Inadvertent time, there is no intention to derive the formula of the maximum number of numbers of 3-dimensional space, that is, the n plane can split the space into how many parts. Plan to use a progressive method to solve the method of proven by mathematics.
Start with the segmentation of 1 dimensional space, 1-dimensional space is a point division line, and the conditions are not coincident. Splitting numbers are simple:
SN = N 1
Then, 2-dimensional space division: 1 linearly divided the plane into 2 parts, 2 straight lines (intersecting) divided into 4 parts, 3 straight lines (two-two-two-phase intersection, no repetition) divide the plane into 7 Some, 4 straight lines (two-two-phase intersection, no repetition) divide the plane into 11 parts ... adopt regression analysis, solving:
Pn = n (n 1) / 2 1
It seems that the number of divided numbers in 3 dimensional space should be 3 times. Then, as soon as soon as possible, it will push the formula between the 3-dimensional space. Unfortunately, I have limited space imagination, and if I can't imagine if 4 planes.
The card is unable to start here, so I use Google to query online to see if anyone has studied this problem. As a result, it was found that Germany's Geometrics Steiner first proposed and resolved this problem. This issue has been classified into modern 100 classic mathematics issues. Unfortunately, I didn't find a specific solution step online, so the next few months continued to be tortured by this problem ... until the last month from the online "Mathematics 100 basic issues" of Shanxi Science and Technology Press The book will be relieved. Now announced the solution process:
Still pushing from the plane first. The above-mentioned maximum number of points is listed above, it will not be repeated. Now, the N straight lines are divided into PN parts, for PN, first try to obtain its recurrent formula. Assume that the plane has been divided into PN-1 in the N-1 straight line, and then the nth line is added to obtain the maximum number of divisions. At this time, it must increase the N-1 intersection, and this newly added straight line must pass through the original N part, and each of these N parts is 2 points. Therefore, the addition of this Nth line has increased the fraction of some of the planes, so it gets a recurring formula:
PN = PN-1 N
According to the session n = 1, 2, 3 ,. . . , N, notice that P0 = 1, add the obtained N type subsequent:
Pn = 1 (1 2 3 ... N) = n (n 1) / 2 1
Secondly consider space. In order to divide the space into the largest score, the above-mentioned plane satisfies a similar condition, that is, any two planes intersect, and there is no more than three plane intersections. This is the most spatial spatial spatial spatial spatial. Assume that the space has been divided into CN-1 by the N-1 plane, and then the nth plane is added to obtain the maximum number of divisions. At this point, the newly added plane and the original plane must produce an N-1 intersection, and any 2 intersection is two or two, and the intersection is not repeated. Projection all the intersections into a plane, so the N-1 intersection added to the newly added plane is divided into PN-1. The projection plane is restored, then this PN-1 graphic portion is divided into two copies of the original space portion thereof. Therefore, the addition of this Nth plane makes the number of parts of the original space increases PN-1, so it has obtained the corresponding recurrent formula: CN = CN-1 PN-1
According to the session n = 1, 2, 3 ,. . . , N, notice that C0 = P0 = 1, add the obtained n type subsequent:
CN = 1 (1 p2 ... PN-1) = 2
According to the integer, the power and formulas are:
= N (N-1) / 2, = N (N-1) (2n-1) / 6,
Solution formula to the above-mentioned CN-1:
CN = 2 N (N-1) (2N-1) / 12 N (N-1) / 4 (N 1) = (N3 5N 6) / 6
Doubt before you can solve it. However, my brain has an new question mark. As the 2-dimensional plane can only observe 1 dimension, the 3-dimensional creature can only observe 2 dimensionally, the spatial dimension in real life is definitely greater than 3 (no matter 4-dimensional time and space in general relativity) Models or 10 or 11-dimensional space models in the M theory in modern physics). 4 Dimensional space or even more dimensional multi-dimensional space can be divided, then its maximum segmentation must be guided.
In the above derivation process, it is important to project the N-dimensional space to N-1 dimension, and then use the N-1 dimensional division formula to calculate the maximum number of divided numbers in n-dimensional space. I used this method to try the maximum number of divided numbers in 4-dimensional space (the N3-dimensional space to divide 4-dimensional space into VN):
VN = VN-1 CN-1 = (N4-2 N3 11N2 14N 24) / 24
However, due to the space of 3D or more, it may not be projected to the 3-dimensional space between 4-dimensional spaces, so that the above formula is only guess. However, there is no bold guess, any technologies can't make breakthroughs on the original basis, isn't it?
