 0.30	High contrast, crack-like	Dried mud, tectonic faults
Procedural Game Map	0.50 - 0.65	Balanced realism	Open-world RPG terrain
Material Roughness	0.20 - 0.45	Micro-scale surface texture	PBR roughness maps
White Noise (Limit)	0.00	Uncorrelated, maximum entropy	Static / random noise

Frequently Asked Questions

The fractal dimension of a Brownian surface is D = 3 − H. A Hurst exponent of 0.5 yields D = 2.5, which is typical for natural mountain terrain. Lower H increases the fractal dimension toward 3.0, producing space-filling roughness.

The Diamond-Square algorithm requires recursive halving of the grid. A side length of 2ⁿ + 1 ensures that midpoints always land on integer coordinates at every subdivision level. Common sizes: 65, 129, 257, 513, 1025.

The square step samples along axis-aligned diamonds, introducing directional bias. This is a known limitation. Mitigation strategies include: adding a small amount of post-process Gaussian blur, using wrapping boundary conditions, or switching to spectral synthesis (FFT-based) for artifact-free results. This tool uses edge-aware averaging to reduce but not eliminate this effect.

Yes. Export the grayscale heightmap PNG. In Unity, import it as a RAW or PNG heightmap in the Terrain component (Terrain > Import Heightmap). In Unreal, use the Landscape tool's Import function. Ensure the resolution matches your terrain actor dimensions. The grayscale values map linearly: black = minimum elevation, white = maximum.

The seed initializes a deterministic pseudo-random number generator (Mulberry32). Identical seed, grid size, and roughness will always produce the same surface. Changing the seed by even 1 produces a completely different terrain. Use seeds to catalog surfaces or share configurations with collaborators.

The maximum grid in this tool is 1025 × 1025 (2¹⁰ + 1), requiring approximately 8.4 MB for a Float64Array. Computation takes 50 - 200 ms on modern hardware. The 3D renderer draws up to ~2 million triangles, which may reduce frame rate on integrated GPUs. For smooth interaction, use 257 or lower for real-time rotation.