This is split into a story and then a problem if you want a concise problem to solve.
Once there was a lady who lived alone on a mountaintop for 20 years. Day in and day out she would meditate and ponder the universe and its strange laws.
One day, she decided to start a journey and teach what she had learned as she had grown wise and most people did not have the privilege of being able to deeply think alone for so long. She headed straight towards the largest city she knew of as the people there were the busiest of all and never stopped to think.
In the city, she came across a young man with his forehead against a wall. There were business papers everywhere and a busted briefcase. From the looks of it, it was thrown with great force. She approached the young man and asked what was wrong.
“Everything is wrong; what is not wrong! People suffer everywhere; people die everywhere; and no one can do anything about it! I mean, how can one person make any difference in a world this big?”
The wise woman looked at him with confusion. She could not understand how he felt so powerless. After a few seconds of thought, she picked up one of the papers on the ground, and drew a line on it, about an inch in length.
|“Does this look big to you?” she asked.|
|“No, but why does that matter?” he responded.|
|“Well, do you think you can change it?”|
|“Yes, but it means nothing. I can’t tell what you are getting at.”|
|“This line looks quite small, and yet it is not. It contains an infinite number of points that are jam packed together. By rearranging these points, I can show you how to make it grow into something twice as big, 10 times as big, as big as you want. Can you see how?”|
|“Umm… I guess, you can just stretch it.”|
|“Right, 0 stays where it is, 1 goes to 10, .2 goes to 2 and everything just moves to create a larger line, but every point in the larger line still came from exactly 1 point in the smaller one, and every point in the smaller one goes to exactly 1 point in the larger one. So, these two are the same size.”|
|“Ok, that’s great; you taught me multiplication. How is that important?”|
|“Well, what if I said you could create a whole line from this, not just a line segment? What if you could make a whole square from it, or a ball? What if you could move the points in this line to at most one other point in all of space and all of time and fill all of it up? You can take any point in space and time: (x, y, z, t), and represent it as a unique point on this line. Then these two are really the same size, and in reality, you can affect something as big as the universe itself.”|
|“Ya right, like you expect me to believe that.”|
So she told him, and went through step-by-step how to describe any point in the universe and for all time as a number between 0 and 1. This greatly excited and encouraged the man, and he picked his papers back up and went on with a new ambition for change.
How can you assign every point in space-time a unique number between 0 and 1? You should assume that space-time can be represented by 4 real numbers which go from -infinity to +infinity and you can use all real numbers between 0 and 1, including 0 and 1, to describe space-time. You don’t have to use every number between 0 and 1, but you can if want the extra challenge.
For some of you who may be more math inclined, this is equivalent to finding a one-to-one function (a function which does not give the identical outputs for two different inputs) which takes any vector from R4 and produces a real number in [0, 1]. For an added challenge, make sure this function is also onto.