Plane and Space
Visualize !
One plane divides space into two parts. Two planes passing through a point divide space into four parts. See it with your two palms as two planes.
The question is to find out, How many parts three planes passing through a point divides the space ? What about four planes ?
One plane divides space into two parts. Two planes passing through a point divide space into four parts. See it with your two palms as two planes.
The question is to find out, How many parts three planes passing through a point divides the space ? What about four planes ?
Labels: mathschallenge, thinktank
posted by Rajesh Lal at
11:19 PM
16 Comments:
Number of planes plus one equals the amount of space parts
if we're talking 3 dimensional space:
3 planes will give you 8 sections.
4 planes will give you 16 sections.
and so on.
The key here is that all planes must pass through a single point.
Since that is the case,
1 plane divides space into 2 parts.
2 planes divide space into 4 parts.
3 planes divide space into 8 parts.
4 planes divide space into 12 parts.
blahsjekjafh ds k
sdf
sdf
fdg
fdg
Depends on the orientation of the planes. If all intersect on a line (and therefore also through a point, many points) its space is divided into 2n pieces. (Visualize looking at all of them from the top, where they all look just like lines. If the 3rd plane is perpendicular to the first two, it's n=1::2; n>1::4n-4
Either there are some trick answers from anonymous guys or the question has impacted on their brains considerably
There is a great book by Edwin Abbott Abbott, called Flatland. It looks like it would take an hour to read, but the story leads to an infinity of joy.
The respondent who said to visualize the planes as lines viewed from above is correct, but obtuse. The respondent with the "many points fudge" didn't read the question properly.
Actually, the question gave the answer:
1 plane = 2 spaces
2 planes = 4 spaces
3 planes = 6 spaces
4 planes = 8 spaces, etc.
the equation is 2 to the power of n where n is the number of planes.
example
1 plane = 2
2 planes = 2*2 = 4
3 planes = 2*2*2 = 8
2 planes = 2*2*2*2 = 16
etc.
This all depends on you interpretation of plane and point, and the dimensional space in which you are working.
Indeed if you think of the plane as a straight line drawn on a piece of paper and have each plane pass through a given point, then each plane will add an additional two areas of space.
However, now think of the plane as a sheet of paper and your areas of space as being on each side.
For 1 plane (sheet of paper) there will always be 2 areas of space (1 on each side) no matter what the orientation of the plane.
For 2 planes there will always be 4 areas of space when both planes pass through the same point unless both planes have exactly the same orientation whcih would probably make them the same plan anyway.
For 3 planes the is no definative answer as we are now dependent on the orientation of the 3rd plane relative to the first 2 planes. If it is parallel in one direction to them, with no rotation then, there will be 6 areas of space. But as soon as we add any element of rotation there will be 8 areas.
The same is true of 4+ planes giving 8, 12 or 14 areas.
What we can say is the increase in the number of areas of space for each additional plane is the number existing planes that the plane intersects which it shares no elements of parallelism with less the number of planes that the plane interesects which it shares and element of parellelism multiplied by 2.
You cannot use the 2 to the power rule. A single plane must be just that and so cannot pass through all areas of space past the third plane. Think of a cube. Cut it in half vertically. Hold the pieces in position then cut in half horizontally from front to back. Again hold the now four pieces in position and cut in half horizontally from left to right. you have eight pieces, but you cannot now make single cut which divides all eight pieces at once.
You will either subdivide 4 (cutting straight down across top front right corner to top back left corner) or 8 (cutting diagonally down from top front right corner to back bottom left corener)
Obviously some here are constrainted by 3 dimensional space, the fourth plane does intersect each of the other planes if it passes thru the given single point, thus 16 is correct for 4 planes. And they are all at right angles to each other.
Consider cutting a pizza. The first two cuts through the center will double the number of slices, since they can cut through all previously made slices. After the second cut, subsequent cuts will cut through at most two other slices (the same number as the second cut), therefore increasing the number of resulting slices by two.
Similarly, with our three-dimensional 'pizza', the first three cuts will double the number of 'slices', resulting in eight parts. Subsequent cuts will only be able to subdivide at most four of the previous parts (the same number as the third cut), therefore, increasing the number of parts by four. So the number of parts for plane cuts, starting with the first, will be 2, 4, 8, 12, 16, 20...
3 planes=8 parts, 4 planes=16 parts in 4 dimensions
Post a Comment
Links to this post:
Create a Link
<< Home