This pattern is clearly different as each subsequent value added to the next term is +1 greater than the value of the term before. Hence from this the next 3 triangle stages may be derived. In the 6th, 7th and 8th stages, the number of dots become respectively: 21, 28, 36. As demonstrated by this table:
Thus, from the above further data, a general statement may be formed from finding , which will now be F(n) to the nth term.
n will be defined as the stage number of the triangle
F(n) will be defined as the function of the nth triangle
In the above the equation: appears twice once as normally viewed and the other time as the reason is that they are being added together and the reason for them being added together is that to get the equation there must be a visualization, that the triangle is half of another triangle thus forming a square.
hence the resulting number of dots would be half the number of the formed squares dots, which also means that the dots of two triangles is equal to one square.
Then,
as there are “n” quantities of “n+1” the equation becomes “n(n+1)”. Then the whole equation is divided by 2 to make the equation representative for the triangle rather than the square.
The final general statement for the triangle will be expressed in terms of n:
if the general statement for the nth triangle were multiplied all together then the equation would become the quadratic function: . In order to test for validity the graphing program “Graph 4.3” was used to demonstrate the validity of the general statement. As demonstrated by the graph below, both equations satisfy the general statement for the triangular numbers.
Figure 1: Graph of general statement of triangles with data of the 8 stages of the triangle
As shown in Figure 1, the R squared value is at 1 as shown on the graph which means that the formula created passes through all the given points perfectly. The R squared value was obtained through the “insert trend line function” of the “Graph 4.3” program. Thus, both the first and second function are valid and therefore both functions are verified. The second function could have been obtained through the use of technology, but instead it was obtained through another medium that of the multiplication of all terms of n in the obtained general statement, but both functions are essentially the same.
Stellar Numbers (Total number of dots in stellar number)
The second shapes that shall be considered are the stellar shapes with 6-vertices.
The first stellar shapes to be considered have 6-vertices. The counted values of the different stages of the stellar shapes are summed up in the table below:
The variables will be defined the same for tables to do with triangluar numbers:
-n will be defined as the stage number of the stellar shape
- as the nth stage of the stellar shape
From the table it can be seen that there is no common difference () in the sequence of terms (). On the other hand the difference between the two terms the value is constant (12n+12),(12n+24),(12n+36)+...+12n. The pattern that follows is represented in this chart:
Un-factorized form:
Factorized form:
This pattern is clearly different as each subsequent value added to the next term is +12
greater than the value of the term before. From this finding the other stages of the stellar shape can be found.
Thus, from the above further data, a general statement may be formed from finding , which will now be F(n) to the nth term.
In order to maintain consistency of the terms, and reduce complexity “1” will temporarily be removed from the equation, then it will be returned into the general statement at the end. This means that all stellar terms will lose “1” from their total value hence,
n will be defined as the stage number of the stellar shape
F(n) will be defined as the function of the nth stellar shape
The reason for this is that in the stellar shapes there is always one dot in the middle from to hence taking out the 1 from the equation does not alter the equation, as the “1” is objective due to its constant presence in all stellar shapes observed.
The same method utilized in the triangle general statement will be utilized here:
The reason for adding F(n)+F(n) is because of the geometric construction of the stellar shape as seen below there are 12 triangles which can be drawn in any stage of the stellar shape with 6 vertices. The 12 triangles all represent and since there are 12 of them this means that it is 0.5 multiplied by 12 which is 6. Hence, in the equation the 2F(n) is to account for this 0.5 as to isolate F(n) it will be divided out.
Then,
It is 12n(n+1) due to F(n) being n times of (12n+12). Then divided by 2 and one is added back in. The general statement here shall be tested by being graphed against the points of the known data as well as the terms generated by the general statement shown below.
Figure 2: Graph of general statement of stellar shapes (6-vertices) and generated data, with data of the 7 stages of the stellar shapes
As it it is seen here the data generated from the general statement does not match the data which is known. The reason for this lies in the creation of the stellar numbers. With the general statement created, true stellar shapes are generated that is instead of starting with a dot, it starts with the first stellar shape, with the dot in the middle. With this in mind, in order to account for this, what must happen is that the “n+1” must change into a “n-1” sign. This is necessary as by moving the graph to the left, the parabola will fit the data as demonstrated below:
Figure 3: Graph of general statement of stellar shapes (6-vertices) and generated data, with data of the 7 stages of the stellar shapes and general statement which fits known data
Based on the work above, the general statement for the 6-vertices star shapes are:
Other Stellar Numbers (5-vertices, 7-vertices)
In order to obtain a general statement other stellar shapes must considered with different vertices therefore: stellar shapes with 5-vertices, 7-vertices shall be considered in order to formulate a general statement. Therefore, the subsequent tables will depict the 3 stages of a 5-vertices and 7-vertices stellar shape. Therefore, a stellar shape generating program shall be used to depict these new shapes.
5-vertices stellar shape
7-vertices stellar shape
5-vertices stellar shape and 7-vertices stellar shape to the nth term:
is the function of the 5-vertices stellar shape to the nth term
is the function of the 7-vertices stellar shape to the nth term
When comparing 3 functions it is clear that they are quite similar:
The only difference seen is that the p- the number of vertices changes.
The graph below illustrates the functions given:
Title: Graph of general statements of 5,6,7 vertices and data on existing 5,6,7 vertices
As seen in the graph, the generated values match the general statement
Hence, a general statement can now be formed, from the patterns viewed above the general statement for all stars with p as vertices and n representing number of stages and representing any stellar number stage with p-vertices:
Conclusion:
Scope and Limitations:
One of the limitations for this equation is for those shapes which do not have the minimum number of vertices required to make a stellar shape. Therefore there is a limit on the number of vertices this limit being , although it is possible to argue that a shape with 2-vertices is a star, the resulting shape does not look like the classical star. However, since the main concern for this investigation is counting the number of dots inside a “stellar” geometric shape, it must be inclusive that they form the shape of a classical star, then the limitation would be as when it is substituted into the equation the result it yields is always 1. The limitations for n on the other hand is that all integers of n must be positive as although negative integers when substituted into the general equation does yield results. To have negative stages of star development would be to discuss the imaginary numbers. Therefore, in the context of having practical applications for the general formula created, ,
Therefore, the general statement:
is true when
