Circles In My Head

Sometimes I wish other people could see things like I do. I want to write a book called "Multiplying Shapes Together." But who would read it? The reason is because I think multiplying a polygon or a circle by itself produces one of the most amazing kinds of symmetry there is, but no one can ever see it because it only exists in the 4th dimension. If you square a circle, the resulting 4-dimensional shape has all the symmetry of a circle twice - but on two perpendicular planar axes. That's two planes that only intersect at one point. Isn't that amazing? The squared circle consist of two solid surfaces. Each surface is a cylinder wrapped around the other so that the ends meet. That can never happen in 3 dimensions. If you try to wrap two surfaces around each other, only one of them can go all the way around without breaking. Imagine living on a planet with two equators that don't intersect.

I actually want to see my friends answer this question more than I want to answer it myself. I have a lot of strange and obscure talents, but not many of them are all that noteworthy. Some of them are noteworthy, and probably shared by many people in the world, but a small fraction of all the people in the world. Most of these are artistic - the ability to imitate almost any confining artistic style with anything given to me, generally speaking. This includes ambigrams, tessalations, caricatures, etc. But an even more obscure, more strange and more unique talent I have is one of the things that distinguishes me from the most people. Meaning, I've never met, talked to, or even heard of someone who has this ability to the extent that I do: the ability to conceptualize 4-dimensional geometric shapes in my mind, and easily perceive their charicteristics. I didn't realize how unique or noteworthy of a talent this was until I tried to talk to people about these shapes and realized that they didn't understand what I was talking about. It's so obvious to me, I don't understand why it's so hard for other people to think of it the way I do.

As for other people's obscure talents, I've seen some of them, but I don't know whether the owners of these talents would want me to post them publicly. I am fascinated by obscure talents, though, so if you have one, I think you should either answer this question in your own journal, or leave a comment after this post.

I was told this was called "cartesian products" but if you look that up on wikipedia, it's a bunch of gibberish to the common mind. I like to think of it as multiplying shapes together. In fourth grade, we played with blocks to illustrate the point that if you multiply two lengths together, you get a rectangle with the area as the product of the lengths:

But I'd like to propose that you can reject the concepts of length and area and if you multiply the segments themselves, you will get the rectangle. On that basis, you can also multiply a circle by a line segment and get a cylinder.

The principle is that you extend each point in the circle into a perpendicular segment of the same length. Or, make each point on the segment into a perpendicular circle of the same size. Either way, you end up with a cylinder.

So, what happens when you square a circle, or multiply a circle by itself? By the same principle, you would make each point in the circle into a perpendicular circle. Or, make each vertical line on the circle into a cylinder. The result is 4-dimensional. Here's a stretched, overlapping version:

So that's what you get when you square a circle: A 4-dimensional cylinder-like round thing. No, it's not a riddle. You can do this with any two shapes - multiply a pentagon by an elipse, or whatever.

Square a 5-pointed star:

I was reading my old journal from high school today. I was in taking calculus, learning about derivatives at the time I wrote this:

"Derivatives tell me something that I’ve wondered about a lot, but I could never describe how I wondered about it." - Feb 22, 2003.

When I read it a moment ago, I remembered exactly what I was talking about five years ago. I could even try to describe it now. Before I knew about calculus, I wondered whether there was a way you could measure the continuity of a curve. Like if you took a piece of a sine wave, and connected it to a piece of a parabola, would there be some kind of disturbance in the curve where they were stuck together, or would it just be like a different contunuous function? Anyway, I learned that there was a way to measure that, and the answer is yes, there would be a disturbance, because if you took the derivative enough times, eventually the one curve would not connect to the other curve.

Do you intuitively know things about 4-dimensional geometry? I do. I don't know anyone else who does. Why do I have this useless and unique talent? Maybe it comes from being able to understand perspectives that are not my own, my ability to extend analogies, and my spatial thinking aptitude.

Anyway, a while back I posted something about Baha'i, and someone of the Baha'i faith responded. For this entry, I would like a similar response, please.