05/28/2013 05:18 pm ET Updated Dec 06, 2017

Geometry, A Love Story

Unless one studies mathematics in university or other related fields, it's unlikely that one knows what university-level mathematics is. This is because mathematics in university is completely different from that in high school - mathematics all of a sudden abstract. Numbers rarely appear anymore and things mathematicians deal with are no longer rooted to reality.

I believe it is a pity that many people pass their lives without ever getting a taste of the wonderful world of abstract mathematics. But it's never too late.

Here is a story about me and geometry.

I fell in love with mathematics for the first time in the algebra class that I took in my first semester at university.

I fell in love more with mathematics when I took my geometry class.

Klein bottle was my lover in my geometry class.

This is what she looks like:

Fig. 1



This is what she looks like when she's feeling low:

Fig. 2


But she reveals her real beauty when she's higher.
She has a borderless beauty in the fourth dimension, just straight out of heaven in the fifth dimension.

Fig. 1 and 2 provide the same information, just represented with different dimensions.

To get from Fig. 2 to Fig. 1, you have to unite the arrows matching their direction. This is a process only possible in three dimensions and hence we have to add a dimension.

But we only live in three dimensions so we will never be able to see what she looks like in four or five dimensions or do things that are only allowed in higher dimensions.
But we have mathematics that can describe exactly what she looks like up there.
In Fig. 1, one part of the bottle awkwardly pokes a hole in the bottle, injects itself and meets the other end (go to the Wikipedia entry to see the whole Klein bottle construction and see what I'm talking about more clearly). In other words, she's not completely homogeneous in three dimensions. You can watch a wonderful video of a cyclist on a Klein bottle and realize that there is a discomforting pole sticking out on some parts.
She's more elegant than that - she does not self-intersect in the fourth dimension and standing on one point on the Kein bottle will look exactly the same anywhere else.

If you have made a Möbius Strip in your earlier mathematics classes, you learned that there is no way to describe the inside or the outside - it is just one long surface. Drawing a Möbius Strip in two dimensions (on a flat surface like on a piece of paper), it impossible to draw it in a way that it does not self-intersect. In other words, there is a unique point (the intersection) on the Möbius Strip in two dimensions but you know that is not the case in three dimensions. Klein bottle is a Möbius Strip with a three dimensional "surface". Perhaps you now understand why it is possible for something to not intersect in a higher dimension.

Like how Romeo could only see Juliet from below the balcony, our conception does not allows us to "see" the Klein bottle in higher dimension. But the nurse provided Romeo with a ladder, the world provided us with mathematics.

Mathematics can tell me how something looks in a world that I will never be able to go to, a world that is only inside our heads. But it is not just a virtual reality we've created, the world of mathematics exits. This world sometimes has three dimensions like our own or infinite dimensions not quite like our own.

This is how I fell in love with mathematics.

I think the ideas in mathematics are absolutely beautiful.

This is the beauty of abstract mathematics.