Question

On a reddit post a person posted a picture of a Mobius strip redditor 1 said Technically there is no height since it both is finite and infinite redditor 2 said we know the width to be finite, would the area be considered infinite despite having a limited amount of space to write on? a möbius strip is magical, but is it on par with a menger sponge? discuss reply as redditor 1 that it is infinite

Answer 1

As redditor 1, I’d say:

That's an interesting point! The Möbius strip definitely has some mind-bending properties. In theory, while it has a finite surface area and a finite width, its fascinating topology means that you can traverse its entire surface without ever encountering an edge. So, in a sense, it has that infinite quality because you can keep going around and around it endlessly.

When we talk about the area, it’s true that there’s a limited physical space you can write on if it’s a physical strip. But in a conceptual or mathematical sense, the surface is continuous, and you can think of it as having an infinite number of points because of that non-orientable property. It kind of stretches our understanding of how we categorize things as “infinite” or “finite.”

Now, comparing it to a Menger sponge, I can see how they both challenge our intuitions about dimensions and space. The Menger sponge has an infinite surface area while being bounded in a finite volume, which is another twist! In a way, both are magical in their own rights, but they really explore different aspects of infinity and space. The Möbius strip captivates with its simple yet profound twist, while the Menger sponge dives deeper into fractal geometry. I’d say they’re both wonders of mathematics, but in different ways!