- Level: International Baccalaureate
- Subject: Maths
- Word count: 2259
Stellar numbers
Extracts from this document...
Introduction
IB 2 Mathematical Portfolio:
Stellar Numbers
Michael Diamond
IB Math SL 2
September 16, 2010
Mr. Hillman
Block G
Table of Contents:
Page 3
-Introduction
-Triangular Numbers
Page 7
-Stellar Numbers
Page 12
-Other stellar numbers
Page 15
-Conclusion
Introduction:
Certain geometric shapes can yield special types of numbers. The simplest examples of these numbers would square numbers 1, 4, 9, 16, which are the squares of the values 1, 2, 3 and 4 as demonstrated by this table and by the formula . Henceforth, all tables generated are from the Numbers program by the macintosh company and all mathematical notation is from the program Mathtype 6.0. Any graphs displayed will be generated from the graph program “Graph 4.3.”
1 | 1 | 3 | 2 |
2 | 4 | 5 | 2 |
3 | 9 | 7 | |
4 | 16 |
In this investigation the following geometric shapes shall be considered for investigation in order to determine how many dots are in each type of shape from triangular figures to stellar (star) figures. With the ultimate goal being a general encompassing statement in order to determine the number of dots in any star with p-vertices in any n-stage.
Triangular Numbers (Total number of dots in a triangle)
The first shapes that shall be considered are the triangular figures:
When the values of the triangles (the number of dots) are input into a table:
Stage of triangle | Image of Triangular shape | Counted number of dots | ||||
1 | 1 | 1 | 2 | 1 | ||
2 | 3 | 3 | 3 | 1 | ||
3 | 6 | 6 | 4 | 1 | ||
4 | 10 | 10 | 5 | |||
5 | 15 | 15 |
The variables will be defined the same for tables to do with triangluar numbers:
-n will be defined as the stage number of the triangle
- as the nth stage of the triangle
The variables will be defined the same for all tables hereafter:
-as the difference between the two terms of the stages and
- as the difference between the two terms of
Middle
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.
Conclusion
When comparing 3 functions it is clear that they are quite similar:
5-vertices function (p=5) | 6-vertices function (p=6) | 7-vertices function (p=7) |
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
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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