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mgf1107lecture25 - This characteristic growth pattern is...

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MGF 1107 – EXPLORATIONS IN MATHEMATICS LECTURE 25 Spiral Growth in Nature When discussing the golden ratio in Lecture 23, it was claimed that it often occurs in nature. While this fact may at first seem coincidental, it can be explained using the concept of a gnomon from Lecture 24, in particular Fibonacci rectangles. Fibonacci Rectangles If we start with a 1 x 1 square, and successively add a square equal in length to the longest side of the previous shape (rectangle in all but the first case), we build a series of rectangles whose sides have the length of two consecutive Fibonacci numbers. Since we know that the ratio of two Fibonacci numbers quickly converges to the Golden Ratio, the difference between the Fibonacci rectangles above and the Golden rectangles discussed in Lecture 24 becomes increasingly small. Now consider a living organism continually growing in a spiral shape inside each successive Fibonacci rectangle.
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