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<p>A generalization of the filled-in Julia set is presented using the multicomplex numbers and an algorithm is presented to visualize these sets in the tridimensional space. There are many ways to visualize these higher dimensional fractals sets on a computer. We therefore introduce an equivalence relation between 3D representations and show that, for the filled-in Julia sets associated to the polynomial <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(z^p + c\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>z</mi> <mi>p</mi> </msup> <mo>+</mo> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation>, there are nine 3D slices when <i>p</i> is an odd integer and four when <i>p</i> is even. These results differ from the recent characterization obtained by Brouillette and Rochon in 2019 and the proofs require different arguments in the context of the filled-in Julia sets.</p>

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Classification of Principle 3D Slices of Filled-in Julia Sets in Multicomplex Spaces

  • 期刊论文

  • 发表日期：

    2026年2月12日

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  • Quentin Charles,
  • Pierre-Olivier Parisé

摘要

A generalization of the filled-in Julia set is presented using the multicomplex numbers and an algorithm is presented to visualize these sets in the tridimensional space. There are many ways to visualize these higher dimensional fractals sets on a computer. We therefore introduce an equivalence relation between 3D representations and show that, for the filled-in Julia sets associated to the polynomial \(z^p + c\) z p + c , there are nine 3D slices when p is an odd integer and four when p is even. These results differ from the recent characterization obtained by Brouillette and Rochon in 2019 and the proofs require different arguments in the context of the filled-in Julia sets.