The Fast Fourier Transform (FFT): Most Ingenious Algorithm Ever?
Why Hacker News recommends it
HN repeatedly calls it a fantastic explanation that made FFT click, with some rewatching it periodically.
What it is
Reducible's visual walkthrough of the Fast Fourier Transform, motivated by polynomial multiplication. It shows how the FFT's key ideas can be discovered by asking the right questions.
Recommended by 7 HN commenters across 8 different months, 2021 to 2026.
What Hacker News said
Strong rec withinboredom · 2021-11 · on “Ask HN: Most interesting, mildly impractical, well-written books on software?”
Not a book, but I truly enjoy this guy’s videos: [this video] . I usually watch them while eating lunch. He does a truly fantastic job.
Strong rec squaredot · 2022-12 · on “Understanding Convolutions in Probability: A Mad-Science Perspective”
As the author mentions, convolutions for signal processing and convolutions for probability are essentially equivalent things in two different contexts. It is interesting that the same concept is behind the multiplication of polynomials. In the discrete case we have the parallel of the vectors representing ... the (discretized) probability density <-> the (discrete) signal intensity <-> the coefficients of the polynomial ... in the respective context. If we look at the discrete case, since the operation of convolution is computationally expensive (O(n^3)), people often rely on the Fast Fourier Transform Algorithm (FFT), that runs in O(n log(n)). An excellent and very enjoyable video on the F
Strong rec kippinitreal · 2023-01 · on “Ask HN: Concepts that clicked only years after you first encountered them?”
That video did the same for me. I also like the reducible video for Fast Fourier Transforms as well as the Veratasium piece that shares some fun history. Reducible - FFTs the most ingenious algorithm ever [this video] Veritasium - The Most Important Algorithm of All Time [LINK]
Strong rec tverbeure · 2024-07 · on “Convolutions, Fast Fourier Transform and polynomials (2022)”
If you haven’t seen it yet, you should watch this video: [this video] It derives the FFT algorithm from polynomial multiplication and it’s fantastic . I rewatch it every 6 months or so.
Strong rec tverbeure · 2025-10 · on “Discrete Fourier Transform”
I almost always prefer text over video, but this video on the same subject is fantastic: [this video] . I watch it at least once every year. Also, the title of the blog post (and by extension this HN post) is IMO not really correct: it's not about the DFT but specifically about the FFT.
Recommends jstx1 · 2021-10 · on “A gentle introduction to the FFT (2002)”
And an FFT explanation by Reducible - [this video] It's in a similar style to 3blue1brown (uses the same visualisation library) but the video focuses specifically on the FFT algorithm, not on Fourier transforms in general. I'm pretty sure Grant himself recommended the Reducible video at some point.
Recommends foooobaba · 2023-02 · on “Polynomial Interpolation”
Related: Fast Fourier Transform (FFT) can be used for polynomial interpolation in O(nlogn) time (if you get to choose the points, and use “complex roots of unity”), where n is the degree of the polynomial, and is used as a step in multiplication of polynomials in O(nlogn) as well. This video covers it well: [this video] Also, chapter 30 of CLRS: [LINK]
Recommends medbar · 2026-04 · on “Understanding the FFT Algorithm (2013)”
For anyone interested and with 30 minutes to spare, I would recommend the youtube video [1] by Reducible for an intuition on how FFT works. [1]: [this video]