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METhomae’s Function-The Rationals Spike
This is a function I met during my first year studying maths.
I remember staring at it, confused and frustrated — it looked like nonsense. Just a strange piecewise rule where "rationals spike" and everything else stays at zero.
I didn’t get it. At all.
Years later, after diving deeper into analysis and learning how to visualize abstract things, I came back to it. And now? I’m honestly proud of what I can see.
This is called the Thomae function, or the "popcorn function."
Every time you pick a rational number — like ½, ⅓, or 7/8 — the function spikes up.
But the "height" of the spike depends on how simple the fraction is: the simpler it is, the taller it goes.
And if the number is irrational — like π or √2 or just any random messy decimal that never repeats — the function stays at zero.
So it’s this strange mix:
🔹 Rationals light up with little vertical spikes
🔹 Irrationals? Silent. Flat. Invisible.
🔹 And all of this inside the same interval, overlapping each other infinitely.
It’s chaotic but precise. A staircase without steps. A skyline with no buildings.
A perfect example of how maths can hide deep structure inside what looks like noise.
It finally makes sense. And being able to animate it, frame by frame — spike by spike — feels like giving shape to something I once thought was unseeable.
#mathematics #physics #maths #math
@merlinomaths
