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David's Pyramid
How it works There are three main aspects to my pyramid. The first aspect is all the numbers that make up all the rows, diagonals, and platforms of my pyramid. The second part would be having all the a’s b’s and c’s in the right spots on the inside and outside of the pyramid. The very last but one of the most important parts of this pyramid is exponents to all the letters and numbers.
1 Aspect
Row 0 1 Row 1 1 1 Row 2 1 2 1 Row 3 1 3 3 1 Row 4 1 4 6 4 1 Pascal’s Triangle How to get the numbers on the outside of each row and platform? The outside numbers are the ones that are highlighted in the color yellow. The outside numbers to the pyramid co inside with the row numbers of Pascal’s triangle. Where row three on Pascal’s triangle is the same as the outside numbers on Platform three of my Pyramid. The diagram on the left shows Pascal’s triangle up to
1 3 3 3 3 1 3 3 1 Row four. And the diagram on the right shows how Platform 3 of my pyramid co-insides with row three of Pascal’s Triangle. This works for all the outside numbers of any Platform on my Pyramid. To get the inside numbers of any Platform is a little bit more tricky. There is a complicated relation ship between ehach number in the pyramid, wich is very close to the samae relation ship as pascal’s triangle. This relation ship can easily be expressed as a formula. The formula is This formula is very easy to use, all you do is plug in all the information into the calculator and you get the answer. Example 1 5 5 10 20 10 10 30 30 10 5 20 x 20 5 1 5 10 10 5 1 X = row- diagonal + 1/ Diagonal *20 X = 4 – 2 + 1/ 3 * 20 X=30 This formula works throughout all of the pyramid.
Second Aspect
The second part is really simple. It’s on how you get all the letters in the right spots on the inside of the pyramid. It goes like this, all the corners of the pyramid are there single letter. All the very outside numbers are a mixture as in the out side numbers between a and b are ab, all the numbers between a and c are ac, all the numbers between b and c are bc. All of the inside are abc’s (theses are the only parts of the pyramid left to fill out.)
Third Aspect
The last part in filling out the pyramid is putting in all of the exponents. This can be easily done with the formula a(-r+p)b(r-d)c(d) This formula is a great way to get all the exponents. After you do this you may observe that all of the exponents of a get less and less as you get farther from it and all the exponents of b get lass and less as you get farther from it and all the exponents of c get less and less as you get farther from it.
The Final Formula
This is the final Formula to the Pyramid It works in the same way as all the other parts of the pyramid that I just explained. Except for the very first formula instead of using r – d +1 /d , we use the formula that is like a modified version of Pascal’s formula for Pascal’s triangle. I then adapted the trinomial formula to my pyramid, so that instead of just being able to find out the coefficient of one term you can find out every coefficient to the trinomial plus the letters and the exponents.