Quartic Julia

Take the usual Julia recipe and turn the dial one notch: fix one complex constant c, then repeat z → z⁴ + c. Each fourth power multiplies the complex argument by four. A direction that might fork in two for a quadratic map can open into four here — the source of the four-way branching.

With the same c, this higher-degree map usually crams more turns into the set than its quadratic cousin. Whether the filled Julia set stays in one piece or breaks apart still depends on c, under the Fatou–Julia classification for higher-degree polynomials. Watch the critical orbit—here governed by the critical point z = 0 and the critical points of the derivative—and it gives the clue to that connectivity.

On screen, it reads less like a cloud than a stamped ornament: fine filigree, rosettes, and seal-like knots recur with four-fold symmetry. The extra degree packs in detail, yet the angular layout stays strikingly orderly.

The theory came first. Pierre Fatou and Gaston Julia developed the classical framework for iterated rational functions in 1917–1918, and it applies to polynomials of any degree. Quartic Julia sets inherit its principles for connectivity and dynamical classification. When fractal-rendering software moved beyond quadratics in the 1980s and 1990s, computer exploration of higher-degree Julia sets became common. The named quartic variant belongs to that line; no single discoverer is attributed to it.

FractalPark uses the standard fourth-degree recurrence as a fixed-parameter Julia-family formula.

Angles multiplied by four leave a recognizable trace: fine seals, branching rosettes, and filaments arranged on a quartic grid. A quadratic Julia may grow a spiral or one pair of arms; here, feature centers often send out four matching arms or lobes.

More detail fits into the same space than in a quadratic Julia, but it arrives in a tidier order. Turn ninety degrees around a prominent point and a motif often returns. The canonical FractalPark image picks out those edges with ivory filigree.