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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

image11.png

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.

...read more.

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-

image01.png

.

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 –

image20.png

In the result box, we can see that the equation of the curve is –

Y = 0.5image02.png

2 +0.5image02.png

 + 1

Therefore, substituting R and n back, I got the generalized rule as follows –

image03.png

Three – dimensional object-

For three-dimensional graphing, I again used the Autograph software.

...read more.

Conclusion

image07.png

Substituting x and y as n and (P) respectively, I got –

image08.png

image15.png

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 –

image09.png

,          

where the image10.png

 term is from a 3d sequence.

...read more.

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