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Stellar Numbers. In this study, we analyze geometrical shapes, which lead to special numbers. We consider various geometrical patterns of evenly spaced dots and derive general statements of the nth special numbers in terms of n.

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Introduction

image30.jpg

Timur Karimov

Anglo – American School of Moscow

IB Mathematics

Internal Assessment Portfolio Practice 1


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Aim: The aim in this task is to consider geometric shapes, which lead to special numbers. The simplest examples of these are square numbers, 1, 4, 9, and 16, which can be represented by squares of side 1, 2, 3 and 4.

Introduction: In this study, we analyze geometrical shapes, which lead to special numbers. We consider various geometrical patterns of evenly spaced dots and derive general statements of the nth special numbers in terms of n. Henceforth, all tables and mathematical formulas are generated by the Microsoft Word. All the stellar and triangular number visual representations are modulated using GeoGebra 4. The rest of the shapes are generated by “insert shape” function of Microsoft Word.

Task 1

The first task is to complete the triangular numbers sequence with three more terms.

image55.png

As we can see, each successive triangle pattern numbered image42.pngimage42.png (n is the stage number) can be derived from the previous one number (image82.pngimage82.png) by simply addingimage42.pngimage42.pngevenly spaced dots on either of the sides of the triangle.image60.pngimage00.png

image01.png

In other words, the number of dots in the triangular number n equals to the number of dots in the triangular number image31.pngimage31.png) plus image42.pngimage42.png. Lets denote this number by Gn

image51.png

Using this formula, we can calculate the number of dots in the triangular number 6:

image52.pngimage52.png

In the triangular number 7:

image53.png

And in the triangular number 8:

image54.png
and etc.

The following table can demonstrate this sequence:

image07.png

image42.pngimage42.png - Number of the Triangular Pattern

image56.pngimage56.png – nth Triangular Number

image57.png

1

1

2

3

2

3

6

3

4

10

4

5

15

5

6

21

6

7

28

7

8

36

8

The last 3 triangular numbers can be graphically demonstrated as following.image16.png

image58.png

image23.png

Using the expression image51.pngimage51.png one can continue the triangular number sequence indefinitely.

Task 2

...read more.

Middle

image81.png

image06.pngimage06.png

To find the number of dots in each stage, one can simplify each shape as a combination of a number of simple stars with 6 vertices and the dot in the middle. For example: the shape S3 consists of two simple stars and one dot in the middle; S4 consists of three simple stars and the dot in the middle, etc. The total number of dots in any particular shape is the sum of the dots of comprising it simple stars, plus one dot.

Lets denote the number of dots of the outer simple star of the shape Sn as Fn (F1 = 0).

For example: F2is to the number of dots in the outer star of S2 and equals 12. The total number of dots is

F2+ 1 = 12

We can also observe a simple sequence of the Fn. From the “Diagram for Increased Number of Dots” above, one can derive that:

Fn= Fn-1+2*6

Thus, each consecutive outer simple star has 12 more dots than the previous one (two per each of the six vertices). We can summarize all these findings in the following table:

image07.png

n – the number of each consecutive stage

Fn – the number of dots in each consecutive outer simple star

The total number of dots equals to image32.pngimage32.png

1

0

0+1=1

2

12

0+12+1=13

3

12+12=24

0+24+12+1=37

4

24+12=36

0+36+24+12+1=73

5

36+12=48

0+48+36+24+12+1=121

6

48+12=60

0+60+48+36+24+12+1=181

Lets denote by Tn the stellar number for the stage Sn.Judging from the table 3.1, we have the following result of the calculations:

image07.png

n – stage number

image83.pngimage83.png– Stellar number (number of dots)

1

1

2

13

3

37

4

73

5

121

6

181

Task 4

Find an expression for the 6-stellar number at stage S7.

Using the table 3.

...read more.

Conclusion

Another example to prove the general statement as a stellar with 7 vertices (p=7) is going to be attempted below:

image45.jpg

image26.png

image27.pngimage29.pngimage27.pngimage27.png

image07.png

n – Stage Number

Tn as calculated manually

image39.pngimage39.png, p=5

1

1

image46.png

2

15

image47.png

3

43

image48.png

4

85

image49.png

Again, as one can notice, the numbers are identical, proving the general statement’s authenticity.

Task 8

Discuss the scope and limitations of the general statement.

We have tested the general statement Tn = image50.pngimage50.png for a sufficient range of stellar structure with number of vertices from 3 to 7. Building the stellar structures with larger number of vertices is quite time consuming and thus further checks have to be done by a sophisticated computer program. However, the logic behind this general statement remains the same for any number of vertices, which leads us to the conclusion that the formula is correct. One has to use it properly, only for the values of n higher than 0 and for the values of p higher than 2, as one cannot imagine stellar shapes with one or two vertices. Technically it is possible to use any, even negative numbers, but we find no logical context for such calculations.

Task 9

Explain how you arrived at the general statement.

There were three key findings that helped me to arrive at the general statement. The first realization was that each new outer star has two more additional dots per each vertex. This allowed me to calculate the number of dots in any outer star using the recurrent formula.

Secondly, I noticed that each formula of the stellar number consists of one inalienable part: the sum of (0+1+2+3…+(n-1)).

Finally I noticed that this part above is in fact the triangular number from the Task 2.

Putting these three observations together, I found the formula, which was later confirmed by a decent number of verifications.

...read more.

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