More recently in math I’ve come to the conclusion that every graph we make is a cross section of something three-dimensional. This is probably wrong, but I can’t help but wonder in my n-dimensional mind, are many of our polar graphs simply cross sections of 3D shapes?

It began with the lemniscate problem. Try graphing the polar equation r = sqrt(4cos(2t)), where “t” is theta. You’re supposed to see a figure eight, much like the infinity symbol, but on any calculator or online graphing tool, you do not see this. In fact, you see something quite broken in the middle. Why? Because you’re taking the square root of cosine while it’s negative, giving an undefined point. However, the strange thing is, this equation doesn’t even form a lemniscate when the formula is presented as r^2 = a^2cos(2t)! So, why was this so perplexing? I thought about it for a while, and then looked up some information.

It turns out that the lemniscate wasn’t always a polar equation. In fact, its true form seems to lie within the bipolar coordinate system, where a curve can use two foci to loop around instead on one as in polar coordinates. On top of this, however, another thought of mine was confirmed.

What was the lemniscate, really? Where did it come from? I thought about how I’d always seem the polar coordinate system – a top down view of a sphere, and then I knew. The lemniscate was really something three-dimensional. And, in fact, it was:

Yes, it was a Toric Section – a cross section of a torus. So, knowing this as I did now, how was I supposed to look at everything else we graphed? I took it with a grain of salt until yesterday, when I came across something interesting.

Take that same polar coordinates graphing tool and graph this: r = 1/cos(t), or in rectangular coordinates, x = 1. Pay special attention to the way the calculator graphs the formula. What I found most interesting: It graphed starting at the x-axis, then went downward. After it reached the bottom of the screen, the line started again at the top of the screen, moving downward until it hit the x-axis again and finished the line. On top of that, when I traced the line, the increments between y-values which each traced point was grew as the cursor neared the bottom or top of the screen. It’s as though something were making the cursor move faster as it went to the bottom or top of this seeming line. I called my teacher over to discuss it.

She immediately pointed out how it graphed interestingly, and said it was because of the theta value. I quickly rebutted (sounding less fluid than this actually does):

“I’m not seeing it that way. Imagining that the polar plane is really a top-down view of a sphere, what would happen if this line could go around and under the sphere? What would we see on this two-dimensional plane? We see the line go down, leave the plane, and magically appear on the other side, as though it had just jumped somewhere. But in reality, the line looped around the bottom of the sphere, taking a shortcut route in the third dimension to get to the other side of the graph.”

She paused and reflected, then said: “Jason, you make me think.”

(Click sphere for larger version.)

I’m completely convinced that most polar equations are like this, having properties of being looped around a sphere or somehow entering into the third dimension. However, it seems hard for many people to see, which is why I tried my best with some very poorly made diagrams. I would have done better with the 3D sphere, but 3dsmax currently doesn’t work on Windows Vista, so I had to make due with Photoshop. Sorry!

At any rate, give this stuff a thought. You might realize that there’s more to math than what you’re learning in school, or what you’ve learned already. It’s natural for me to see things in the third (and higher) dimensions. I’d like to apply this experiment to other polar graphs such as curves. Perhaps they’re cross sections of three dimensional shapes. You never know, and I’m certainly excited to find out.