- #1

- 346

- 3

## Main Question or Discussion Point

I have always wondered if spacetime is actually smooth, or if it is granular such as a quantized spacetime.

I remember learning that if you draw the smallest possible square around a circle, you will, of course, get a perimeter, 8 times the radius of the circle. Then if you draw in the largest possible squares in each of the four corners between the inside of the large square and the outside of the circle, you will get closer to the area of the circle. But you won't get closer to the circumference.

If you kept on cutting out the largest possible square in the spaces between the outer edge of the circle and the inside of the original square, you will approach the circle's area, but the perimeter stays constant at 8r.

Now here's the point. Let's say that the length of a quantum of spacetime is d (let's just forget about Planck lengths for now). No matter what shape the quantum of this spacetime is in, there will still have to be an n number of them in the x direction and the same number in the y direction in getting around a circle. This would mean that any circle that we think is a circle actually has a circumference 8r.

But, I did a simple experiment. I took a piece of measuring paper and wrapped it around a disk. When I took the same piece of measuring paper and measured the diameter of the disk, as you probably have guessed, the length of the piece of paper was about 6.24r.

I even laid down the measuring paper and rolled the disk over the length to make sure that the paper didn't somehow lose length when curling around the disk. The results were the same.

How can such a difference between 8r and 6.24r be explained?

Is this evidence of infinitesimals of spacetime?

I remember learning that if you draw the smallest possible square around a circle, you will, of course, get a perimeter, 8 times the radius of the circle. Then if you draw in the largest possible squares in each of the four corners between the inside of the large square and the outside of the circle, you will get closer to the area of the circle. But you won't get closer to the circumference.

If you kept on cutting out the largest possible square in the spaces between the outer edge of the circle and the inside of the original square, you will approach the circle's area, but the perimeter stays constant at 8r.

Now here's the point. Let's say that the length of a quantum of spacetime is d (let's just forget about Planck lengths for now). No matter what shape the quantum of this spacetime is in, there will still have to be an n number of them in the x direction and the same number in the y direction in getting around a circle. This would mean that any circle that we think is a circle actually has a circumference 8r.

But, I did a simple experiment. I took a piece of measuring paper and wrapped it around a disk. When I took the same piece of measuring paper and measured the diameter of the disk, as you probably have guessed, the length of the piece of paper was about 6.24r.

I even laid down the measuring paper and rolled the disk over the length to make sure that the paper didn't somehow lose length when curling around the disk. The results were the same.

How can such a difference between 8r and 6.24r be explained?

Is this evidence of infinitesimals of spacetime?