How to wrap $f(x)=x$ around the singular circle properly?

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I was watching this video: https://youtu.be/spUNpyF58BY

About Fourier Series, which explained we can wrap a graph around a circle to get a form of $e^{if(t)}$ to have $|e^{if(t)}|=f(t)$, I give a reference to my explanation in here:

How from Fourier Transform with imaginary numbers we get the one with real numbers?

I tried to wrap $f(x)=x$ around a circle, and I assumed that the answer to $|e^{...}|$ would be $x$. Because we wrap $f(x)=x$ around the circle. But I was wrong, $|e^{ix}|$ is $1$.

Why? Why is that? How can we wrap ANY graph around a circle to get the correct result, what's the proper way to do so?

EDIT: I found the following link, but it's off topic and talks only about programmatically doing the plot: https://mathematica.stackexchange.com/questions/64534/how-to-wrap-a-plot-around-a-circle

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Dear me...

The wrapping is true for periodic function. In your example a period of $1$ is hidden. At $t=1$, $f(1)$ equals $1$, hence indeed still it is true and WYSIWYG (What You See Is What You Get).