table of Contents
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1. introduction
From the point of view of the philosophy of mathematics, for workers engaged in a professional direction, it must first be emphasized that to raise a good question. Drawn by the outstanding problem is the logic instructive. Today, I try to write the question here is solved before large numbers of people, but to write down the significance of the problem still missing.
Mito how to use a piece of tofu cut into nine?
Obviously, the mathematical modeling job is to abstract common problem and combined with logical, then the language of logic to reasoning. We'll call this into question, Mito up to tofu cut into many pieces, and then a little abstract, to convert it into a maximum of three planes three-dimensional space can be divided into a number of sub-space.
2. segmentation of the three-dimensional space for
Simple, space we live in a time that is three-dimensional space. Fixed point in three-dimensional space \ (O \) , any straight line drawn \ (X \) , then drawn from an arbitrary point in a straight line \ (X \) line perpendicular \ (Y \) , the two mutually perpendicular generating a straight plane, through the point \ (O \) be in a straight line and the plane \ (of Oxy \) perpendicular to the straight line we call \ (Z \) , so that, \ (Oxyz \) constitute a to \ (O \ ) as the origin of the rectangular coordinate system space. Based on this space Cartesian coordinate system, we can say construct a three-dimensional space on a mathematical sense.
Of course, we introduce the coordinate system just to introduce three-dimensional space coordinate system and in fact will not play its role. Then begin formal discussions cutting three-dimensional space.
With a plane, i.e. two-dimensional space, to cut a three-dimensional space, the intuitive yet, we can see the following cases
| The number of plane | The largest number of sub-space |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
Of course, the number of sub-spaces of three planes can be cut out most of us are also envisioned, a value of 8, but the more planar notation clearly not the best way.
In fact, we can see the effect of increasing each plane segmentation.
0 is the number of two-dimensional space, the three-dimensional space as a whole; increased a first plane, a two-dimensional space is divided into two; the second plane is increased, if the first and second planes do not intersect, will cut a subspace, while the second plane intersects the first, second plane cutting the two sub-space, apparently, the cutting method we have to take is to increase the number of new space cut up subspace . We know that each subspace is cut into two new planes, while the old space and the cutting plane into sub-sub plane is not Unicom, then obviously, the number of sub-planes would mean an increase in the number of new sub-space. To make up the sub-space increases, this plane is equivalent to a maximum number of cutting lines turned out sub-planes. As a result, we put this problem into divided on the issue has become a two-dimensional space.
3. division conducted a two-dimensional space
Before we follow the way of the definition of two-dimensional plane, but does not introduce straight \ (z \) , so we get the plane \ (Oxy \) . Let's look at a simple cut numbers.
| The number of straight line | The most number of sub-plane |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 7 |
We found that, when the largest increase in the number of new straight line to be cut up to sub-planes. Similarly, a new straight line dividing plane will quilt. Only up to a point of intersection between two straight lines, so most of the points situation is split each line of this line intersects with the previous.
Set of \ (n-1 \) Total subplanes number of cuts is \ (of S_ {n--. 1} \) , the newly added number of planes promoter is \ (B_ {n-} \) , there is a formula \ (of S_ n-of S_ {} = {n-+}. 1-n-B_ {} \) . Obviously, the number of newly added sub-planes equals the number of new straight line is divided into segments, namely \ (n-\) ( \ (. 1-n-\) number of intersections with a straight line intersecting the old to the new sub-planes, and \ (N- 1 \) point to the straight line is divided into \ (n-\) above).
Then we get the following term formulas:
\ [S_ {} = n-of S_ {}. 1-n-n-+ \]
\ [S_ {} = 0. 1 \]
As is evident Gaussian summation will smile
\ [S_ { n} = \ sum_ {i =
0} ^ n {i} +1 = {{{n (n + 1)} \ over {2}} + 1} \] this is what we find in a two-dimensional cutting dimensional space can get the maximum number of sub-space.
4. Go back to the three-dimensional space
Set of \ (n-1 \) space the total number of the sub-cuts for \ (of S_ {n--. 1} \) , the newly added number of spatial promoter is \ (B_ {n-} \) , there is a formula \ (of S_ n-of S_ {} = {n-+}. 1-n-B_ {} \) .
We have previously seen, the \ (n-\) space the maximum number of sub-planes that can be cut, i.e., the newly added number of sub-spaces, as before \ (n-1 \) generating sub-plane to this plane intersecting subspaces the maximum value, i.e., before each subspace generating line of intersection therewith, the maximum value of the line of intersection of the cutting plane. That \ (n-1 \) of straight lines maximum cutting plane.
In turn obtained from §3, \ (n-B_ {} = {{n-(. 1-n-)} \ {2}} over \) $
then we get the following term formulas:
\ [n-of S_ {} = {N- of S_ {{{+}. 1 n-(. 1-n-)} \}} over {2}. 1 + \]
\ [. 1 of S_ {0} = \]
simple sum
\ [S_ {n} = \ sum_ {i = 0 } ^ n {{{n (
n-1)} \ over {2}} + 1} = {{n ^ 3 + 5n + 6} \ over {6}} \] this we solve the two-dimensional space dimensional cutting problem, then the next there is a four-dimensional space, five-dimensional space, even to the \ (n \) dimensional space, this time we can not rely on a simple induction, but to use some algebra to solve the problem.