The task of this assessment is to consider and investigate geometric shapes that lead to stellar numbers. One of the simplest forms of this would be of square numbers, terms 1, 2, 3, 4, and 5 in terms of geometric shapes lead to the special numbers of 1, 4, 9, 16, and 25. To find these outcomes, simply all that is needed to be done is to square the nth term of the sequence. For example, 12=1, 22=4, 32=9, 42=16, and 52=25, therefore the resulting formula for square numbers would be n2=Tn where T=total amount of n. For this investigation though, the geometric shapes of equilateral triangles are studied. Triangular numbers are the amount of dots that evenly fit inside the triangle. After solving the sequence of the triangular numbers and finding a formula to solve for all equilateral triangles, the knowledge attained from triangular numbers will then be used to find and solve the correlation between triangular numbers and stellar (star) numbers.
Here is a sample of the first 5 terms in the triangular numbers sequence:
The row of numbers above the triangles is the number of each term or n.
The rows of numbers below the triangles are triangular numbers of each term or n.
` n1 n2 n3 n4 n5
Here is a sample of the first 4 stages of p-stellar shapes that are being investigated. P indicates the amount of vertices in the stellar shape. In this case the stellar shapes being investigated are 6-stellar, however after deriving a formula, the stellar number of any p vertices may be found.
Sn is the different stellar shape terms, and the numbers aligned in a row below S1-S4 are the p-stellar numbers, in this case the 6-stellar numbers.
1 13 37 73
n1 n2 n3 n4 n5
The 2 triangles on the right are n2 and n3
From n1 to n2, 2 dots are added making a 2nd row of dots, and from n2 to n3, 3 dots are added making a 3rd row of dots.
This is repeated for every term in the triangular sequence; adding... [continues]
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