Sine, Cosine, Tangents and Fast Fourier Transforms

Normally we don't really think much about Sine, Cosine or Tangent functions... Maybe you did a bit in Calculus, but most people don't really touch upon anything else then early transcendentals in Calculus. I believe Fourier Transforms are classified as an "Advanced Transcendental" method in Calculus. But are still relatively simple concept to grasp... I know I keep saying otherwise complex shit is "simple", but, hey maybe I'm right and this is actually simple? 🤔

Sound and Vibrations

You know when people talk about vibrations and "good vibrations". Maybe one day we'll be analytical enough to figure out what are "good" and "bad" vibrations. The best way to identify distribution of sound is through Fourier Transforms, since most sound waves travel in sinusoidal waves. Most sinusoids are relatively simple in periods, the domain and range can easily be formatted to adhere to reasonable sinusoid that can fit into a Fourier Transform too.

Time Complexity and Data Sampling

As you probably have noticed. I do all my code samples on my blog in Javascript for legibility. However, there aren't really any data structures we can optimize to reduce time complexity for a Fourier Transform in Javascript in comparison to C. Fourier Transforms perform at O(n^2) worst case and O(n * log n) best case. Data fed into a Fourier Transforms is binary 1 or 0. Making interpretation of data relatively simple once you recieve outputs.

Riemann's Hypothesis and Fourier Transforms

Riemann's Hypothesis was discovered in 1856 and has been unsolved ever since. I find it's application of Fourier Transforms absolutely fascinating in using trivial and non-trivial zeroes to discover secure and safe prime numbers. I am not going to discuss any potential postulations of Riemann's hypothesis on this blog post, but, if I do magically discover how Riemann's Zeta Function is actually supposed to work. You may see me on TV getting a Millenium Prize, after I write a blog post about it as an Axiom Proof. (Just maybe 😂)

Basic Implementation

Fast Fourier Transforms end up computing as either "Discrete Fourier Transform" or an "In place Fourier Transform". Here's a basic code sample of a "Discrete Fourier Transform":

Personally, I think Riemann is onto something here. If you test out this Fourier Transform method you could easily tune the inputs to produce prime numbers. Now if only we could figure out a proper tuned function that produced secure and safe primes effiecently. Hmm... Hopefully you learned something!