Figure 2.1
The relationship between Tn and n is demonstrated in the picture above and the table below.
The relationship between the number of vertical transversals and the number of parallelograms formed is shown below.
The similarity of these sequences is demonstrated in graph 1.1.
Graph 1.1
The relationship between triangular numbers and the number of parallelograms formed when the number of vertical transversals is increased is clearly shown by the similarity of both the lines in the graph above. This similarity must indicate that the general formula of Pascal’s theory of triangular numbers is similar to the general formula of the relationship between the number of vertical transversals and the number of parallelograms formed.
To discover the general formula of the relationship of the number of vertical transversals and the number of parallelograms, it is necessary to test the general formula of triangular numbers (shown in figure 2.1) on the relationship between the number of vertical transversal and the number of parallelograms.
∴
∴
This proves that the general formula for triangular numbers does not directly apply to the general formula for the relationship between the number of vertical transversal and the number of parallelograms. This is because the general state for triangular numbers start at Tn = 1 whereas the general formula for the relationship between the number of vertical transversal and the number of parallelograms needs to start at 2, this is because a parallelogram is not formed when there is only one vertical transversal, shown below.
To combat this effect the equation needs to be changed to, this is because a parallelogram is only created when there is two or more vertical transversal.
∴ The general formula is:.
Where v is the number of vertical transversal and P is the number of parallelograms formed.
However, this equation assumes that the number of horizontal lines stays constant at 1.
This formula is proved below.
∴
3 = 3
Furthermore,
The above examples justify that is the general formula for the relationship between the number of vertical transversals and the number of parallelograms. However, to develop a general formula that encompasses the variables of the number of vertical transversals and horizontal lines, an equation needs to be developed that deduces the number of parallelograms that is formed when horizontal lines vary. This equation needs to assume that the number of vertical transversals remain constant. The following figures demonstrate how additional parallelograms are formed when an increasing number of horizontal lines are added.
Figure 3.1 Figure 3.2
Figures 3.1 and 3.2 demonstrate how horizontal lines (black) are added to a constant number of vertical transversals to form an increasing number of parallelograms. This relationship is extrapolated in the table below.
The general formula for this relationship would be identical to the general formula for the relationship between the number of vertical transversals and the number of parallelograms. This is due to the relationship of the number of horizontal lines and the number of parallelograms is the same as the relationship between the number of vertical transversals and the number of parallelograms. This is demonstrated in figure 4.1.
Figure 4.1
Rotated 90 degrees:
Figure 4.1 clearly illustrates how horizontal lines are identical to vertical transversals. Furthermore, this relationship is further clarified in graphs 2.1 and 2.2 by the identical equations on both graphs. Therefore, from these two observations, the same general formula that was used in the relationship between the number of vertical transversals and the number of parallelograms can be used in the relationship between horizontal lines and the number of parallelograms.
Graph 2.1
Graph 2.2
∴ The general formula for the relationship between the number of horizontal lines and the number of parallelograms is .
Where P is the number of parallelograms formed and h is the number of horizontal lines. This general formula does not encompass the variable of vertical transversals. The equation assumes that the number of vertical transversals stays constant at 2. To incorporate the variable of the number of vertical transversals it is imperative to merge both general statements. To do so, we need to multiply both equations together. This is because as a new vertical or horizontal transversal is added, another group of parallelograms is added. Therefore, adding another horizontal or vertical transversal effectively multiplies the number of parallelograms.
∴ The General formula that encompasses both variables of the number horizontal and vertical transversals is
Where P is the number of parallelograms produced, v is the number of vertical transversals and h is the number of horizontal lines.
The formula encompasses both variables of the number horizontal and vertical transversals was incorporated into an excel spread sheet. The results are shown in figure 5.1.
Figure 5.1
To further validate this formula the numbers that were produced from figure 4.1 need to be tested. This is shown in figure 6.1 and 6.2.
Figure 6.1
The above figure has parallelograms A1, A2, A3, A4, A1 U A2, A3 U A4, A1 U A3, A2 U A4 and A1 U A2 U A3 U A4. In total, figure 6.1 has 9 parallelograms; this coincides with the general formula’s answer.
Figure 6.2
Figure 6.2 has parallelograms A1, A2, A3, A1 U A2, A2 U A3, and A1 U A2 U A3. In total, figure 6.2 has 6 parallelograms. This coincides with the general formula’s answer.
∴ Therefore the general statement for the relationship between vertical transversals, horizontal lines and the number of parallelograms is
Where P is the number of parallelograms produced, v is the number of vertical transversals and h is the number of horizontal lines.
However, this equation can only be used when ≥ 2 and ≥ 2. This is because a parallelogram is only created when 2 or more of each transversal are present. This is further demonstrated in figures 7.1 and 7.2.
Figure 7.1 Figure 7.2
Furthermore, the variables, and have to be natural numbers. The equation does not work when the variables are fractions, negative numbers or imaginary numbers.
The generalisation of the formula that was found was arrived at because a varying number of vertical transversals, intersect with a constant number of horizontal lines to produce Av parallelograms. Furthermore, a varying number of horizontal lines, , intersect with a constant number of vertical transversals to produce Ah parallelograms. Therefore a varying number of vertical transversals, , intersect with a varying number of horizontal lines, , to produce Av × Ah parallelograms.
Bibliography
Date Accessed: 22/05/09
