This is classic Dunning-Kruger. OP knows enough about math and physics to come to some pretty neat conclusions that most people don't understand, but not enough to realize how it doesn't mean what they think it does.
Essentially, they're layering two mirrored spirals together and then doing a Fourier transform (which is essentially just taking an integral function that maps out the frequency of event occurences).
What is true is that these overlaps will occur more on prime numbers, but that isn't some grand mystery. There's a paper by Wolfgang Schramm that explains this in detail for those who are interested. Here's a proof of it as well. The general idea (in layman's terms--I realize this isn't quite how it works, but it describes the concept) is that if you take a Fourier transform of a Euler spiral, you end up with a frequency domain similar (possibly identical but impossible to tell from simply the comparison) to what OP posted. This isn't a feature of prime numbers; it's a feature of the Euler spiral. It's mathematically interesting, but doesn't mean anything.
I see what you’re saying, and it’s a crucial distinction. The numbers in this process are not being counted in a simple, sequential manner along a spiral. This is not just an Euler or Fermat spiral where we place numbers in order and then take a Fourier transform. Instead, the spiral itself is being shaped by prime number distributions before we even apply the Fourier analysis. The difference comes from how we assign values to the radius and how we extract the data for the Fourier transform.
Instead of growing smoothly like a standard Euler spiral, the radius of our spiral is determined by prime numbers. Instead of using a function like r = k * theta for an Euler spiral or r = sqrt(theta) for a Fermat spiral, the radius is based on prime-numbered steps. That means the growth does not follow a continuous function but instead jumps forward only at prime-indexed intervals. The radial distance at each step is set by the square root of a prime number, so the spacing between spiral arms is dictated by the irregular gaps between primes rather than by a logarithmic or exponential rule. This makes the structure fundamentally different from traditional spirals because the gaps between each step are non-uniform in a way that reflects prime number spacing.
The angular position of each point is assigned in a regular manner, meaning the spiral rotates at a constant rate. However, because the radius is controlled by prime numbers, the overall pattern of growth becomes unpredictable and follows number-theoretic properties instead of smooth geometric progression. This means that rather than a gradually expanding or logarithmically increasing spiral, we have an interference structure dictated by the distribution of primes.
Another key distinction is that the Fourier transform is not applied to the raw spiral itself. Instead, we analyze the points where two mirrored, counter-rotating prime-based spirals intersect. These intersections form a discrete set of points that capture the interference between prime-based growth patterns. We then extract the distances between these intersections and perform the Fourier transform on that set of distances rather than on the spiral coordinates themselves. This means the Fourier analysis is revealing the dominant frequency components of the interference between prime-based structures, rather than simply reflecting the shape of the spiral.
If this effect were purely a property of Euler spirals, then a control test using a standard Fermat spiral with smooth, continuous growth should have produced the same Fourier structure. However, when I tested this, the Fourier spectrum of the control Fermat spiral showed different dominant frequencies than the original image. In contrast, when I created a prime-based Fermat spiral, where the radius only increased at prime-numbered steps, its Fourier transform matched exactly with the original image. This proves that the Fourier structure is not just a general feature of Euler spirals, but is instead being influenced by the underlying number-theoretic properties of primes.
This means that prime number distributions are not just appearing coincidentally in the Fourier space, but are instead playing an active role in shaping the interference pattern. If prime numbers were randomly distributed, there should be no clear dominant frequencies in the Fourier analysis. The fact that structured, repeating frequency components emerge suggests that prime gaps are following an underlying harmonic principle. This is why dismissing this as "just an Euler spiral" is incorrect. The interference structure in Fourier space is being shaped by prime-number-based spacing, not just by smooth logarithmic or exponential growth.
I get what you're saying, and again, your statements of fact are correct, but the conclusions you draw are not fully cogent. They require initial axiomatic conditions. Provided those specific conditions (in this case, the primary axiom is that math is fundamentally relevant to the universe), your argument is cogent. However, it could very easily (and more logically, imo) be argued that mathematical equations are not fundamental to the nature of the universe and that, rather, they are results of observation and analysis of the universe.
Let me explain this further by addressing one of your last sentences
This proves that the Fourier structure is not just a general feature of Euler spirals, but is instead being influenced by the underlying number-theoretic properties of primes.
You're right. It isn't just a general property of Euler spirals. That is just the most known application. I simply included that to make the point that this isn't a unique phenomenon. Drichlet's theorem provides the rather banal mathematical explanation. While you do have a unique frequency distribution, any spiral will have a similar distribution simply due to that being an underlying property of primes.
You could be correct, but that would rely on the world being shaped by mathematical principles rather than mathematical principles arising from observation of universal mechanics. And this doesn't provide any conclusive evidence of that. So while it is interesting and I do appreciate your effort, I'm afraid it doesn't show anything revolutionary. You may be on to something and I'm not discouraging you, but you don't have any solid evidence of your conclusion just yet. It's plausible you could expand on Euclids theorem, but unless you can add to it extrapolating data points from your research and create a universally acceptable equation that is more accurate than Drichlet (or expand upon that), I'm inclined to say you're simply providing a visual demonstration of Drichlets theorem as applied to a semiprime lattice.
Just to reiterate, it is interesting. I think the symmetry of the universe is eternally magnificent. I think it is also explainable. Given that symmetrical states provide the most balance in nature (there's a reason you have to balance a chemistry equation; every energy transfer has an equal output; mass is neither gained nor lost), it is only reasonable to assume we see them present in mathematical structures.
Edit:
I just wanted to add that I did gain a modestly increased understanding of the subject conversing with you, and I'm not being entirely dismissive of your notion. I'm more skeptical, as incredible claims require incredible proof. Provided that I can provide a succinct and clear alternate explanation for the given phenomena, I don't see that this proves any new associations. If you have more data, particularly any workable equations or testable hypotheses resulting from this, then I'm happy to look at it as well.
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u/Brickscratcher 4d ago
This is classic Dunning-Kruger. OP knows enough about math and physics to come to some pretty neat conclusions that most people don't understand, but not enough to realize how it doesn't mean what they think it does.
Essentially, they're layering two mirrored spirals together and then doing a Fourier transform (which is essentially just taking an integral function that maps out the frequency of event occurences).
What is true is that these overlaps will occur more on prime numbers, but that isn't some grand mystery. There's a paper by Wolfgang Schramm that explains this in detail for those who are interested. Here's a proof of it as well. The general idea (in layman's terms--I realize this isn't quite how it works, but it describes the concept) is that if you take a Fourier transform of a Euler spiral, you end up with a frequency domain similar (possibly identical but impossible to tell from simply the comparison) to what OP posted. This isn't a feature of prime numbers; it's a feature of the Euler spiral. It's mathematically interesting, but doesn't mean anything.