This Hacker News discussion revolves around a blog post exploring how the mathematical constant π can vary when using different distance metrics, moving beyond the standard Euclidean metric. The conversation touches on the nature of mathematics, physics, the accessibility of complex concepts, and the validity of mathematical definitions.
Here's a breakdown of the key themes:
The Nature of Mathematical Understanding and Accessibility
One prominent theme is the experience of learning and understanding abstract mathematical concepts. Some users, like "isoprophlex," express a sense of wonder and "provocative" discovery when encountering mathematical snippets, highlighting a difficulty in discerning whether results are trivial or profound. This sentiment is echoed and expanded upon by "BrandoElFollito," who compares his experience with physics to a "toolbox" where some tools felt wondrous and others like "voodoo" (citing renormalization as an example). He suggests that a "special (brilliant) mind" and significant patience are required for deep mathematical understanding, especially in the initial stages.
This idea of accessibility and the gap between expert and novice understanding is further illustrated by "saghm." He relates the discussion of math to physics, pointing out that some physics concepts, like conserved energy, can feel like "deus ex machina" to a novice because they seem to be defined to make conservation work rather than being derived from deeper principles. The reliance on technical terminology, as seen in "drivebyhooting"'s attempt to explain energy conservation, can alienate those who lack foundational knowledge: "You've already lost me by step two; I have no clue what 'force is the negative gradient of potential' means or what a 'conservation force field' is." This led to a discussion about how to best explain complex mathematical or scientific ideas to a broader audience, with "whyandgrowth" initially feeling that certain explanations in the blog post (and a linked article) didn't sufficiently aid non-experts. However, others, like "dpassens," defended the author, pointing out that the linked paper itself served as a resource for deeper understanding, prompting "whyandgrowth" to acknowledge the point.
The Special Role of the Euclidean Metric and Cartesian Coordinates
A significant portion of the discussion focuses on why the standard Euclidean metric (L2 norm) and Cartesian coordinates are so fundamental and prevalent in mathematics and physics. "srean" elaborates on this extensively, suggesting that the "squared Euclidean" metric, despite not being a true metric, minimizes sums of squares and is key to algorithms like K-means. He highlights the Euclidean metric's symmetry properties, pointing out that its level sets (circles) are invariant to coordinate system rotations, unlike the level sets of metrics discussed in the blog post. This symmetry is directly linked to the Pythagorean theorem: "The squared Euclidean distances decomposes nicely along orthogonal (perpendicular) directions." The contrast is made with the metrics from the blog post, which "decompose as, for lack of a better name, Fermat's last theorem decomposition."
"neutronicus" and "shiandow" further support this by mentioning the Parallelogram Law and the connection to inner products, respectively, as foundational properties of the L2 norm. "JadeNB" offers a nuanced perspective, suggesting that while orthogonal coordinates can be found for the L2 norm, the norm itself can be singled out by its symmetries or adherence to the parallelogram law, independent of coordinates. "nerdponx" connects this to statistics, noting that the Gaussian distribution, an exponential of a squared Euclidean distance, feels "magical and universal." This leads to the evocative statement: "In my opinion it holds the same place in mathematics that water holds in biology and chemistry."
Generalizing Metrics and the Definition of Pi
The core of the blog post's exploration, and much of the subsequent discussion, is the generalization of distance metrics, particularly the L_p norms. Users explore the implications of these generalizations on the value of π. "omneity" and "bubblyworld" clarify that the parameter n in the blog post refers to the exponent in the L_p norm, which defines the "circle" for that metric. They explain that changing n leads to different shapes and thus different values of π.
The discussion also delved into the mathematical properties of these metrics, with "brewmarche" questioning if the L_p norm is indeed a metric for $0 < p < 1$. "jaza" raises a fundamental point about the definition of "circle" and "pi," arguing that if you step outside Euclidean geometry and its definition of distance, the "pi's" discussed are not the true π, but rather values specific to those altered systems: "My conclusion therefore isn't 'we have the best pi', but is rather 'we have the only pi', because pi is simply not applicable, as soon as you alter the rules of there being a 2-dimensional plane and there being real-world distance, that the definition of pi depends on." "omnicognate" counters by defining "circle" and "pi" more abstractly based on distance and ratio, showing how the blog post's approach is consistent within its own defined framework.
The Nature of Numbers and Computation
A brief but interesting tangent arises regarding the definition of a "number." "mixedbit" questions if π is more of a computation than a number, drawing a parallel to Fibonacci numbers at infinity. "srean" and "probablypower" engage in a philosophical debate about what constitutes a number, with "srean" suggesting that computable real numbers, like π, are indeed numbers. "teiferer" points out the flaw in the "computation guarantees number" argument, explaining that convergence doesn't automatically mean the limit shares properties of the sequence. However, "Sharlin" and "creata" chime in, defining real numbers through formalisms like Cauchy sequences, which directly address the idea of approximation and convergence, reinforcing π's status as a number under modern mathematical definitions.
Connections to Other Mathematical and Scientific Concepts
The discussion also highlights broader connections and further avenues of exploration:
- Superellipses and Superquadrics: "kragen" provides significant context by linking the curves discussed in the post to "Superellipses" and "Superquadrics," mentioning Piet Hein's work and Martin Gardner's writings. This adds a rich historical and artistic dimension to the mathematical concepts.
- Dimensional Analysis and Hypervolumes: "cubefox"'s exploration of the volume of n-balls (hypervolumes) and its peak at fractional dimensions, as well as the surface area of n-spheres, introduces a related but distinct area of inquiry. "evanb" and "cubefox" then engage in a debate about the meaningfulness of comparing volumes across dimensions without proper scaling (e.g., taking the nth root of the volume), underscoring the importance of careful dimensional analysis.
- Metrics on Spheres: "GMoromisato" and "srean" discuss the metric on the surface of a sphere, and how the value of π varies based on the size of the circle relative to the sphere's radius. This illustrates that π's constancy is itself dependent on the underlying geometry and metric.
- Mathematical Fonts and Rendering: A minor but amusing theme involves observations about the rendering of the Greek letter π in the Hacker News font, highlighting the nuances of digital presentation. "MontagFTB" and "8organicbits" discuss the use of MathJax and browser settings in relation to font rendering.
Overall, the discussion reveals a community grappling with the beauty, complexity, and accessibility of mathematics, with a particular focus on how foundational concepts like the Euclidean metric and the constant π can be explored and understood through generalization and different geometric perspectives.