Why the space is three dimensional

Because 4D space is not determinate.

Let’s construct a magic square. “In recreational mathematics, a magic square of order n is an arrangement of n^2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant.”

For order 3, only one configuration exists. To sum up 3 rows, 3 columns, and 2 diagonals, we need 8 different combinations. And it is exactly what we can get!  Only 5 appears 4 times so it has to sit in the center. 2/4/6/8 appear 3 times so they are at the corners. 1/3/7/9 appear 2 times and they are at the edges. Then the square is totally determined.

8 3 4
1 5 9
6 7 2

We can always add an arbitrary number to all the numbers in the square, eg,

18 13 14
11 15 19
16 17 12

The story totally changes from 3D to 4D. We only need 10 combinations but there are 86 of them! You should imagine more features or restrictions can be imposed, eg, the sum 34 can be found in each of the quadrants, the center four squares, and the corner squares, and more, just like Albrecht Dürer did 500 years ago!

16  3  2 13
5 10 11  8
9  6  7 12
4 15 14  1


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