![]() Notice that the errors trend towards zero, indicating that our approximations are getting closer and closer to the true solution. ![]() The absolute errors between our Fibonacci ratio approximations and the golden ratio itself. Upon running the algorithm, we find that it takes 39 iterations for our approximation errors to converge to $$\varepsilon$$.įigure 1. We want to approximate $$\varphi$$ to a high degree of accuracy thus, we will design the algorithm to run until we converge on the computer’s epsilon error ($$\varepsilon$$), i.e., the last digit recorded on my 64-bit computer. Lets cook up a simple sequence of operations exploiting the last feature of the Fibonacci numbers discussed above. Hmm, this seems like an important property if we, say, wanted to approximate the golden ratio!Īpproximating the golden ratio with Fibonacci numbers The ratio of consecutive terms closely mirrors $$\varphi$$, and the approximations get increasingly more accurate. Each consecutive term is the sum of the previous two, i.e., the third term is also one, because 0 1=1. The first two Fibonacci numbers are zero and one. “ was an Italian mathematician from the Republic of Pisa, considered to be ’the most talented Western mathematician of the Middle Ages’”. Greek mathematicians determined $$\varphi$$ was an irrational number all the way back in the fifth century BC (woa).įibonacci numbers are closely related to the golden ratio. ![]() The golden ratio was (probably) first discovered in Ancient Greece through applications in geometry, which the Greeks emphasized. The golden ratio is a mathematical phenomenon between two numbers, say, a and b. This algorithm lets us peer into the underlying relationship between the Fibonacci sequence and the golden ratio, illuminating a number of interesting patterns. I will approximate the golden ratio by iteratively taking ratios of consecutive terms in the Fibonacci sequence. Using Numerical Mathematics to Approximate the Golden Ratio
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