WebGL Error. Please use a modern browser/device with WebGL enabled.
Settings & Info
Number of collinear pieces
n
−
+
Complex Parameter c (Drag disk in viz)
Re(c)
−
+
Im(c)
−
+
|c|
−
+
arg(c) °
−
+
Parameter \(c\) defines the specific fractal sets. Drag the small colored disk in the visualizer or use controls here. Must have \(|c| > 1\). Tap/Double-click on the visualization background to move \(c\).
Algorithm Tuning
Iteration Levels (m) \(\mathcal{R}_n\)
Iteration Levels (m) \(\mathcal{M}_n\)
Queue Size
←
→
Thickness \(\mathcal{R}_n\)
−
+
Thickness \(\mathcal{M}_n\)
−
+
Advanced Features
'Iteration Levels (m)': Render points hitting target between m_min and m_max. 'Queue Size': Affects detail/speed (max adapted to device). 'Thickness': Target trap thickness scale (0-100%). Rn/Ecn thickness is effectively halved. Mn status check always uses 100% thickness. Advanced features modify coloring/traps. If neither "Only" trap toggle is active, automatic selection based on c's region is used.
Appearance
Palette Cycle
↻
Grayscale Cycle
↻
Custom Palette
Custom Palette Colors
Select 8 colors for the 'inside set' palette (C1-C8). Ensure sufficient contrast for accessibility. Changes are saved locally.
Visible Panels
Click buttons to toggle visibility. Double-click a panel button to isolate it. Use \(\subsetneq\) / \(\subseteq\) buttons for special views.
Presets
n=4c=...|c|=...arg=...°Zoom=1.0xMn Status: ...
Collinear Fractals (Preview) | B. Espigule, UdG (2025) | Complex Trees
Share Current View
Copy URL to share the current view parameters:
URL Copied!
Include in Share URL (if different from default):
Core parameters (n, c, panels) and advanced features are always included if different from default.
Embed in your website (iframe):
Embed Code Copied!
About Collinear Fractals & How to Cite
Introduction
This tool explores Collinear Fractals \(E(c,n)\) and associated parameter spaces \(\mathcal{R}_n\) and \(\mathcal{M}_n\), defined for integer \(n \ge 2\) and complex \(c\) with \(|c| > 1\). They arise from the Iterated Function System (IFS):
\(E(c,n)\) is the unique non-empty compact attractor: \(E(c,n) = \bigcup_{t \in A_n} f_t(E(c,n))\).
Polynomials with integer coefficients are fundamental objects in mathematics, serving as building blocks in various areas such as algebra, number theory, and geometry. The roots of these polynomials often exhibit fascinating and intricate patterns in the complex plane. In this paper, we uncover a deep connection between these roots and a class of self-similar sets that we call collinear fractals. These fractals are generated by repeatedly applying simple mathematical transformations involving a complex parameter \(c\) and an integer \(n\). Exotic elements of the family include some self-affine tiles with a collinear digit set independently studied in [1]. Remarkably, our set \(\mathcal{M}_n\) of parameters \(c\) for which the corresponding fractal is connected can be identified with the set of roots of polynomials with integer coefficients restricted from \(-n+1\) to \(n-1\). By exploring this bridge between algebra and geometry, we provide new insights into long-standing mathematical questions, demonstrating how the algebraic properties of polynomials shape the geometric structure of fractals and give rise to complex and beautiful sets.
The concept of visualizing roots of polynomials is not new, and numerous mathematical explorations have arisen from this idea, particularly in blogs and online mathematical discussions [2,3]. Research on the so-called Littlewood polynomials, whose coefficients are \(\pm 1\), produced some of the earliest \(\mathcal{M}_n\)-like imagery; see, for instance, the work of Peter and Jonathan Borwein [4,5]. Similarly, polynomials with coefficients restricted to \(\{0, 1\}\), known as Newman polynomials, were thoroughly investigated in the seminal work of Odlyzko and Poonen [6]. Other studies related to roots of polynomials include the Thurston’s Master Teapot [7–9], Algebraic Number Starscapes [10,11], and the eigenvalues of Bohemian Matrices [12].
Visualized Panels
Toggle panel visibility using the buttons (bottom or Settings). Click \(\subsetneq\) or \(\subseteq\) to show comparisons (click again for difference view). Double-click a panel button (\(\mathcal{R}_n\), \(\mathcal{M}_n\), etc.) to isolate it.
The small colored disk (c) overlaid on the view shows the precise location of the parameter \(c\). Its size on screen is invariant under zoom. The layout uses fixed horizontal divisions when multiple panels are shown, with color-negated lines separating them (lines are hidden in difference modes).
The connectedness locus \(\mathcal{M}_n\) is the set of parameters \(c\) for which \(E(c,n)\) is connected. We have shown (Proposition 1) that \(E(c,n)\) is connected iff neighboring pieces intersect, which is equivalent to \(2c \in E(c, 2n-1)\). The loci are nested: \(\mathcal{M}_n \subset \mathcal{M}_{n+1}\) (Proposition 3). We established bounds: \(\mathcal{M}_n \subset \mathcal{M}_n^* \subset B(1 + \sqrt{n - 1}, 0) \cup [-n, n]\) (Proposition 4), where \(\mathcal{M}_n^*\) is the convexity set. Also, \(\{c : 1 < |c| < \sqrt{n}\} \subset \mathcal{M}_n\) and \((-n, -1) \cup (1, n) \subset \mathcal{M}_n \cap \mathbb{R}\) (Proposition 5). Nakajima's work implies \(\mathcal{M}_n\) is connected and locally connected (Theorem 1).
Our main result in [13] (Theorem 2) concerns the region \(\mathcal{X}_n = \{ c \in \mathbb{C} \setminus (\mathbb{R} \cup \mathbb{D}) : |c \pm 1| \leq \sqrt{2n} \}\). We proved that \(\mathcal{M}_n \cap \mathcal{X}_n \subset \operatorname{clos}(\operatorname{int}(\mathcal{M}_n))\) for all \(n \ge 2\). This relies on a covering property (Lemma 3): for \(c \in \mathcal{X}_{(n+1)/2}\), the rectangle \(R(c,n)\) (defined by vertices \(\pm \psi(c,n), \pm \overline{\psi(c,n)}\) where \(\psi(c,n) = c(n+1)/(1 \pm c)\)) satisfies \(R(c,n) \subset \bigcup_{t \in A_n} (t + c^{-1}R(c,n))\). This implies \(R(c, 2n-1) \subset E(c, 2n-1)\) for \(c \in \mathcal{X}_n\). Using this and Rouché's theorem (Lemma 4: \(\mathcal{M}_n = \operatorname{clos}(\widehat{\mathcal{M}}_n)\), where \(\widehat{\mathcal{M}}_n\) are roots of polynomials with coefficients in \(D_n\)), we show that points in \(\widehat{\mathcal{M}}_n \cap \operatorname{int}(\mathcal{X}_n)\) are interior to \(\mathcal{M}_n\).
Furthermore, for \(n \ge 21\), we showed that \(1 + \sqrt{n-1} < -1 + \sqrt{2n}\), which implies \(B(1+\sqrt{n-1}, 0) \subset B(\sqrt{2n}, -1) \cap B(\sqrt{2n}, 1)\). Since \(\mathcal{M}_n \setminus \mathbb{R} \subset B(1+\sqrt{n-1}, 0)\), it follows that \(\mathcal{M}_n \setminus \mathbb{R} \subset \mathcal{X}_n\) for \(n \ge 21\). Therefore, for \(n \ge 21\), \(\mathcal{M}_n \setminus \mathbb{R} \subset \operatorname{clos}(\operatorname{int}(\mathcal{M}_n))\), proving the generalized Bandt's conjecture.
The covering rectangle \(R(c,n)\) has vertices \(\pm \psi(c,n)\) and \(\pm \overline{\psi(c,n)}\), where \(\psi(c,n) = c(n+1)/(1 \pm c)\). The 'Thickness' parameter scales the height (imaginary part) of this rectangle in the algorithm, while the width (real part) remains fixed. The parallelogram trap \(P(c,n)\) has vertices \(\pm(n-1 \pm \kappa(n') c^{-1})\), where \(\kappa(n') = 1 + \lfloor -2 - 2\sqrt{n'} + n' \rfloor\) for \(n' = (n+1)/2 > 7\) and \(\kappa(n')=1\) otherwise. The BFS algorithm automatically selects between \(R(c,2n-1)\) (if \(c\) is in the region \(\mathcal{X}_{n}\)) and \(P(c,2n-1)\) (otherwise), unless overridden by the "Only" toggles.
Rendering uses a GPU-accelerated Breadth-First Search (BFS) on the inverse Iterated Function System (IFS) maps \(g_t(z) = c(z-t)\). Layout uses fixed horizontal divisions with color-negated separators (hidden in difference modes). Algorithm tuning controls affect the BFS behavior: iteration range (`m_min` to `m_max`), queue size limits (max adapted to device capabilities), and target checking thickness (0-100%). Thickness for \(\mathcal{R}_n\) and \(E(c,n)\) is effectively halved (100% UI = 50% actual height scaling). Thickness for \(\mathcal{M}_n\) and \(\frac{1}{2}E(c,2n-1)\) uses the dedicated `thicknessME` parameter (100% UI = 100% height scaling for rectangle, factor 2 for parallelogram). The B&W view modes use simplified radius checks. Points inside the unit disk for parameter spaces (\(\mathcal{R}_n, \mathcal{M}_n\)) are darkened in normal color mode. Subset comparison difference modes render the inner set inverted and the outer set normally. '\(\mathcal{M}_n\) Status' uses a separate JavaScript BFS (in a Web Worker) to check if \(2c \in E(c, 2n-1)\) up to the iteration limit (`m_max2`), *always using 100% thickness for this check*, and reports the sequence of inverse maps applied to the initial point. Custom colors can be configured in Settings. The 'c' marker disk size is invariant under zoom. Advanced features (Darken Boundary Hits, Rectangle/Parallelogram Trap selection) modify coloring and trap logic.
Remarks
The connection \(2c \in E(c, 2n-1)\) for \(c \in \mathcal{M}_n\) illustrates the asymptotic self-similarity between \(\mathcal{M}_n\) and \(E(c, 2n-1)\). New research extended the regular-closed property for \(n<21\) using our parallelogram trap method (in preparation [39]). Future research could explore investigating the absolute continuity of measures on \(E(c,n)\).
References from the Paper
Akiyama, S.; Loridant, B.; Thuswaldner, J. Topology of planar self-affine tiles with collinear digit set. J. Fractal Geom. 2021, 8, 53–93.
Christensen, J.D. Plots of Roots of Polynomials with Integer Coefficients. 2006. Available online: http://jdc.math.uwo.ca/roots/ (accessed on 25 November 2024).
Baez, J.C.; Christensen, J.D.; Derbyshire, S. The Beauty of Roots. Not. Am. Math. Soc. 2023, 70, 1495–1497.
Bailey, D.H.; Borwein, J.M.; Calkin, N.J.; Girgensohn, R.; Luke, D.R.; Moll, V.H. Experimental Mathematics in Action; A K Peters: Natick, MA, USA, 2007.
Borwein, P.; Erdélyi, T.; Littmann, F. Polynomials with coefficients from a finite set. Trans. Am. Math. Soc. 2008, 360, 5145–5154.
Odlyzko, A.M.; Poonen, B. Zeros of Polynomials with 0, 1 Coefficients. L’Enseignement Mathématique 1993, 39, 317.
Bray, H.; Davis, D.; Lindsey, K.; Wu, C. The shape of Thurston’s Master Teapot. Adv. Math. 2021, 377, 107481.
Lindsey, K.; Wu, C. A characterization of Thurston’s Master Teapot. Ergod. Theory Dyn. Syst. 2023, 43, 3354–3382.
Lindsey, K.; Tiozzo, G.; Wu, C. Master teapots and entropy algorithms for the Mandelbrot set. Trans. Am. Math. Soc. 2024, doi:10.1090/tran/9346.
Harriss, E.; Stange, K.E.; Trettel, S. Algebraic Number Starscapes. Exp. Math. 2022, 31, 1098–1149.
Dorfsman-Hopkins, G.; Xu, S. Searching for rigidity in algebraic starscapes. J. Math. Arts 2022, 16, 57–74.
Sendra, J. Bohemian Matrices: Past, Present and Future. In Proceedings of the Communications in Computer and Information Science, Waterloo, ON, Canada, 2–6 November 2020; Volume 1414.
Espigule, B.; Juher, D.; Saldaña, J. Collinear Fractals and Bandt’s Conjecture. Fractal Fract. 2024, 8, 725.
Solomyak, B. Measure and dimension for some fractal families. Math. Proc. Camb. Philos. Soc. 1998, 124, 531–546.
Bandt, C.; Hung, N.V. Fractal n-gons and their Mandelbrot sets. Nonlinearity 2008, 21, 2653–2670.
Himeki, Y.; Ishii, Y. \(\mathcal{M}_4\) is regular-closed. Ergod. Theory Dyn. Syst. 2020, 40, 213–220.
Calegari, D.; Koch, S.; Walker, A. Roots, Schottky semigroups, and a proof of Bandt’s conjecture. Ergod. Theory Dyn. Syst. 2017, 37, 2487–2555.
Calegari, D.; Walker, A. Extreme points in limit sets. Proc. Am. Math. Soc. 2019, 147, 3829–3837.
Bousch, T. Sur Quelques Problèmes de Dynamique Holomorphe. Ph.D. Thesis, Université de Paris-Sud, Orsay, France, 1992.
Bousch, T. Connexité locale et par chemins holderiens pour les systemes itérés de fonctions. Unpublished work, 1993; pp. 1–23.
Nakajima, Y. Mandelbrot set for fractal n-gons and zeros of power series. Topol. Appl. 2024, 350, 108918.
Barnsley, M.F.; Harrington, A.N. A Mandelbrot set for pairs of linear maps. Phys. D Nonlinear Phenom. 1985, 15, 421–432.
Indlekofer, K.H.; Járai, A.; Kátai, I. On some properties of attractors generated by iterated function systems. Acta Sci. Math. 1995, 60, 411.
Bandt, C. On the Mandelbrot set for pairs of linear maps. Nonlinearity 2002, 15, 1127.
Solomyak, B.; Xu, H. On the ‘Mandelbrot set’ for a pair of linear maps and complex Bernoulli convolutions. Nonlinearity 2003, 16, 1733.
Solomyak, B. Mandelbrot set for a pair of linear maps: the local geometry. Anal. Theory Appl. 2004, 20, 149–157.
Solomyak, B. On the ‘Mandelbrot set’ for pairs of linear maps: asymptotic self-similarity. Nonlinearity 2005, 18, 1927–1943.
Shmerkin, P.; Solomyak, B. Zeros of \(\{-1, 0, 1\}\) Power Series and Connectedness Loci for Self-Affine Sets. Exp. Math. 2006, 15, 499–511.
Bandt, C.; Hung, N. Self-similar sets with an open set condition and great variety of overlaps. Proc. Am. Math. Soc. 2008, 136, 3895–3903.
Hare, K.G.; Sidorov, N. Two-dimensional self-affine sets with interior points, and the set of uniqueness. Nonlinearity 2015, 29, 1.
Hare, K.G.; Sidorov, N. On a family of self-affine sets: Topology, uniqueness, simultaneous expansions. Ergod. Theory Dyn. Syst. 2017, 37, 193–227.
Shmerkin, P.; Solomyak, B. Absolute continuity of complex Bernoulli convolutions. Math. Proc. Camb. Philos. Soc. 2016, 161, 435–453.
Silvestri, S.; Pérez, R.A. Accessibility of the Boundary of the Thurston Set. Exp. Math. 2023, 32, 405–422.
Bandt, C.; Keller, K. Self-Similar Sets 2. A Simple Approach to the Topological Structure of Fractals. Math. Nachrichten 1991, 154, 27–39.
Hata, M. On the structure of self-similar sets. Jpn. J. Appl. Math. 1985, 2, 381.
Beaucoup, F.; Borwein, P.; Boyd, D.; Pinner, C. Power series with restricted coefficients and a root on a given ray. Math. Comput. 1998, 67, 715–736.
Espigule, B. Asymptotic self-similarity between collinear fractals E(c,2n-1) and M_n. Available online: https://youtu.be/11NZDHNahJs (accessed on 12 June 2024).
Espigule, B., Juher, D., & Saldaña, J. (2025). Collinear fractals and the internal structure of Mn. (In preparation)
Guided Tour
Citation
If you use this tool or findings from it in your research, please cite the following paper:
@Article{EJS2024,
AUTHOR = {Espigule, B. and Juher, D. and Saldaña, J.},
TITLE = {{Collinear Fractals and Bandt's Conjecture}},
JOURNAL = {Fractal and Fractional},
VOLUME = {8},
YEAR = {2024},
NUMBER = {12},
ARTICLE-NUMBER = {725},
URL = {https://www.mdpi.com/2504-3110/8/12/725},
ISSN = {2504-3110},
DOI = {10.3390/fractalfract8120725}
}Copied!
For the web application itself:
Espigule, B. (2025). Collinear Fractals Explorer [Web application]. Complex Trees Research Project. Retrieved from complextrees.com/collinearCopied!
Upcoming work:
Espigule, B., Juher, D., & Saldaña, J. (2025). Collinear fractals and the internal structure of Mn. (In preparation)
Funding
This work was supported by grants PID2023-146424NB-I00 (MICINN, Spain) and 2021 SGR 00113 (Generalitat de Catalunya). B.E. acknowledges support from the IFUdG 2022–2024 fellowship program (Universitat de Girona & Banco Santander).