Karl Weierstrass’s function revolutionized mathematics by demonstrating a curve that is continuous everywhere but differentiable nowhere. This “monster” function, built from an infinite sum of cosine waves with increasingly higher frequencies and smaller amplitudes, visually appears jagged and chaotic at every scale. Its existence challenged the prevailing assumption that continuous functions were mostly smooth, with only isolated points of non-differentiability. Weierstrass's discovery exposed a deep rift between intuition and mathematical rigor, ushering in a new era of analysis focused on precise definitions and rigorous proofs, impacting fields from calculus to fractal geometry.
In the realm of mathematical analysis, a field dedicated to the study of change and continuity, there exists a peculiar function known as the Weierstrass function, often dramatically dubbed "Weierstrass's Monster." This function, formulated by the eminent German mathematician Karl Weierstrass in 1872, presented a groundbreaking challenge to the conventional understanding of calculus, specifically the concept of differentiability. It possesses a truly remarkable property: while being undeniably continuous, meaning its graph can be drawn without lifting the pen, it is nowhere differentiable. This means that at no point on the curve can a tangent line be drawn, as the function experiences infinitely rapid oscillations at every scale.
Before Weierstrass's discovery, mathematicians generally believed that continuous functions were differentiable almost everywhere, meaning they possessed derivatives at all but a few exceptional points, such as sharp corners or cusps. Weierstrass's function shattered this assumption, demonstrating the existence of functions that are continuous everywhere yet differentiable nowhere. This revelation sent shockwaves through the mathematical community, prompting a re-evaluation of the foundations of calculus and a deeper exploration of the subtle relationship between continuity and differentiability.
The Weierstrass function is typically represented as an infinite sum of cosine functions with carefully chosen frequencies and amplitudes. These individual cosine waves, while smooth and differentiable on their own, combine in a way that produces the function's characteristic jaggedness and non-differentiability. This intricate construction demonstrates the power of infinite series to generate highly complex and counterintuitive mathematical objects.
The discovery of the Weierstrass function was a pivotal moment in the history of mathematics. It exposed limitations in the intuitive understanding of continuity and differentiability, highlighting the importance of rigorous mathematical proof. It also spurred the development of more precise definitions and a deeper understanding of the subtleties of real analysis. Furthermore, the Weierstrass function served as a catalyst for the development of new branches of mathematics, such as fractal geometry, which deals with complex shapes exhibiting self-similarity at different scales. The "monster" ultimately helped mathematicians tame the wild and intricate landscape of functions, leading to a richer and more nuanced understanding of the mathematical universe. While initially perceived as a pathological example, it ultimately enriched and expanded the field, demonstrating the beauty and sometimes surprising nature of mathematical exploration.
Summary of Comments ( 81 )
https://news.ycombinator.com/item?id=42810103
HN users generally express fascination with the Weierstrass function and its implications for calculus. Several comments dive into the history and significance of the function, appreciating its role in challenging intuitive notions of continuity and differentiability. Some discuss its relation to fractals and Brownian motion, while others highlight the beauty of mathematical discoveries that defy expectations. A few commenters provide additional resources, including links to visualizations and related mathematical concepts like space-filling curves. Some debate the accessibility of the original Quanta article, suggesting ways it could be more easily understood by a broader audience. A recurring theme is the wonder inspired by such counterintuitive mathematical objects.
The Hacker News post titled "Weierstrass's Monster," linking to a Quanta Magazine article about the Weierstrass function, has a moderate number of comments discussing various aspects of the function and its historical context.
Several commenters focus on the function's counterintuitive properties. One commenter highlights the disconnect between intuitive notions of smoothness and the mathematical definition of continuity, noting that the Weierstrass function is continuous everywhere but differentiable nowhere. This leads to a discussion about how mathematical definitions can sometimes defy our intuition based on real-world experience. Another user echoes this sentiment by explaining how the function subverts the expectation that a continuous curve should have a tangent at every point, pointing out the "infinite jaggedness" at all scales.
Another thread discusses the historical significance of the function. A commenter notes that the Weierstrass function challenged the prevailing assumptions of 19th-century mathematicians about the relationship between continuity and differentiability, opening up new avenues of research in analysis. This sparked a brief exchange about other mathematical "monsters" and their role in advancing mathematical understanding. Someone further elaborates on the historical impact, mentioning that the function's discovery contributed to a more rigorous and formal approach to mathematical analysis, moving away from purely intuitive arguments.
A few comments delve into the practical implications or applications of the function. While some acknowledge that direct real-world applications might be limited, one commenter suggests its potential relevance to modeling phenomena exhibiting fractal-like behavior, where roughness persists at arbitrarily small scales.
Some comments offer additional resources or related concepts. One user points to a 3Blue1Brown video that visually explains the concept of the Weierstrass function. Another commenter connects the function to Brownian motion, mentioning the similar nowhere-differentiable nature of Brownian paths.
Finally, some comments provide technical clarifications or corrections. One commenter points out a subtle distinction between nowhere differentiability and infinite jaggedness, emphasizing that infinite jaggedness is not a formally defined mathematical concept. Another user clarifies the historical context, pointing out that while Weierstrass's example is the most famous, he was not the first to discover such a function.