FractalPark
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Buffalo

A folded Burning Ship relative: squared coordinate terms take absolute values, while the cross term keeps x signed and opens broad horns with mirrored inner gates.

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Buffalo fractal with curved horned lobes, mirrored gates, and twisted filaments
The Buffalo: broad horns, mirrored interior gates, and twisted filament edges.

Overview

Buffalo moves the Burning Ship’s folds to new places. It squares the real and imaginary parts separately, takes absolute values of those squared magnitudes, and keeps x signed in the cross term. The result is distinct from both Burning Ship and the simpler Celtic variants.

The parameter set opens into broad, curved horns. Inside sit mirrored archways and eclipse-shaped pockets, often rimmed with twisted filaments. Fold across the real axis and the upper and lower halves closely answer one another.

Burning Ship favors rectangular, flame-like ridges. Buffalo rounds its large outer lobes, then saves the sharp turns for inner gates and filament edges.

The Mathematics

Buffalo iteration

z(n+1) = absolute x(n) squared minus absolute y(n) squared + 2 i x(n) absolute y(n) + c

The squared coordinate magnitudes make the real part: xₙ₊₁ = |xₙ|² − |yₙ|² + Re(c). The imaginary part pairs signed xₙ with folded yₙ: yₙ₊₁ = 2·xₙ·|yₙ| + Im(c). The difference from Burning Ship sits exactly there: Buffalo folds squared terms one by one, rather than folding raw coordinates before squaring. Keeping xₙ signed preserves a directional imbalance that Burning Ship’s fully folded cross term erases.

Try c = −0.5. Starting from z₀ = 0, the first iteration gives z₁ = |0|² − |0|² + 2i·0·|0| − 0.5 = −0.5. The second iteration: z₂ = |−0.5|² − |0|² + 2i·(−0.5)·|0| − 0.5 = 0.25 − 0.5 = −0.25. The orbit settles into a bounded oscillation between negative real values. With the same c, Burning Ship gives (|−0.5| + i|0|)² − 0.5 = 0.25 − 0.5 = −0.25 at the second step; the paths already differ because it folded the real part before squaring.

History

After Burning Ship appeared in 1992, Buffalo surfaced in the fractal exploration community as one of several variants. It belongs to folded quadratic maps, where shifting an absolute-value operation to another spot in the recurrence reshapes the escape-time geometry.

Buffalo has no single published introduction. It appears to have come from online fractal-software communities, where programs such as Ultra Fractal and Fractal Extreme let people define custom iterations. “Buffalo” likely refers to the large horn-like lobes, resembling a buffalo-head silhouette.

The earliest documented reference is the theory.org fractal dynamics page, which calls Buffalo a Burning Ship derivative. FractalPark uses the HPDZ Buffalo iteration rather than the original Buffalo formula described there.

Visual Characteristics

Broad horns establish the outline, with smooth outer curves cut by sharp inner detail. Inside are mirrored eclipse-like gates and twisted filament edges. The set reflects across the real axis.

Remix and Examples

Start from the canonical Document

Open the same approved formula state used by this guide, then change the view, coloring, transforms, or animation in the interactive Explorer.

Remix in ExplorerOpen Formula Editor

Frequently Asked Questions

Which part of the orbit is folded?

Buffalo takes absolute values of the squared coordinate magnitudes. In the cross term it folds only the imaginary component and keeps x signed. So its folds act after squaring on each coordinate’s magnitude; Burning Ship instead folds both raw components first and squares one complex number.

Can Buffalo produce Julia sets?

Yes. Fix c and vary the starting point z₀ to reveal Buffalo Julia sets. As in the parameter set, the chosen c controls the balance of smooth and angular features. Both connected and disconnected Julia structures appear.

References