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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.”image00.png

image05.png

image62.png

image06.png

image63.png

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:

image01.png

When the values of the triangles (the number of dots) are input into a table:

Stage of triangle

Image of Triangular shape

image05.png

image04.png

Counted number of dots

image06.png

image07.png

image08.png

image09.png

1

1

1

2

1

image10.png

image50.png

2

3

3

3

1

image12.png

image58.png

3

6

6

4

1

image14.png

image59.png

4

10

10

5

image60.png

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 triangleimage04.png

The variables will be defined the same for all tables hereafter:

-as the difference between the two terms of the stages and image61.pngimage16.pngimage04.png

- as the difference between the two terms of image18.pngimage17.png

...read more.

Middle

image67.png

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.image68.pngimage69.png

image70.png

 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,

image71.png

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:

image72.png

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.image02.png

Figure 1: Graph of general statement of triangles with data of the 8 stages of the triangle

image03.png

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.

...read more.

Conclusion

image48.png

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)

image44.png

image49.png

image45.png

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 statementimage51.png

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:image04.png

image52.png

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, , image54.pngimage55.pngimage53.png

Therefore, the general statement:

image56.png

is true when

image55.png

image57.png

...read more.

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