Let's say you stick a paint roller diagonally in a tray of paint, so that the intersection between the cylindrical roller and the paint -- if you can imagine the paint cutting through the roller -- is an ellipse. When you "unroll" the cylinder onto the wall, the ellipse produces a sinusoidal wave. Hmmm ....
So you would think that mathematicians had pretty much settled such questions as, What happens if you draw this or that shape on a cylinder or a cone, and then "unroll" it onto a plane? Or what happens if you draw a shape onto a plane, and then "roll" it into a cylinder or a cone? What will that shape look like if you view it from the "ceiling," that is, straight above the apex of the cone, or straight down the verticle axis of the cylinder? You'd be surprised what happens to pretty simple things like a line or a circle ... I found a long article on it, published only in 2007, and the authors say that in fact these questions are very far from being exhausted...
I like such things ... I mean, that the fields are still so wide open; and they aren't idle, because they have implications for many areas of math that you'd think were not related....
Anyhow, the word MATHEMATICS comes from ancient Greek MATHEMA, what IS LEARNED (the -M- inside has a passive force). It's interesting that THAT kind of learning took the word over, as if there were something essential about it, something that got right to the heart of what it means to learn. Plato thought so; see the MENO. Its root is an augmentation of a simpler root having to do with thought and states of MIND: cf. English MIND, Latin MENS, Greek MNEME = MEMORIAL, EPITAPH ... Cousins are all over the Indo European map.
I like the Greek word for Jesus' disciples: MATHETAI ... which is well rendered by Latin DISCIPULI, meaning LEARNERS = PUPILS, with a diminutive sense. In Old English, the LEARNING was emphasized, with a sense of subordination: LEORNUNGCNIHTAS = LEARNING-KNIGHTS, that is, APPRENTICES ... But in Luther's German, the word is JUENGER, = YOUNGSTERS, that is, JUNIORS, SCHOOLBOYS ...