Practical Problem-triangulation
Week Three Practical Problem
Questions
A. Determine T(n) for n = 6, 7, & 8.
T(6)=(2*6*10*14)/(6-1)!=14
T(7)= (2*6*10*14*18)/(7-1)!=42
T(8)= (2*6*10*14*18*22)/(8-1)!=132
B. Do you detect a pattern to these numbers? This pattern may arise out of the numbers or the manner in which you generated triangulations. (A closed-form function for T(n) is relatively straightforward, but is fairly nontrivial to construct; you will not have to explore that here.)
The pattern for these numbers is described using the formula: T(n)=2*6*10...(4n-10)/(n-1)!. You add four to the previous amount of the upper part of the equation and then multiply.
C. How would T(n) change if you ignored the vertices’ distinctness? That is, if you remove the labels, and say two triangulations are identical if one can be transformed into the other via a rotation or a reflection, how does this change T(n) for n = 4, 5, 6, 7, & 8?
I think that the amount of triangulations would be lower as the amount of sides increases compared to what the solutions are now. This is because if the line reflect or can be rotated on to each other, which some of them do, then they would be counted as one making the overall product decrease. Considering them separately makes for more triangulations.
D. What effect does relaxing the convexity restriction have on T(n)? See how T(n) changes for n = 4, 5, & 6. Do you see a pattern?
Relaxing the restrictions on these three polygons does not in effect really make any difference. Their solutions would still be the same because none of them can rotate or reflect onto each other causing none of the lines to be identical.
Questions
A. Determine T(n) for n = 6, 7, & 8.
T(6)=(2*6*10*14)/(6-1)!=14
T(7)= (2*6*10*14*18)/(7-1)!=42
T(8)= (2*6*10*14*18*22)/(8-1)!=132
B. Do you detect a pattern to these numbers? This pattern may arise out of the numbers or the manner in which you generated triangulations. (A closed-form function for T(n) is relatively straightforward, but is fairly nontrivial to construct; you will not have to explore that here.)
The pattern for these numbers is described using the formula: T(n)=2*6*10...(4n-10)/(n-1)!. You add four to the previous amount of the upper part of the equation and then multiply.
C. How would T(n) change if you ignored the vertices’ distinctness? That is, if you remove the labels, and say two triangulations are identical if one can be transformed into the other via a rotation or a reflection, how does this change T(n) for n = 4, 5, 6, 7, & 8?
I think that the amount of triangulations would be lower as the amount of sides increases compared to what the solutions are now. This is because if the line reflect or can be rotated on to each other, which some of them do, then they would be counted as one making the overall product decrease. Considering them separately makes for more triangulations.
D. What effect does relaxing the convexity restriction have on T(n)? See how T(n) changes for n = 4, 5, & 6. Do you see a pattern?
Relaxing the restrictions on these three polygons does not in effect really make any difference. Their solutions would still be the same because none of them can rotate or reflect onto each other causing none of the lines to be identical.