Part 2 – Use this one weird trick to show that the number of points in a plane is the same as the number of points on a line, and 3D space, and any number of dimensions
You have a point on a plane, and it has an X and Y coordinate, such as:
You can take the X and Y coordinates, and combine them to make a single, unique number by doing the following:
Take the first decimal place of the X coordinate, and follow it with the first decimal place of the Y coordinate, in this case: 0.32, then the second decimal place of the X and Y coordinates: 0.3250, then the third, fourth and so on to as much precision as you care to have: 0.32502689… this, like any number, can be represented by a point on a line. You can do this for every point on the plane and each point will map to a unique point on a line. Similarly, you can take a point on a line and pair up the numbers to make a unique point on a plane. So, now you have a 1:1 function that maps all the points on a plane to points on a line and vice versa.
Now you can add a third dimension.
X:0.07982 Y:0.92580 Z:0.10458
Combine the decimal places in triplets, so in our example the result will be: 0.091720954885208…
You can keep adding dimensions, and no matter how many dimensions you have, you can use this trick to map all the points in 1:1 fashion to a line.
The amazing conclusion is that the number of points on a line of any length is the same as the number of points on a plane, which is the same as the number of points in any dimensional space.