- Level: International Baccalaureate
- Subject: Maths
- Word count: 1051
In this portfolio, I am required to investigate the number of regions obtained by making cuts in one, two and three dimensional objects
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Introduction
MATHS PORTFOLIO
HOW MANY PIECES?
INVESTIGATION QUESTIONS –
A line segment is a finite one-dimensional object. Find a rule which relates the maximum number of segments (S) obtained from n cuts. Comment on your results.
A circle is a finite two-dimensional object. Investigate the maximum number of regions (R) obtained when n chords are drawn.
A cuboid is a finite three-dimensional object. Investigate the maximum number of parts (P) that are obtained with n cuts.
If a finite four-dimensional object exists and the procedure is repeated what would you expect to find?
In this portfolio, I am required to investigate the number of regions obtained by making cuts in one, two and three dimensional objects. By finding a rule in all the three cases, I need to develop a rule for a four-dimensional object by searching for a definite pattern.
The data is as follows-
One – dimensional object –
No of cuts made (n) | No of segments formed (S) |
1 | 2 |
2 | 3 |
3 | 4 |
4 | 5 |
5 | 6 |
In the data table, we can see that in all the 5 cases, the number of segments formed is 1 more than the number of cuts made.
Middle
After drawing all the sketches and finding the number of regions obtained, I proceeded to find the pattern of the sequence.
First I found the difference between the terms of the sequence of number of regions.
i.e – 2, 4, 7, 11, 16
The differences are - 2, 3, 4, and 5
And the difference is equal to the number of chords drawn.
Therefore,
Number of regions = Number of regions obtained + number of chords drawn
obtained in the previous sketch
We can also consider the number of regions obtained as a function of the number of chords drawn.
Therefore, the recursive rule can be stated as follows-
.
However, to find the generalization, I need to eliminate the function and prepare an equation only with two variables – n and R. Therefore, I plotted the figures in the data table in the XY data set of geometry software Autograph to find the equation of the line. The results was –
In the result box, we can see that the equation of the curve is –
Y = 0.5
2 +0.5
+ 1
Therefore, substituting R and n back, I got the generalized rule as follows –
Three – dimensional object-
For three-dimensional graphing, I again used the Autograph software.
Conclusion
Substituting x and y as n and (P) respectively, I got –
After getting the equation, I also found the value for 5 cuts-
(P) = 2(15) – 8 + 4
(P) = 26
Therefore, the final results obtained from all the 3 dimensions are –
No of cuts (n) | No of segments in 1d (S) | No of regions in 2d (R) | No of parts in 3d (P) |
1 | 2 | 2 | 2 |
2 | 3 | 4 | 4 |
3 | 4 | 7 | 8 |
4 | 5 | 11 | 15 |
5 | 6 | 16 | 26 |
The sequence of the values for 1d line is - 2, 3, 4, 5, and 6.
And, the sequence of the values for 2d line is – 2, 4, 7, 11, and 16.
The sequence of the difference in values for 2d circle is – 2, 3, 4, and 5 which is the same as the sequence of values for 1d line.
Therefore, the values of a 2d figure can be simply obtained by just adding the previous term and the respective term for a 1d figure.
i.e. 11 (answer needed) = 7 (previous answer) + 4 (answer in 1d for the previous number of cuts)
The same thing is applicable between 2d and 3d figure.
i.e. 26 (answer needed) = 15 (previous answer) +11 (answer in 2d for the previous number of cuts)
Therefore, the same should be applicable for a 4d figure as well.
Hence, the recursive formula for a 4d figure can be written as –
,
where the
term is from a 3d sequence.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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