assignment 12

We wish to create a spreadsheet composed of the fibonacci numbers and then investigate various ratios of these numbers. In the first column of our spreadsheet we create the fibonacci numbers,that is those numbers beginning with 1 and 1 where each subsequent number is the sum of the previous two numbers. This is easily done through the use of formulas in the spreadsheet to create as many of these numbers as we desire .

The second column of numbers is generated by the ratio of adjacent numbers in the first column. This produces an interesting result in that these ratios approximate a well known number called the golden ratio. Greeks were aware of this ratio of numbers as being particularly architecturally pleasing to the eye. We see this ratio exemplified in one of the most well known Greek antiquities, the Parthenon.

The third column is generated by the ration of every other entry in the first column. This produces a constant number which is the Golden ratio plus one.

An interesting question to ask is whether these ratios are particular to the fibonacci sequence or might other numbers work just as well in the first column. Lets try an example:

This is interesting. The same ratios are generated. This result begs the question as to whether the same ratio is generated by any two beginning numbers. Lets generalize slowly and start by investigating any two numbers such that the second number is 4 more than the first number. This would produce a spreadsheet which looks like this:

Careful inspection of these ratios in column b shows that the numerical part of the expression is exactly the fibonacci sequence and the fractional algebraic part of the expression is approaching 0 for all x. (The denominator is increasing in size quickly while the numerator is decreasing.) So it appears that for any two numbers one of which is 4 greater than the other, which generate a sequence formed by the addition of the previous two numbers, the golden ratio is approximated. Let try a positive and negative number just to further convince ourselves:

Or we will also try two negative numbers. The results are the same. We may generate a further generalization spreadsheet by choosing the first two numbers x and x+n. The result is closely akin to the generalized result above where the second number is 4 greater than any number x. I leave the production of that spreadsheet to your enjoyment.

Return