$Newton’s fractal (which Newton knew nothing about)$

Newton’s fractal (which Newton knew nothing about)

Item: Newton’s fractal (which Newton knew nothing about)
Rating: 5
Author: PostSingularity Institute

by 3Blue1Brown (Grant Sanderson)

Formal & Physical Sciences Mathematics PBKCalculus and mathematical analysis PBKJDifferential calculus and equations

🎙 3Blue1Brown (Grant Sanderson) 👥 8.6M 📅 October 12, 2021 ⏱ 26 min 👁 3.2M 📄 science communication 🧭 2026-08-28

Available in: English (current) Français

Keywords

Newton-Raphsonfractalcomplex planepolynomialiteration

Summary

This video by 3Blue1Brown explores the fractal patterns that emerge when applying Newton’s method to find roots of polynomials in the complex plane. It begins with the practical importance of root-finding in engineering, using the example of rendering Bézier curves in computer graphics. The method is introduced as an iterative algorithm that approximates roots by repeatedly using linear approximations. When applied to complex numbers, the basins of attraction for different roots form intricate fractal boundaries. The video demonstrates that these fractals appear for any polynomial with three or more distinct roots, and explains why the boundaries must be infinitely complex: the boundary of each basin is shared by all others, preventing any smooth segments. This property is illustrated with a thought experiment involving colored blobs. The video also touches on related topics such as the insolubility of the quintic, Voronoi diagrams, and fractal dimension. It concludes with a cliffhanger hinting at connections to the Mandelbrot set, setting up a follow-up video.

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Critical Evaluation

Value of the Information & Strength of the Argument

The video provides significant educational value by demystifying a complex mathematical phenomenon through clear visualizations and intuitive explanations. The argumentation is solid: it builds from a simple algorithm to a deep property of fractal boundaries, proving that the complexity is not accidental but necessary. The use of interactive examples and step-by-step reasoning strengthens the logical flow. The video also connects abstract mathematics to real-world applications, enhancing its relevance.

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Title / Content Match

The title accurately reflects the content: the video explores the fractal generated by Newton's method, a concept Newton himself did not know about.

Quality & Reliability

9/10

High-quality mathematical exposition with rigorous reasoning, clear visualizations, and references to reputable sources. The content is accurate and well-explained, with a strong pedagogical approach.

Chapters

Cited Sources

Newton's Fractal - Interactive Lesson — Interactive companion to the video, allowing users to explore the fractal.
Simon Tatham's page on Newton's Fractal — Article discussing Newton's fractal and its properties.
Polynomial Roots and Newton's Method — Blog post exploring polynomial roots and Newton's method.
The Beauty of Bézier Curves — Video by Freya Holmér on Bézier curves, relevant to the application example.
The insolubility of the quintic — Video discussing the insolubility of the quintic, mentioned in the video.
The math behind rasterizing fonts — Video on font rasterization, related to the polynomial application.
Mathologer on the cubic formula — Video by Mathologer on the cubic formula, referenced for background.
Manim - Mathematical Animation Engine — Custom Python library used for animations in the video.
Manim Community Edition — Community-maintained version of the animation library.
3Blue1Brown FAQ — FAQ section explaining the use of Manim.

Concurring Sources

Newton's method - Wikipedia — General reference for the algorithm, consistent with the video's explanation.
Fractal - Wikipedia — General concept of fractals, supporting the video's classification.

External References

Contribution & Novelties

The video offers a fresh perspective on Newton’s method by visualizing its behavior in the complex plane, revealing the fractal nature of basins of attraction. It provides a clear explanation of why these fractals must exist, based on the property that the boundary of each basin is shared by all others. This insight is not commonly presented in standard mathematics education.

Pour aller plus loin :

Newton’s method (Wikipedia) — Foundational algorithm, directly relevant.
Fractal dimension (Wikipedia) — Concept used to measure the boundary’s complexity.
Basin of attraction (Wikipedia) — The regions colored in the fractal.
Mandelbrot set (Wikipedia) — Appears in the cliffhanger, related to iterative dynamics.

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Radar Profile

The radar profile shows very high scores across all dimensions, indicating a well-rounded, highly informative, and technically sound video. The lowest score is in technical level (8), but this is still high, reflecting the video's accessibility without sacrificing depth.

Reliability 9/10

💬 Très positif. Sur les 30 commentaires analysés, les spectateurs expriment une admiration massive pour la clarté des explications, la beauté des visualisations, et l'impact pédagogique du contenu, avec plusieurs témoignages personnels sur l'influence positive de la chaîne.