A high-dimensional guide for mathematicians

Or, just as we can expand the face of a cube into six squares, we can expand the three-dimensional boundary of a cube to obtain eight cubes, as shown in Salvador Dalí’s 1954 painting Suffer (Corpus hypercube).

We can imagine a cube by expanding its faces. Similarly, we can start to imagine a hypercube by expanding its bounding cube.

All these add up to form an intuitive understanding that abstract space is n-If there is n Degrees of freedom inside (like those birds), or if it needs n Coordinates describe the location of a point. However, as we will see, mathematicians find that dimensions are more complicated than these simple descriptions suggest.

Formal research on higher dimensions appeared in the 19th century and became quite complex within a few decades: the 1911 bibliography contained 1,832 n aspect. Perhaps because of this, at the end of the 19th century and the beginning of the 20th century, the public became obsessed with the fourth dimension. In 1884, Edwin Abbott wrote a popular satirical novel Plain, Using the analogy of two-dimensional creatures encountering three-dimensional characters to help readers understand the fourth dimension. A 1909 Scientific american The essay contest entitled “What is the fourth dimension?” received 245 entries and competed for a prize of US$500. Many artists, such as Pablo Picasso and Marcel Duchamp, incorporated the idea of ​​the fourth dimension into their works.

But during this time, mathematicians realized that the lack of a formal definition of dimensions was actually a problem.

George Cantor’s most famous discovery is Infinity has different sizes, Or base. At first, Cantor believed that the set of points in line segments, squares, and cubes must have different bases, like a line of 10 points, a grid of 10 × 10 points, and a cube of 10 × 10 × 10 points. Different number of points. However, in 1877, he discovered that there was a one-to-one correspondence between points in line segments and points in squares (and cubes of all dimensions), indicating that they had the same cardinality. Intuitively, he proved that lines, squares, and cubes all have the same number of infinitesimal points, despite their different sizes. Cantor wrote to Richard Dedekin, “I saw it, but I don’t believe it.”

Cantor realized that this discovery threatened intuition n-Dimensional space required n Coordinates, because each point n-A dimensional cube can be uniquely identified by a number in an interval. Therefore, in a sense, these high-dimensional cubes are equivalent to a one-dimensional line segment. However, as Dedekin pointed out, Cantor’s function is highly discontinuous-it essentially divides a line segment into infinite parts and then recombines them into a cube. This is not the behavior we want the coordinate system to do; it is too disorderly to help, such as providing unique addresses for buildings in Manhattan but assigning them randomly.

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