Mathematically, the potential of any point on a straight line or line segment is the same as that of any point on a plane, which can be regarded as as as many points on a straight line and a plane, or even as many points on a three-dimensional figure. We can set up a function to correspond straight lines with points on the plane one by one. Such a function is called piano function, and its curve is called peano curve or Hilbert curve.

Set potential

In modern set theory, with? Potential? Describe the size of a set. For a finite set, the number of elements in the set is the potential of the set. For example, the potential of the set {e, i, 0, 1} is 5; For infinite sets, we need to use projection method to determine the potential of the set.

In middle school, we thought that all infinity was the same and could not be compared with infinity. In fact, when we involve deeper set theory, we will find that infinity is hierarchical. It was19th century that the German mathematician Cantor first discovered infinite levels, and he founded the theory of super-finite numbers.

Suppose there are two sets A and B, and the elements in these two sets can be mapped one by one, then we think that the potentials of these two sets are equal; If A can be mapped to some elements in B, but all elements in B cannot be mapped to A, then the potential of B is greater than A, or B has more elements than A).

For example:

The set of natural numbers is {0, 1, 2, 3}

The set of nonnegative even numbers is {0,2,4,6}

Although non-negative even elements are part of natural numbers, they can be mapped one by one (0? 0, 1? 2,2? 4,3? 6), so the number of natural numbers and non-negative even numbers is equal.

Using Cantor's diagonal rule, it can also be proved that the number of natural number set and rational number sets is equal; But we can't correspond natural numbers with irrational numbers one by one, which shows that irrational numbers have greater potential than natural numbers.

The potential of the midpoint of a straight line

Points on a straight line or a line segment form a set. It is easy to prove that there are as many points on a straight line with infinite extension at both ends as on a line segment with arbitrary length by using the function y=tan[(x/a- 1/2)? ], the line segment with length a (excluding both ends) can be in one-to-one correspondence with the points on the infinite straight line, and the potential of the line segment at both ends remains unchanged.

We can also establish a function to correspond the potential of a line segment with the potential of a plane one by one. Take a square with a side length of 1 and a line segment with a length of 1 as examples. The simplest method is to write the point coordinates on the square as (0.abcd ..., 0.xyzw ...), and then map the point coordinates on the line segment as 0.axbyczdw? .

We can easily prove that the potentials of square points of any size are all equal, so no matter how long your line segment is, it is equal to the number of any plane points.

hilbert curve

Hilbert curve refers to a function whose domain is [0, 1], and its function curve traverses all points in the unit square; This function was first put forward by Italian mathematician piano, which maps points on a straight line with points on a plane one by one.

After further expansion, it can be concluded that there are as many points on a straight line as on a plane, even equal to the number of points in a three-dimensional space.