# Stellar numbers

IB 2 Mathematical Portfolio:

Stellar Numbers

Michael Diamond

IB Math SL 2

September 16, 2010

Mr. Hillman

Block G

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.”

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:

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

From the table it can be determined that there is no common difference () in the sequence of terms () meaning, that an arithmetic sequence formula cannot model the values produced by subsequent triangles, as if it could then the difference between each term would be constant e.g. (n+1,n+2,n+3...n etc.) and neither can it be geometric as there is no common ratio for the values. On the other hand the difference between the two terms  the value is constant as it is: (n+1),(n+2),(n+3)...n. The pattern that follows is represented in this chart:

This pattern is clearly different as each subsequent value added to the next term is +1 greater than the value of the term before.  Hence from this the next 3 triangle stages may be derived. In the 6th, 7th and 8th stages, the number of dots become respectively: 21, 28, 36. As demonstrated by this table:

Thus, from the above further data, a general statement may be formed from finding , which will now be F(n) to the nth term.

n will be defined as the stage number of the triangle

F(n) will ...