- Level: International Baccalaureate
- Subject: Maths
- Word count: 1130
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.
1 | 2 | 3 | 4 | 5 | |
Shape | |||||
Number of Dots | 1 | 3 | 6 | 10 | 15 |
We can calculate the term by adding to the previous number of dots. For example, the value for equals to 15 +6 which is 21. The same applies for the other values of .
The next three terms are represented by this diagram:
6 | 7 | 8 | |
Shape | |||
Number of dots | 21 | 28 | 36 |
Middle
2nd diff.: 0 0 0 0 0
If the general formula of linear sequences is , then will be 0.5 and will be 0.
The expression of this sequence will be
Therefore the expression of triangular numbers will be:
We can check this by trying the expression with the 9th term..The difference between 45 and 36 is 9, therefore the expression is correct because this difference fits with the previous ones.
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 value of is the number of vertices the star has and this is also called -stellar number.
6-stellar numbers
These are stellar numbers of stars with 6 vertices.
1 | 2 | 3 | 4 | |
Shape | ||||
Number of dots | 1 | 13 | 37 | 73 |
5 | 6 | |
Shape | ||
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 () because the 2nd
Conclusion
1
2
3
4
5
6
Seq.
1
11
31
61
101
151
5
20
45
80
125
180
Seq.-
-4
-9
-14
-19
-24
-29
We now have a linear sequence ( ). 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 now, it will not work. We have to add +1 to it.
The expression for S7 is
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.
General Statement:
The functions of these two stellar numbers are very similar as in this graph.
After knowing the expressions for stellar numbers of 2 different vertices, we found that the common units in the expression are
And we found that the coefficient of and is the number of vertices that the star has.
The general statement for every star with p vertex is:
Where is greater than 2 and 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.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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