The blog post explores the interconnectedness of various measurement systems and mathematical concepts, examining potential historical links that are likely coincidental. The author notes the near equivalence of a meter to a royal cubit times the golden ratio, and how this relates to the dimensions of the Great Pyramid of Giza. While acknowledging the established historical definition of the meter based on Earth's circumference, the post speculates on whether ancient Egyptians might have possessed a sophisticated understanding of these relationships, potentially incorporating the golden ratio and Earth's dimensions into their construction. However, the author ultimately concludes that the observed connections are likely due to mathematical happenstance rather than deliberate design.
"An Infinitely Large Napkin" introduces a novel approach to digital note-taking using a zoomable, infinite canvas. It proposes a system built upon a quadtree data structure, allowing for efficient storage and rendering of diverse content like text, images, and handwritten notes at any scale. The document outlines the technical details of this approach, including data representation, zooming and panning functionalities, and potential features like collaborative editing and LaTeX integration. It envisions a powerful tool for brainstorming, diagramming, and knowledge management, unconstrained by the limitations of traditional paper or fixed-size digital documents.
Hacker News users discuss the "infinite napkin" concept with a mix of amusement and skepticism. Some appreciate its novelty and the potential for collaborative brainstorming, while others question its practicality and the limitations imposed by the fixed grid size. Several commenters mention existing tools like Miro and Mural as superior alternatives, offering more flexibility and features. The discussion also touches on the technical aspects of implementing such a system, with some pondering the challenges of efficient rendering and storage for an infinitely expanding canvas. A few express interest in the underlying algorithm and the possibility of exploring different geometries beyond the presented grid. Overall, the reception is polite but lukewarm, acknowledging the theoretical appeal of the infinite napkin while remaining unconvinced of its real-world usefulness.
The website "Explorable Flexagons" offers an interactive introduction to creating and manipulating flexagons, a type of folded paper polygon that reveals hidden faces when "flexed." It provides clear instructions and diagrams for building common flexagons like the trihexaflexagon and hexahexaflexagon, along with tools to virtually fold and explore these fascinating mathematical objects. The site also delves into the underlying mathematical principles, including notations for tracking face transitions and exploring different flexing patterns. It encourages experimentation and discovery, allowing users to design their own flexagon templates and discover new flexing possibilities.
HN users generally praise the interactive flexagon explorer for its clear explanations and engaging visualizations. Several commenters share nostalgic memories of making flexagons as children, spurred by articles in Scientific American or books like Martin Gardner's "Mathematical Puzzles and Diversions." Some discuss the mathematical underpinnings of flexagons, mentioning group theory and combinatorial geometry. A few users express interest in physical construction techniques and different types of flexagons beyond the basic trihexaflexagon. The top comment highlights the value of interactive explanations, noting how it transforms a potentially dry topic into an enjoyable learning experience.
This post explores the connection between quaternions and spherical trigonometry. It demonstrates how quaternion multiplication elegantly encodes rotations in 3D space, and how this can be used to derive fundamental spherical trigonometric identities like the spherical law of cosines and the spherical law of sines. Specifically, by representing vertices of a spherical triangle as unit quaternions and using quaternion multiplication to describe the rotations between them, the post reveals a direct algebraic correspondence with the trigonometric relationships between the triangle's sides and angles. This approach offers a cleaner and more intuitive understanding of spherical trigonometry compared to traditional methods.
The Hacker News comments on Tao's post about quaternions and spherical trigonometry largely express appreciation for the clear explanation of a complex topic. Several commenters note the usefulness of quaternions in applications like computer graphics and robotics, particularly for their ability to represent rotations without gimbal lock. One commenter points out the historical context of Hamilton's discovery of quaternions, while another draws a parallel to using complex numbers for planar geometry. A few users discuss alternative approaches to representing rotations, such as rotation matrices and Clifford algebras, comparing their advantages and disadvantages to quaternions. Some express a desire to see Tao explore the connection between quaternions and spinors in a future post.
"Slicing the Fourth" explores the counterintuitive nature of higher-dimensional rotations. Focusing on the 4D case, the post visually demonstrates how rotating a 4D cube (a hypercube or tesseract) can produce unexpected 3D cross-sections, seemingly violating our intuition about how rotations work. By animating the rotation and showing slices at various angles, the author reveals that these seemingly paradoxical shapes, like nested cubes and octahedra, arise naturally from the higher-dimensional rotation and are consistent with the underlying geometry, even though they appear strange from our limited 3D perspective. The post ultimately aims to provide a more intuitive understanding of 4D rotations and their effects on lower-dimensional slices.
HN users largely praised the article for its clear explanations and visualizations of 4D geometry, particularly the interactive slicing tool. Several commenters discussed the challenges of visualizing higher dimensions and shared their own experiences and preferred methods for grasping such concepts. Some users pointed out the connection to quaternion rotations, while others suggested improvements to the interactive tool, such as adding controls for rotation. A few commenters also mentioned other resources and tools for exploring 4D geometry, including specific books and software. Some debate arose around terminology and the best way to analogize 4D to lower dimensions.
This post explores the problem of uniformly sampling points within a disk and reveals why a naive approach using polar coordinates leads to a concentration of points near the center. The author demonstrates that while generating a random angle and a random radius seems correct, it produces a non-uniform distribution due to the varying area of concentric rings within the disk. The solution presented involves generating a random angle and a radius proportional to the square root of a random number between 0 and 1. This adjustment accounts for the increasing area at larger radii, resulting in a truly uniform distribution of sampled points across the disk. The post includes clear visualizations and mathematical justifications to illustrate the problem and the effectiveness of the corrected sampling method.
HN users discuss various aspects of uniformly sampling points within a disk. Several commenters point out the flaws in the naive sqrt(random()) approach, correctly identifying its tendency to cluster points towards the center. They offer alternative solutions, including the accepted approach of sampling an angle and radius separately, as well as using rejection sampling. One commenter explores generating points within a square and rejecting those outside the circle, questioning its efficiency compared to other methods. Another details the importance of this problem in ray tracing and game development. The discussion also delves into the mathematical underpinnings, with commenters explaining the need for the square root on the radius to achieve uniformity and the relationship to the area element in polar coordinates. The practicality and performance of different methods are a recurring theme, including comparisons to pre-calculated lookup tables.
Summary of Comments ( 13 )
https://news.ycombinator.com/item?id=43207962
HN commenters largely dismiss the linked article as numerology and pseudoscience. Several point out the arbitrary nature of choosing specific measurements and units (meters, cubits) to force connections. One commenter notes that the golden ratio shows up frequently in geometric constructions, making its presence in the pyramids unsurprising and not necessarily indicative of intentional design. Others criticize the article's lack of rigor and its reliance on coincidences rather than evidence-based arguments. The general consensus is that the article presents a flawed and unconvincing argument for a relationship between these different elements.
The Hacker News post titled "The Meter, Golden Ratio, Pyramids, and Cubits, Oh My" has generated a moderate number of comments, most of which express skepticism and amusement at the original article's attempt to connect the meter to the Great Pyramid of Giza via the golden ratio and cubits.
Several commenters point out the historical inaccuracy of the claims. One commenter highlights that the meter's definition has changed over time, initially being related to the Earth's circumference and only later linked to a physical artifact. This debunks the idea of a pre-planned connection to ancient Egyptian measurements. Another commenter mentions the imprecision inherent in measuring the pyramid itself, making any exact correspondence with the meter highly improbable. The variability in historical cubit lengths is also raised, further undermining the argument for a precise relationship.
Another line of discussion centers on the perceived "pyramid inch" and its alleged relationship to British Imperial units. Commenters dismiss this connection as coincidental and highlight the convoluted logic required to arrive at such a conclusion. The tendency to find patterns where none exist is also discussed, referencing the phenomenon of pareidolia.
Some commenters approach the topic with humor, joking about the prevalence of such theories and the fascination with hidden connections. One commenter sarcastically suggests a connection between the size of their foot and the circumference of Jupiter. Another uses the opportunity to plug a book debunking similar historical myths.
A few commenters attempt to engage with the mathematical aspects, discussing the golden ratio and its properties. However, these discussions generally reinforce the skepticism towards the original article's claims, emphasizing the lack of evidence for any meaningful connection.
In summary, the comments on Hacker News largely reject the premise of the linked article. They point out historical inaccuracies, methodological flaws, and the general implausibility of the proposed connections. The overall tone is one of skepticism, occasionally tinged with humor and amusement at the article's attempts to find profound meaning in numerical coincidences.