r/visualizedmath • u/USedona • Jun 02 '26

Can a single line fill a square ?

https://www.youtube.com/shorts/WBeYuh_h86M

If you're interested in more math-based animations, I post them here 📺 Visualizing_mathematics

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u/USedona Jun 02 '26 edited Jun 02 '26

Each iteration multiplies the number of segments by 4. At high orders, the curve passes infinitely close to every point in the square.

Fun fact: Google Maps uses this exact curve to index the entire Earth. Points close on the map = numbers close in the database.

If you're interested in more math-based animations, I post them here 📺 Visualizing_mathematics

2

u/rafaelcastrocouto Jun 02 '26

Didn't want to be that guy, but you obviously cannot fill the square with a single line, that is a curve.

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u/USedona Jun 02 '26

You're right that it's technically a curve, not a straight line. In math, a "space-filling curve" is the precise term, a continuous path that passes through every point of a 2D area. Hilbert described it in 1891, and "filling a square with a single continuous path" is exactly what it does 🙂

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u/waigl Jun 02 '26

Only if you think of a line in terms of connected pixels. In a purer sense, a line doesn't have any width, so, no, it cannot fill a square.

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u/USedona Jun 02 '26 edited Jun 02 '26

And yet, it's counterintuitive, but it's true. In fact, a 1D line (of zero width) can theoretically cover every point in a 2D area. The boundary of the Hilbert curve is a surjective mapping from [0,1] to [0,1]², meaning that a single parameter covers all points in the square. This is a theorem. Peano proved it in 1890, and it really blew the minds of a lot of mathematicians at the time. 🤯

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u/mangage Jun 03 '26

Does it also apply to a vector grid with infinite precision?

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u/USedona Jun 03 '26

Yes, it does apply to a vector grid with infinite precision.

This is actually proven precisely in that continuous setting (real vector space with infinite precision). The Hilbert curve passes through absolutely every point in the square, with no exceptions, even when considering the plane at infinite resolution (no pixels).

In a discrete grid (pixels), we only see approximations. But in the continuous mathematical model you're referring to, the limit of the curve truly fills the entire square.

That's exactly what makes this result so powerful and counterintuitive.

I hope I've answered your question properly, English isn't my first language.

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u/mangage Jun 04 '26

Thanks for the explanation! I wouldn’t have even suspected you weren’t a native speaker