A journey directed by a mathematician through high levels

Alternatively, just as we can unveil the face of a cube in six squares, we can unveil the three-dimensional boundary of a Tesract to get eight cubes, as Salvador Daly demonstrated in his 1954 painting. Crucifixion (Corpus Hypercubus).

We can imagine the cube with its face exposed. Similarly, we can begin to imagine by exposing the border cubes of a tesseract.

All of this adds to an intuitive understanding that is an abstract space n-If there is dimension n This includes the level of freedom (as the birds had), or if needed n Coordinates to describe the position of a point. Yet, as we will see, mathematicians have discovered that dimension is more complex than this simple description.

High-level formal studies appeared in the 19th century and became more sophisticated over the decades: a 1911 bibliography contains 1,832 references to geometry. n Dimensions. Perhaps as a result, in the late 19th and early 20th centuries, the public became fascinated by the fourth dimension. In 1884, Edwin Abbott wrote the popular satirical novel Flat land, Which used a two-dimensional animal to face a character from the third dimension as an analogy to help readers understand the fourth dimension. A 1909 Scientific American “What is the fourth dimension?” Essay competition in the title? 245 have been submitted for the $ 500 prize. And many artists, such as Pablo Picasso and Marcel Duchamp, incorporated fourth-dimensional ideas into their work.

But at this point, mathematicians realized that the lack of a formal definition for dimensions was actually a problem.

George Cantor is best known for his discovery that infinity comes in different shapes or cardinities. At first Cantor believed that the points of a line segment, a square and a cube must have different cardinities, such as a line of 10 points, a 10 × 10 grid point and a 10 × 10 × 10 cube point with a different number of points. However, in 1877 he discovered one-to-one correspondence between the points of a line segment and the points of a square (and cubes of all dimensions in the same way), showing that they have the same cardinality. Spontaneously, he proved that lines, squares and cubes all have infinitely small dots of the same dimension, despite their different dimensions. “I see it, but I don’t believe it,” wrote Cantor Richard Dedaikind.

Cantor realized that this discovery threatened the intuitive notion n-Dimensional space required n Coordinates, because each point in a n-Dimensional cubes can be uniquely identified by a number from a distance, so that in a sense these high-dimensional cubes are equivalent to one-dimensional line segments. However, as Dadekind noted, Cantor’s work was highly isolated – it essentially divided a line segment into infinitely many parts and reassembled them to form a cube. This is not a behavior we want for a coordination system; It would be very chaotic to be as helpful as assigning unique addresses to Manhattan but assigning them randomly.

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