No of segments = No of cuts made + 1
Two – dimensional object –
For a circle, it was hard to find the maximum number of regions because chords can cut different number of regions in the way they are drawn and because I needed maximum number of regions, I had to draw chords in a specific manner.
The diagrams are as follows –
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. In my first try, I proceeded normally, entered various equations in the form x=a, x=z, z=y etc but when I added the values in graph, I couldn’t get a proper best fit curve. This meant that the values were incorrect. Even after trying again, and checking the number of parts from various angles, I couldn’t get a different answer.
After few more trials, I noticed that equations such as x=a, x=b, x=c, x=v where one side is x and the other side is any variable other than y and z, were all same. After this, I understood what next I had to do and so I started mixing these equations like x = a + b etc and got better results as follows –
When n=1,
P = 2
When n=2,
P = 4
When n=3,
P = 8
When n=4,
Top view-
Number of parts = 7
Bottom view-
Number of parts = 7
Part covered by planes on all sides.
Total number of parts =
7 + 7 + 1
= 15
Data table –
The first difference sequence is 2, 4, and 7.
To find the recursive formula, I took the answer as 15 and tried to find the different values which could lead to 15. The working is as follows –
15 (answer needed) = 4 (term of first difference sequence) + 11
However 4 was the previous term as (15-8) is 7. Therefore I wrote 4 as (8-4) i.e. the previous term minus the previous to previous term and 11 as the first difference (7) plus the previous n term.
i.e. 15 = (8-4) + (8+3)
.
Taking
together, I got –
.
However the generalized formula was still left so I added these values in the XY data set of Autograph software. This time I took the best curve fit of order 3 because there were three dimensions. The resulting equation I got was –
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 –
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.