Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations

What people said

Strong rec shezi · 2021-08 · Hacker News · on “Ulam Spiral”

Here is one of the most enlightening explanations for these kinds of spirals I have seen by 3blue1brown: [this video] Note that the setup is slightly different,since in that video they plot numbers along their polar coordinates. The results are very similar, since both create spirals with predictable angular offsets.

Recommends excessive · 2019-11 · Hacker News · on “Unexpected Beauty in Primes”

The first time I saw these, I thought it hinted at some deep insight into the nature of primes. However, after watching the following video, I think it just says something about modular arithmetic and rational approximations of 2 Pi. [this video] The video is only about Archimedian spirals, but I suspect a similar analysis would apply to Ulam spirals too. We really like to find patterns.

Recommends hinkley · 2020-07 · Hacker News · on “Why do electron shells have set limits? (1999)”

In this vein, I really enjoyed 1brown3blue breaking down why prime numbers form patterns when graphed in polar coordinates. [this video]

Recommends lupire · 2021-03 · Hacker News · on “The Prime Hexagon”

3B1B has a semi related video on how working with prime numbers can help you see properties that are also found in composite numbers, since the sequence of primes overlaps with many other sequences. [this video]

Recommends andrewflnr · 2021-03 · Hacker News · on “The Prime Hexagon”

For IMO a more meaningful phenomenon, this video by 3Blue1Brown explains why primes plotted as (p, p) in polar coordinates produce interesting patterns. [this video]

Recommends zimpenfish · 2024-08 · Hacker News · on “Breakthrough a step toward revealing hidden structure of prime numbers”

Numberphile[1] and 3b1b[2] (with a particularly good explanation of why it happens) have done good videos on the prime spiral. [1] [LINK] [2] [this video]

Recommends godelski · 2024-08 · Hacker News · on “Ask HN: What is the best software to visualize a graph with a billion nodes?”

> But after a fairly small size, there's almost no utility in visualizing graphs. I want to stress this point and go a bit further. It can be worse as people have pareidolia[0], a tendency to see order in disorder. Like how you see familiar shapes in the clouds. There is a danger in that with large visualizations such as these that instead of conveying useful information, you counterproductively convince someone that something that isn't true is! Here's a relevant 3B1B video where this is kinda discussed. There is real meaning but the point is that it is also easy to be convinced of things that aren't true[1]. In fact, Grant's teaching style is so good you might even convince yourself of the

Cites mdiesel · 2020-09 · Hacker News · on “How Many Decimals of Pi Do We Really Need? (2016)”

I've got Pi memorised more precisely than 355/113 as well for no good reason, and for engineering applications it doesn't serve any purpose. But the fraction is weirdly close and that's interesting for other (more academic) reasons, such as explaining why plotting primes in polar coordinates looks like a pattern: [this video]

Cites sargstuff · 2022-04 · Hacker News · on “Λ-2D: Exploration of Drawing as Programming Language, Ideas from Lambda Calculus”

Warning: Bit of a 'Hitch hikers guide to the galaxy"/conway game of life yak shave for a reply. Nice start to providing a simple way to visually interact/experiment/manipulate different multiple Neural network concepts by loading/transforming any traditional code (working or not) & reshuffling/tesselating the structual representation of the code to generate a turing tape path via color execution paths. Switching to pi calculus could provide the theory to support coroutine/coprossing/ threading" multiple interacting tape paths/color combinations/path diffs/scaled recusion to perform assembly language instructions. Raster graphics without the yak ( [LINK] ) Feed back between tape & "traditiona

Cites curtisblaine · 2023-11 · Hacker News · on “Footsteps of pi”

I guess the Ulam spiral is more an artefact of the cartesian plane than an "hidden pattern": [this video]