Mathematics · computation · visual exploration

Bernat
Espigulé

Mathematical researcher & creator of Complex Trees

I study self-similar sets and the spaces of parameters that shape them. My work connects proofs and computation with interactive maps, mathematical images, and sculpture.

PhD candidate · Universitat de Girona · Catalonia

Explore the work View CV

The Beauty of Roots: a gold and blue map of polynomial roots with fine branching detail and marked algebraic landmarks. — The Beauty of Roots · 2019Mathematical cartography ↗

September 2026 Thesis deposited at the Universitat de Girona. Defence pending.The doctoral thesis

From a parameter to a world

Explore the mathematics.

All projects & materials ↗

Three-dimensional angular chart of the four-real-dimensional parameter family, coloured by finite fingerprint components.

Interactive atlas · Research preview

Connectedness in four dimensions

Move through planar binary similarity families and inspect their attractors, contacts, and finite numerical signatures.

Explore ↗Read the guide

$Collinear fractals explorer with linked parameter and dynamical planes.$

Research software · 2026

Collinear Fractals Explorer

Follow a parameter into its self-similar set. Explore connectedness, finite capture, and restricted polynomial roots.

Explore ↗Source

A sequence of branching Golden Trees forms.

Geometry · sculpture · animation

Mathematical forms

See how a family of branching rules becomes an image, an animation, or a physical object.

Explore ↗Works & records

Recent research

Publications & preprints.

Browse all works ↗

2024

Collinear Fractals and Bandt's Conjecture

Journal article

Bernat Espigule, David Juher, Joan SaldañaFractal and Fractional 8(12), 725

DOI ↗arXiv ↗PDF ↗

Cite

@article{espigulejuhersaldana2024bandt,
  author = {Espigule, Bernat and Juher, David and Salda{\~n}a, Joan},
  title = {Collinear Fractals and {Bandt}'s Conjecture},
  year = {2024},
  journal = {Fractal and Fractional},
  volume = {8},
  number = {12},
  pages = {725},
  doi = {10.3390/fractalfract8120725},
  url = {https://doi.org/10.3390/fractalfract8120725}
}

2026

Finite capture and the closure of roots of restricted polynomials

Preprint

Bernat Espigule, David JuherarXiv:2603.07397

arXiv ↗PDF ↗Explorer ↗

Cite

@misc{espigulejuher2026capture,
  author = {Espigule, Bernat and Juher, David},
  title = {Finite capture and the closure of roots of restricted polynomials},
  year = {2026},
  eprint = {2603.07397},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.2603.07397},
  url = {https://arxiv.org/abs/2603.07397}
}

2026

Infinitely many holes in connectedness loci for collinear affine iterated function systems

Preprint

Bernat EspigulearXiv:2606.01467

arXiv ↗PDF ↗

Cite

@misc{espigule2026holes,
  author = {Espigule, Bernat},
  title = {Infinitely many holes in connectedness loci for collinear affine iterated function systems},
  year = {2026},
  eprint = {2606.01467},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.2606.01467},
  url = {https://arxiv.org/abs/2606.01467}
}

2026

Explicit interpolations among the Sierpiński, Rauzy, and Apollonian gaskets

Preprint

Bernat EspigulearXiv:2606.23278

arXiv ↗PDF ↗Explorer ↗

Cite

@misc{espigule2026gaskets,
  author = {Espigule, Bernat},
  title = {Explicit interpolations among the {Sierpi\'nski}, {Rauzy}, and {Apollonian} gaskets},
  year = {2026},
  eprint = {2606.23278},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.2606.23278},
  url = {https://arxiv.org/abs/2606.23278}
}

Doctoral thesis · 2026

Collinear Fractals
and Connectedness Loci

Topology, Finite Capture,
and Restricted Polynomial Roots

Bernat Espigulé PonsUdG

The doctoral thesis

Topology, finite capture,
and polynomial roots.

My doctoral research brings together the geometry of collinear self-similar sets, the topology of their connectedness loci, and roots of polynomials with restricted integer coefficients.

Supervised by David Juher and Joan Saldaña.
Corrected circulation edition · 15 September 2026.

Read about the thesis ↗Cite the thesis

Research & experience

A path through mathematics.

Trained in physics and mathematics, I have worked in research, scientific computing, museum content, and teaching.

Full academic & professional CV ↗

2022–present: Doctoral research · Universitat de GironaSelf-similar sets, connectedness loci, and restricted polynomial roots. Thesis deposited; defence pending.
2021: MSc Advanced Mathematics · Universitat de BarcelonaMaster’s thesis completed in 2018; degree awarded in 2021.
2013–2017 2020–2021: Consultant · Wolfram ResearchMathematical computation, interactive content, and technical communication.
2012: BSc Physics · Universitat Autònoma de BarcelonaInternational study at UC Santa Barbara and Heidelberg.

Ideas in motion

Talks & communication.

Talks & resources ↗

Mar 2026

Finite capture and the closure of roots of restricted polynomialsInstitut Henri Poincaré · Paris

Watch recording ↗

2020–2021

History of MathematicsInteractive learning journeys · Wolfram Research & MoMath

Visit the exhibition ↗

A collection of branching mathematical forms from the Fractal Arboretum.

From a theorem to something you can see.

Animations, workshops, and sculpture open different ways into the same mathematical questions.

Explore the art portfolio ↗Teaching & workshops

BernatEspigulé

Explore the mathematics.

Connectedness in four dimensions

Collinear Fractals Explorer

Mathematical forms

Publications & preprints.

Collinear Fractals and Bandt's Conjecture

Finite capture and the closure of roots of restricted polynomials

Infinitely many holes in connectedness loci for collinear affine iterated function systems

Explicit interpolations among the Sierpiński, Rauzy, and Apollonian gaskets

Topology, finite capture,and polynomial roots.

A path through mathematics.

Talks & communication.

From a theorem to something you can see.

Bernat
Espigulé

Topology, finite capture,
and polynomial roots.