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Perpendicular Celtic

Un mapa cuadrático de la familia Perpendicular: pliega la parte real antes de elevar al cuadrado, vuelve a plegar el resultado real y la frontera se anuda en trenzas.

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Fractal Perpendicular Celtic con bandas trenzadas, arcos espejados y lazos tipo porcelana
Perpendicular Celtic: bandas entrelazadas y arcos espejados, con pliegues marcados en el borde y lazos en el interior.

Resumen

Perpendicular Celtic pertenece a la familia Perpendicular de variantes del Burning Ship, junto con Perpendicular Burning Ship y Perpendicular Buffalo. Su movimiento es un doble pliegue: pliega la componente real antes de elevar al cuadrado y luego vuelve a plegar la componente real del resultado. La componente imaginaria se salta ese segundo pliegue.

Cada iteración redirige la frontera dos veces. Empiezan a aparecer formas trenzadas y entrelazadas —tan parecidas a los nudos celtas que explican el nombre. Arcos espejados, lazos tipo porcelana y bandas de tiempo de escape giran en ángulo recto en los pliegues marcados.

En algunas regiones es simétrico respecto a los ejes real e imaginario, a diferencia de algunas variantes del Burning Ship. La razón es la disposición perpendicular: los pliegues actúan de forma ortogonal, un eje a la vez en lugar de ambos al mismo tiempo.

Las matemáticas

Perpendicular Celtic iteration

Fold the real input, square it, fold the resulting real part, then add c

First fold the real input: construct z' = |xₙ| + iyₙ. Square it: (z')² = (|xₙ|² − yₙ²) + i(2·|xₙ|·yₙ). Then fold the real part of that result: zₙ₊₁ = ||xₙ|² − yₙ²| + i(2·|xₙ|·yₙ) + c.

The double fold reflects the real component twice per iteration: before the square, negative x mirrors to positive; after it, a negative resulting real part mirrors to nonnegative. The imaginary component folds only indirectly, through |xₙ| in the cross term.

Consider c = −0.5. For the Mandelbrot set: z₁ = −0.5, z₂ = −0.25, bounded. For the Perpendicular Celtic: z₁ = ||0|² − 0²| + i(2·|0|·0) − 0.5 = 0 − 0.5 = −0.5, then z₂ = ||−0.5|² − 0²| + i(2·|−0.5|·0) − 0.5 = |0.25| − 0.5 = −0.25, same orbit. But try c = 0.3i: the double fold redirects the orbit in a way that neither the Mandelbrot set nor the Burning Ship reproduces.

Historia

Perpendicular Celtic belongs to the “Perpendicular” Burning Ship family, with Perpendicular Burning Ship, Perpendicular Buffalo, and related forms. Like the others, it came from the fractal art community rather than academic research.

“Celtic” names the visual resemblance to Celtic knotwork made by the double fold. “Perpendicular” names the folds’ orthogonal placement: first on the real coordinate before squaring, then on the real part of the result.

It appears in formula browsers supporting custom iterations, including Ultra Fractal, Fractal Extreme, and Mandelbrowser. The earliest documented instances are in online fractal forums and fractal wiki pages cataloging Burning Ship variants.

Características visuales

Braided bands, mirrored arches, and porcelain-like loops gather at folds where escape-time color turns at right angles. From far out, the structure is symmetric across both axes; deeper in, fine asymmetries appear.

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Preguntas frecuentes

Why are absolute values used twice?

The folds act at different moments: the first absolute value folds the real input before squaring; the second folds the squared result’s real part. The real component is therefore reflected twice per iteration, changing how the orbit responds to the parameter c. Compared with the single-fold Burning Ship, the boundary becomes finer and more braided.

Is it simply a rotated Burning Ship?

No. It is related to the Burning Ship through absolute-value folds, but it places them differently in the quadratic step. The Burning Ship folds both raw coordinates before squaring. Perpendicular Celtic folds the real coordinate before squaring and then folds the result’s real part again. That two-stage order gives the orbit a different geometry and a distinct symmetry.

Referencias