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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