Yo, this is super interesting. Infinities!!
Well this is one of the coolest things I’ve ever learned about.
So I’m in Complex Variables this semester, right? Today we talked about how to take limits of complex numbers as well as the closely related topic of infinity.
I’m assuming most people who read this regularly (or just happen to stumble upon it) know at least a little about infinity in the context of real numbers. Mainly, if we represent the reals on a number line, we have a direction that goes off towards negative infinity and a direction that goes off to positive infinity. But does this translate to complex numbers?
Well, not really. When we deal with complex numbers, we deal with the complex plane: a 2-D space with one axis representing the real part of a number and the other axis representing the imaginary part of a number. That is, one way we can think of a complex number is as a set of coordinates on the complex plane. For example, if I had the number z = 3 + i2, I could represent it with the coordinates (3,2) and plot it like this:
Since we’re now dealing with a plane, we actually have infinitely many directions that can be thought of as infinity—basically, any direction out from the origin.
So how do we define infinity in the complex plane to allow us to, among other things, take limits involving the point at infinity?
Answer: The Riemann sphere!
The Riemann sphere is a stereographic projection of the complex plane onto the unit sphere at the point (0,0,1). Piccy from Wiki:
So what does this do? Well, for each point A on the complex plane, there exists a line that intersects both the point A and the north pole of the sphere. This line hits the sphere itself at α, point unique to the position of point A in the complex plane—that’s how the plane is “mapped” to the sphere and that unique mapping point is called the “projection” of point A.
The further out you go on the complex plane—that is, the further away from the sphere you go, those projection points get closer and closer to the north pole itself. However, no point is projected directly onto the north pole. So we can think of the north pole as being the image of all the points in the complex plane that are at infinity.
Isn’t that cool? It’s a way to reduce an “infinite number of infinities” to a single point.
We didn’t have time to talk about how we’re going to use it yet, but just that idea is super cool and required a blog post.
I set out to re-draw this old drawing to make it nicer/more tattoo-worthy, but then I went nutso with the rainbow and this resulted.
BOOM! Are your eyes bleeding yet?
Um…okay. This is a little doohickey that I wrote in about three minutes while under the influence of Ibuprofen, Zoloft, antibiotics, and iron. Here we go:
The Story of Infinity: How it Came to Be
Once upon a time, long ago in Wisconsin, a lonely Anglo-Saxon whitesmith named Raphael was sitting in a field, waiting for a message from his African-American blacksmith friend, Pete.
While waiting for Pete, the whitesmith felt a feeling. He leapt to his feet, immediately fell to his knees, and began to pray.
“God,” he said, looking up to the sky “if you really are God, please kill Pete and let me have his blacksmithing business, for he is much more profitable than me. Thank you.” He sat back down in the grass, and, after another moment or two, fell asleep.
Suddenly, a great rumble came from the sky above. Raphael awoke. He looked up into the sky, and before him hovered God. God was wearing pink. Raphael questioned this, saying, “Almighty one, why have you chosen such a feminine color?”
God replied in a manly voice, “God has no gender. God is a wonderful being of unquestionable holiness and awesomeness. God is also a sheep. Look closer.” And it was revealed that God was truly a sheep.
“God,” said Raphael “You heard my prayer. Will you kill Pete?”
“Why?” God asked. “Pete is your friend. He is a successful black blacksmith.”
“That is the point,” Raphael said. “He is more successful than me. I do not like it. Plus, his wife is more attractive than mine, and it would be much simpler to have an affair with her if Pete were out of the way.”
God considered. “Alright,” came the answer. “Here is how it will work: I will leave you and you shall sleep. In the dream, a beautiful woman will appear. All you have to do is reach her and kiss her. As soon as you have done that, Pete will be dead.” God then leapt from the field and, with his mighty sheep legs, reached the Kingdom of Heaven in a single bound.
Raphael relaxed in the grass and instantly fell asleep. As he slept, he began to dream. In his dream, he was in a wide field of wheat. A sign was posted next to him that read Finity Field. As he looked around, he saw a beautiful woman in the distance. He judged the length between them to be about a mile. “All I have to do is run to her,” Raphael reasoned “and Pete will be dead.” So he began to run. An hour later, however, though Raphael had run over five miles, the beautiful woman still seemed to be the same length away from him. He kept running. Two hours later, he was exhausted and nowhere closer to the beautiful woman than he had been before.
Irate, he awoke. “God!” he called.
God appeared. “Yes, Raphael?”
“Why have you deceived me? It is not possible to reach the beautiful woman in Finity Field.”
“Why Raphael,” God replied “it is most certainly possible. Just be patient. Run for a longer period of time.”
God left Raphael and he went back to sleep. In his dream, he took Gods advice and ran for two days straight. Still, however, the beautiful woman remained the same distance away from Raphael she was before. But Raphael would not give up. He was determined to kill Pete. So he ran, and ran, and ran, and ran.
Meanwhile, Pete reached the field in which Raphael lay dreaming. “Raphael?” He shook his friends shoulder. Raphael did not awake; he was deep into his dream. So Pete turned to God.
“God?” he said to the heavens. “What is wrong with my friend?”
Again, God appeared, and though Pete wondered about God being a sheep dressed in pink, he said nothing of this and pointed to the sleeping Raphael. “Raphael is being punished.” God explained to Pete. “You see, Raphael wished death upon you because you are more successful than you and your wife is more attractive than his wife. I told him that in his dream, if he would reach a beautiful woman and kiss her, his wish would be granted. However, since I am the Almighty one, I can trick him with my power. He will never reach the beautiful women because of his selfish thoughts. He is in Finity…forever. “
And so, infinity became the word for forever. Also, whitesmithing became unpopular and was virtually destroyed a month or so after Raphael’s demise. Pete flourished, and so did blacksmithing.
The disturbing part is, though, that this story isn’t nearly as whacked out as the stories I come up with when I’m not on drugs.