This blog post, "Portrait of the Hilbert Curve (2010)," delves into the fascinating mathematical construct known as the Hilbert curve, providing an in-depth exploration of its properties and an elegant Python implementation for generating its visual representation. The author begins by introducing the Hilbert curve as a continuous fractal space-filling curve, emphasizing its remarkable ability to map a one-dimensional sequence onto a two-dimensional plane while preserving locality. This means that points close to each other in the linear sequence are generally mapped to points close together in the two-dimensional space. This property makes the Hilbert curve highly relevant for diverse applications, such as image processing and spatial indexing.
The post then meticulously dissects the recursive nature of the Hilbert curve, explaining how it's constructed through repeated rotations and concatenations of a basic U-shaped motif. It illustrates this process with helpful diagrams, showcasing the curve's evolution through successive iterations. This recursive definition forms the foundation of the Python code presented later.
The core of the post lies in the provided Python implementation, which elegantly translates the recursive definition of the Hilbert curve into a concise and efficient algorithm. The code generates a sequence of points representing the curve's path for a given order (level of recursion), effectively mapping integer indices to corresponding coordinates in the two-dimensional plane. The author takes care to explain the logic behind the coordinate calculations, highlighting the bitwise operations used to manipulate the input index and determine the orientation and position of each segment within the curve.
Furthermore, the post extends the basic implementation by introducing a method to draw the Hilbert curve visually. It utilizes the calculated coordinate sequence to produce a graphical representation, allowing for a clear visualization of the curve's intricate structure and space-filling properties. The author discusses the visual characteristics of the resulting curve, noting its self-similar nature and the increasing complexity with higher orders of recursion.
In essence, "Portrait of the Hilbert Curve (2010)" provides a comprehensive and accessible introduction to this fascinating mathematical concept. It combines a clear theoretical explanation with a practical Python implementation, enabling readers to not only understand the underlying principles but also to generate and visualize the Hilbert curve themselves, fostering a deeper appreciation for its elegance and utility. The post serves as an excellent resource for anyone interested in exploring fractal geometry, space-filling curves, and their applications in various fields.
The blog post "Bad Apple but it's 6,500 regexes that I search for in Vim" details a complex and computationally intensive method of recreating the "Bad Apple" animation within the Vim text editor. The author's approach eschews traditional methods of animation or video playback, instead leveraging Vim's regex search functionality as the core mechanism for displaying each frame.
The process begins with a pre-processed version of the Bad Apple video. Each frame of the original animation is converted into a simplified, monochrome representation. These frames are then translated into a series of approximately 6,500 unique regular expressions. Each regex is designed to match a specific pattern of characters within a specially prepared text buffer in Vim. This buffer acts as the canvas, filled with a grid of characters that represent the pixels of the video frame.
The core of the animation engine is a Vim script. This script iterates through the sequence of pre-generated regexes. For each frame, the script executes a search using the corresponding regex. This search highlights the matching characters within the text buffer, effectively "drawing" the frame on the screen by highlighting the appropriate "pixels." The rapid execution of these searches, combined with the carefully crafted regexes, creates the illusion of animation.
To further enhance the visual effect, the author utilizes Vim's highlighting capabilities. Matched characters, representing the black portions of the frame, are highlighted with a dark background, creating contrast against the unhighlighted characters, which represent the white portions. This allows for a clearer visual representation of each frame.
Due to the sheer number of regex searches and the computational overhead involved, the animation playback is significantly slower than real-time. The author acknowledges this performance limitation, attributing it to the inherent complexities of regex processing within Vim. Despite this limitation, the project demonstrates a unique and inventive application of Vim's functionality, showcasing the versatility and, perhaps, the limitations of the text editor. The author also provides insights into their process of converting video frames to regex patterns and optimizing the Vim script for performance.
The Hacker News post titled "Bad Apple but it's 6,500 regexes that I search for in Vim" (linking to an article describing the process of recreating the Bad Apple!! video using Vim regex searches) sparked a lively discussion with several interesting comments.
Many commenters expressed amazement and amusement at the sheer absurdity and technical ingenuity of the project. One commenter jokingly questioned the sanity of the creator, reflecting the general sentiment of bewildered admiration. Several praised the creativity and dedication required to conceive and execute such a complex and unusual undertaking. The "why?" question was raised multiple times, albeit rhetorically, highlighting the seemingly pointless yet fascinating nature of the project.
Some commenters delved into the technical aspects, discussing the efficiency (or lack thereof) of using regex for this purpose. They pointed out the computational intensity of repeatedly applying thousands of regular expressions and speculated on potential performance optimizations. One commenter suggested alternative approaches that might be less resource-intensive, such as using image manipulation libraries. Another discussed the potential for pre-calculating the matches to improve performance.
A few commenters noted the historical precedent of using unconventional tools for creative endeavors, drawing parallels to other esoteric programming projects and "demoscene" culture. This placed the project within a broader context of exploring the boundaries of technology and artistic expression.
Some users questioned the practical value of the project, while others argued that the value lies in the exploration and learning process itself, regardless of practical applications. The project was described as a fun experiment and a demonstration of technical skill and creativity.
Several commenters expressed interest in the technical details of the implementation, asking about the specific regex patterns used and the mechanics of syncing the searches with the audio. This demonstrated a genuine curiosity about the inner workings of the project.
Overall, the comments reflect a mixture of amusement, admiration, and technical curiosity. They highlight the project's unusual nature, its technical challenges, and its place within the broader context of creative coding and demoscene culture.
Summary of Comments ( 5 )
https://news.ycombinator.com/item?id=42744932
Hacker News users generally praised the visualization and explanation of Hilbert curves in the linked blog post. Several appreciated the interactive nature and clear breakdown of the curve's construction. Some comments delved into practical applications, mentioning its use in mapping and image processing due to its space-filling properties and locality preservation. A few users pointed out its relevance to Morton codes (Z-order curves) and their applications in databases. One commenter linked to a Python implementation for generating Hilbert curves. The overall sentiment was positive, with users finding the post educational and well-presented.
The Hacker News post titled "Portrait of the Hilbert Curve (2010)" has a modest number of comments, focusing primarily on the mathematical and visual aspects of Hilbert curves, as well as some practical applications.
Several commenters appreciate the beauty and elegance of Hilbert curves, describing them as "mesmerizing" and "aesthetically pleasing." One points out the connection between the increasing order of the curve and the emerging visual detail, resembling a "fractal unfolding." Another emphasizes the self-similarity aspect, where parts of the curve resemble the whole.
The discussion also touches on the practical applications of Hilbert curves, particularly in mapping and image processing. One comment mentions their use in spatial indexing, where they can improve the efficiency of database queries by preserving locality. Another comment delves into how these curves can be used for dithering and creating visually appealing color gradients. A further comment references the use of Hilbert curves in creating continuous functions that fill space.
A few comments delve into the mathematical properties. One commenter discusses the concept of "space-filling curves" and how the Hilbert curve is a prime example. Another explains how these curves can map a one-dimensional interval onto a two-dimensional square. The continuous nature of the curve and its relationship to fractal dimensions are also briefly mentioned.
One commenter highlights the author's clear explanations and interactive visualizations, making the concept accessible even to those without a deep mathematical background. The code provided in the article is also praised for its clarity and simplicity.
While there's no single overwhelmingly compelling comment, the collective discussion provides a good overview of the Hilbert curve's aesthetic, mathematical, and practical significance. The commenters generally express admiration for the curve's properties and the author's presentation.