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If you select any pair of points inside or on the boundary of the polygon, and join them with a line segment, that line segment will remain inside or on the boundary of the polygon; it will never cross the boundary and be outside the polygon. The final question asks you to consider what sort of effect the loosening of this restriction might have on your efforts.
Below is a series of diagrams showing the ways in which the first few polygons may be triangulated. At the start of this task, consider the vertices of the polygon as distinct; that is, they are distinguished from one another, perhaps by a label, letter, or number. The possible triangulations T(n) of an n-gon, for n = 3, 4, and 5, are illustrated here:
T(3) = 1 (A triangle is its own …show more content…
The next in sequence would be the hexagon, and we would have 2*6*10*(24-10), or 2*6*10*14/5! Which equals 1680/120 =14.
The pattern is contained in the formula. When you carry out numerator you find that the next number in sequence is always an increase of 4. This seems to work for all polynomials.
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?
Using the formula given below, which I believe is designed to eliminate duplication, we would find that T(4) would still be 2, T(5) would still be 5, but the pattern would end there. T(6) = 9, T(7)= 14, and T(8) = 20.
There is a definite pattern to the numbers. Each additional vertex increases the number of triangle by the n-2. That is, when figuring a hexagon, where we had 5 sides with a pentagon, that number is increased by (6-2) for a total of 9 sides. 9 sides plus (7-2) totals 14, and this pattern seems to
Below is a series of diagrams showing the ways in which the first few polygons may be triangulated. At the start of this task, consider the vertices of the polygon as distinct; that is, they are distinguished from one another, perhaps by a label, letter, or number. The possible triangulations T(n) of an n-gon, for n = 3, 4, and 5, are illustrated here:
T(3) = 1 (A triangle is its own …show more content…
The next in sequence would be the hexagon, and we would have 2*6*10*(24-10), or 2*6*10*14/5! Which equals 1680/120 =14.
The pattern is contained in the formula. When you carry out numerator you find that the next number in sequence is always an increase of 4. This seems to work for all polynomials.
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?
Using the formula given below, which I believe is designed to eliminate duplication, we would find that T(4) would still be 2, T(5) would still be 5, but the pattern would end there. T(6) = 9, T(7)= 14, and T(8) = 20.
There is a definite pattern to the numbers. Each additional vertex increases the number of triangle by the n-2. That is, when figuring a hexagon, where we had 5 sides with a pentagon, that number is increased by (6-2) for a total of 9 sides. 9 sides plus (7-2) totals 14, and this pattern seems to