Just roll a paper strip into a loop and glue the ends together, and you get a cylinder — which of course has a clearly separate "outer surface" and "inner surface." But what happens if you twist it once before gluing the ends together? Remarkably, you get a shape where the distinction between the outer and inner surface disappears completely. This is called a Möbius strip.
The easiest way to check this for yourself is to actually make one out of paper. Twist one end of a long paper strip 180 degrees and glue it to the other end, then keep drawing a line along the surface with a pencil. Draw it in one continuous stroke, without lifting the pen, and the line connects all the way to what you originally thought was the "back side" — and it takes exactly two full loops around the whole strip, not one and a half, before it meets back up at the starting point. Since there's only one face, there's only one edge too — an ordinary paper loop has two separate edges, a top and a bottom, but if you trace a Möbius strip's edge with your finger, it runs continuously as just one single edge.
Take this idea one step further and you get the Klein bottle. It's made by forming a long tube shape, then passing one end (the neck) through the inside of the tube's side wall and joining it to the opposite end from within. Made this way, the very distinction between an "inner space" and an "outer space" disappears. Pour water in, and it doesn't collect anywhere — it just keeps spreading across the entire surface.
Here's an interesting fact. A Klein bottle actually can't be made in 3-dimensional space without self-intersecting (without passing through itself). The way the neck appears to pass through the side wall is a kind of "shadow" effect, caused by forcing a shape that can only naturally exist in 4-dimensional space into 3 dimensions. It's similar to how drawing a 3D object on 2D paper makes lines appear to overlap.
The Möbius strip and the Klein bottle are concepts that came out of mathematicians asking, "can a surface be given a direction?" A surface where inside and outside can be clearly separated, like a cylinder or a sphere, is called two-sided, and one that can't be separated this way, like the Möbius strip, is called one-sided. This distinction becomes a crucial starting point in a field called topology, which studies the fundamental properties of shapes that don't change even when they're stretched or bent.
On our activity page, you can draw a line directly on a 3D-rendered Möbius strip and see for yourself that it takes two full loops to reach the starting point, and the "Auto-draw two loops" button lets you watch that same process play out as an animation. On the Klein bottle tab, use the slider to fill it with water and watch with your own eyes as the water spreads across the entire surface with no boundary between inside and outside.