How to Wrap the Earth in Paper

Can we wrap the surface of the Earth with paper? If every place on the surface was physically covered with paper, then this is possible.

Can we wrap the surface of the Earth with just one piece of paper? If this piece of paper is large enough to cover the entire surface area of the Earth, then this is possible.

Can we wrap the surface of the Earth with just one 8.5" x 11" piece of paper (0.01" thick)? If we were able to fold this piece of paper in half 54 times, then this is possible.

If one were to visualize the folding of the paper parallel to the side that is 8.5", the following changes in dimensions can be seen.

0 folds = 8.5" x 11" x 0.01"
1 fold = 8.5" x 5.5" x 0.02"
2 folds = 8.5" x 2.75" x 0.04"
3 folds = 8.5" x 1.375" x 0.08"

It can be seen that the 8.5" remains the same because we are not folding that side. The 11" keeps getting cut in half with every fold because that is the side we are always folding. The 0.01" keeps doubling because the thickness gets doubled from what it previously was, with every fold. The thickness of the paper will double while the length (which was originally 11") will get halved with every fold. By the time we get into the 50+ folds, the dimensions will look like the following.

52 folds = 8.5" x 2.4x10^-15" x 4.5x10¹³"
53 folds = 8.5" x 1.2x10^-15" x 9.0x10¹³"
54 folds = 8.5" x 6.1x10^-16" x 1.8x10¹⁴"

That thickness is getting very high and the length (which was originally 11") is getting very low. Looking at the thickness, it would take light over 4 hours to travel that distance! Looking at the length, the diameter of an average atom is almost 6.5 million times larger than that length!

In other words, the piece of paper is getting VERY long and is VERY thin. It is so thin, that it probably will not even be visible any longer since it is almost 6.5 million times smaller than the diameter of an atom!

The surface area of the Earth is about 509,600,000,000m. For our 54 folds, if we calculate the area we can cover by multiplying the thickness by the 8.5" width, we will get a value of about 987,884,000,000m. We can maybe start at the South Pole and wrap our way around and around in circles until we reach the North Pole. The thickness we would achieve with 54 folds would be more than enough to be able to cover the entire surface area of the Earth, even accounting for skyscrapers, mountains and volcanoes! We might even have enough paper left over to tie a bowtie knot with the ends of the paper.

Of course this cannot be physically accomplished. When you get to the point where the paper you are trying to fold has a width (which you can now call the new thickness since it is so thin) that is smaller than an atom, you won't even be able to see the paper, but it will be there. According to mathematics, this is possible. How unfortunate to see mathematics get bullied by science and mother nature.