• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Stellar shapes

Extracts from this document...

Introduction

International Baccalaureate Diploma

The Sultan's School

"STELLAR NUMBERS" Portfolio

Candidate Name: Tamim Al-Tamimi

Contents

Introduction                                                                3

Triangular numbers                                                4

Stellar Numbers                                                        7

6 stellar numbers                                                7

5 stellar numbers                                                10

General Statement                                                12

Introduction

In this portfolio, I am going to show the expressions of general patterns and numbers for stellar and triangular numbers.

The aim of this portfolio is to “consider geometric shapes which lead to special numbers”.

Triangular shapes were imported from the internet and edited by paint program. 6-stellar diagrams were imported from Google images and edited by paint.      5-stellar shapes were created by paint program.

All the graphs are created by the online program http://functiongrapher.com/.

Triangular numbers

The first type of numbers we are considering are triangular numbers. This table shows the first 5 terms of triangular numbers.

image00.png

1

2

3

4

5

Shape

image01.png

image28.png

image32.png

image35.png

image40.png

Number of Dots

1

3

6

10

15

We can calculate the image47.pngimage47.png term by adding image00.pngimage00.png to the previous number of dots. For example, the value for image02.pngimage02.png equals to 15 +6 which is 21. The same applies for the other values of image00.pngimage00.png.

The next three terms are represented by this diagram:

image00.png

6

7

8

Shape

image24.png

image25.png

image26.png

Number of dots

21

28

36

...read more.

Middle

2nd diff.:        0        0        0        0        0

If the general formula of linear sequences is image09.pngimage10.pngimage10.png, then image36.pngimage37.pngimage37.png will be 0.5 and image38.pngimage39.pngimage39.png will be 0.

The expression of this sequence will be image41.pngimage41.png

Therefore the expression of triangular numbers will be: image42.pngimage42.png

We can check this by trying the expression with the 9th term.image44.pngimage44.pngimage45.png.The difference between 45 and 36 is 9, therefore the expression is correct because this difference fits with the previous ones.image43.png

The graph below shows that the expression is correct.

Stellar numbers

The next type of numbers we are considering is stellar numbers. Stellar numbers are sequences of dots. The 1st term is 1 dot. After that, this dot is surrounded by a star in the 2nd term. The previous star is surrounded by a larger star in the following terms. The image46.pngvalue of image23.pngimage23.pngis the number of vertices the star has and this is also called image23.pngimage23.pngimage48.png-stellar number.

6-stellar numbers

These are stellar numbers of stars with 6 vertices.

image00.png

1

2

3

4

Shape

image49.png

image50.png

image51.png

image52.png

Number of dots

1

13

37

73

image00.png

5

6

Shape

image53.png

image54.png

Number of dots

121

181

To know the expression of this sequence, we must know which difference is constant.

Sequence: 1,        13,        37,        73,        121,        181

1st diff.:        12        24        36        48        60

2nd diff.:        12        12        12        12

This is a quadratic sequence (image55.pngimage55.png) because the 2nd

...read more.

Conclusion

1

2

3

4

5

6

Seq.

1

11

31

61

101

151

image06.png

5

20

45

80

125

180

Seq.- image07.pngimage08.pngimage08.png

-4

-9

-14

-19

-24

-29

We now have a linear sequence (image09.pngimage10.pngimage10.png ). This expression will be added to the first part of the stellar number sequence.

Seq.:        -4,        -9,        -14,        -19,        -24,        -29…

1st Diff.:        -5        -5        -5        -5        -5

If we try the expression image11.pngimage12.pngimage12.png now, it will not work. We have to add +1 to it.

The expression for S7 is image13.pngimage11.pngimage14.pngimage14.png

We can check that this is true by subtracting 151 from 211 and the result should fit in the 1st differences row. The difference is 60 and fits in the 1st differences row. We can also count the stars in the diagram.image15.png

image11.pngimage17.pngimage17.pngimage16.png

General Statement:

The functions of these two stellar numbers are very similar as in this graph.image18.png

After knowing the expressions for stellar numbers of 2 different vertices, we found that the common units in the expression are image19.pngimage11.pngimage20.pngimage20.png

 And we found that the coefficient of image21.pngimage21.png and image00.pngimage00.png is the number of vertices that the star has.

The general statement for every star with p vertex is:

image22.png

Where image23.pngimage23.png is greater than 2 and image00.pngimage00.png is greater than or equal to 1.

This statement works only for real numbers and positive numbers because you can’t have a negative star or half a star.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Maths Internal Assessment -triangular and stellar numbers

    However, as stellar numbers must take the form of a star; p = (infinity) cannot be applied to the general statement as the shape would be a circle. As stated in the previous limitation, that p= (infinity) will be a circle; this indicates that the higher the 'p' value is;

  2. Stellar Numbers. In this study, we analyze geometrical shapes, which lead to special numbers. ...

    3.1, we can calculate the total stellar numbers for p=3 and p=4. n - the number of each consecutive stage The total number of dots equals to , p=3 The total number of dots equals to , p=4 1 0+1=1 0+1=1 2 0+6+1=7 0+8+1=9 3 0+6+12+1=19 0+8+16+1=25 4 0+6+12+18+1=37 0+8+16+24+1=49

  1. Stellar numbers

    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

  2. Arithmetic Sequence Techniques

    Using the sequence 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 find the relationship between each pair of results. I. (3 + 11) and (5 + 9) II. (7 + 17) and (11 + 13) III. (5 + 19)

  1. Stellar Numbers Portfolio. The simplest example of these is square numbers, but over the ...

    S1=1 S2=3 S3=6 S4=10 S5=15 S6=21 S7=28 S8=36 S9=45 S10=55 The first pattern that I noticed when looking at this information was that the order looks like this when broken down: 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5, ... , 1+2+3+4+5+6+7+8+9+10 The first attempt I made was saying the general term was simply (n+1).

  2. Stellar numbers. This internal assessment has been written to embrace one of the ...

    The pattern of the sum of y shown in Table 2 can be represented by the following summation formula: y=i=1ni=1+2+3+?+n-1+n or y=i=1ni=n+n-1+n-2+?+2+1 1 2 3 4 ? n n n-1 n-2 n-3 ? 1 n+1 n+1 n+1 n+1 ? n+1 Chart 1 The two sigma notations can be combined and added together.

  1. Stellar Numbers. In this folio task, we are going to determine difference geometric shapes, ...

    Thus, we are now able to create a table of demonstrating the side numbers and Square numbers as the following. Side no. (n*) 1 2 3 4 5 6 7 8 Square no. (y) 1 3 6 10 15 21 28 36 Consider about the triangular number, we can work

  2. Math Portfolio Stellar Numbers. This task is an investigation of geometric numbers, the ...

    Where n Validation The general term can be validated by substituting the values of 'n' by various natural numbers as follows: General term = Substitute n with 2 = = =3 The number of dots for 'n=2' is 3 which is identical to the value projected in the table of values.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work