- #1

Mashiro

- 5

- 0

- TL;DR Summary
- Question on if exists a fractal of a line such that after infinite iterations it could cover any given point on a plane.

Currently, as far as I know, the two main ways to express any given point on a plane is through either cartesian plane or polar coordinates. Both of which requires an ordered pair of two numbers to express a point. However, I wonder if there exists such a system that could express any given point on a plane using only one number.

Intuitionally, I think of lines. I know there exists fractals of a line that could theoretically fill a plane after infinite iterations. Therefore, I believe we can construct a system of expressing a plane based on such fractal:

1. The fractal must pass through all points, therefore arbitrarily define any point as origin, denoted as zero.

2. Based on zero, define a positive direction.

3. To express any given point, simple trace from zero, alone the fractal. The distance from your point to the origin would be the unique number describing the point on the plane.

However, some problems are also raised:

1. Does there exist such rule so that a fractal could pass through every single point on a plane? No matter rational or not.

2. If the theory is correct, due to the fact that set of irrational numbers is a higher level of infinity compared to the set of rational numbers, there is a 100% chance of meeting an "irrational point" by choosing arbitrarily. Is this going to be problematic?

If everything about this theory works out, could we apply the same method to a higher dimension (for instance 3-dimensional), and express any given point in a three dimensional space with two numbers? Or perhaps even one.

I am currently a sophomore and my knowledge about mathematics is basic. I might have made stupid mistakes anywhere above. Please point them out to me if you spot any.

Intuitionally, I think of lines. I know there exists fractals of a line that could theoretically fill a plane after infinite iterations. Therefore, I believe we can construct a system of expressing a plane based on such fractal:

1. The fractal must pass through all points, therefore arbitrarily define any point as origin, denoted as zero.

2. Based on zero, define a positive direction.

3. To express any given point, simple trace from zero, alone the fractal. The distance from your point to the origin would be the unique number describing the point on the plane.

However, some problems are also raised:

1. Does there exist such rule so that a fractal could pass through every single point on a plane? No matter rational or not.

2. If the theory is correct, due to the fact that set of irrational numbers is a higher level of infinity compared to the set of rational numbers, there is a 100% chance of meeting an "irrational point" by choosing arbitrarily. Is this going to be problematic?

If everything about this theory works out, could we apply the same method to a higher dimension (for instance 3-dimensional), and express any given point in a three dimensional space with two numbers? Or perhaps even one.

I am currently a sophomore and my knowledge about mathematics is basic. I might have made stupid mistakes anywhere above. Please point them out to me if you spot any.