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The golden ratio can be most simply defined as the solution to x-1=1/x. It is often represented in mathematics by the lowercase Greek letter phi (φ). φ is an irrational number approximately equal to 1.618. It turns out that φ has a tremendous number of interesting mathematical properties and can be expressed in a variety of different mathematical expressions that, at first glance, are seemingly unrelated. The mathematical applications are far reaching, especially in geometry. The relationship of φ to Fibonacci sequence, as we have seen above, is approximate. It turns out that dividing a number in the Fibonacci sequence by its immediate predecessor will give the approximate value of φ. As we divide each number in the sequence by the preceding number, these approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase. Dividing the 25,001 number in the Fibonacci sequence by the 25,000 number yields a result that is accurate for φ out to at least 10,000 significant digits!

The golden ratio can be most simply defined as the solution to x-1=1/x. It is often represented in mathematics by the lowercase Greek letter phi (φ). φ is an irrational number approximately equal to 1.618. It turns out that φ has a tremendous number of interesting mathematical properties and can be expressed in a variety of different mathematical expressions that, at first glance, are seemingly unrelated. The mathematical applications are far reaching, especially in geometry where figures with 5 sides are involved. Another of the ways φ can be expressed is (1 + √5)/2.

The Fibonacci sequence is a simple mathematical sequence that was described by Leonardo Fibonacci (c. 1170– c. 1250). The sequence begins with 0, 1. Each Fibonacci number thereafter is the sum of its two immediate predecessors (0+1 = 1, 1+1 = 2, 1+2 = 3, 2+3 = 5, etc., ad infinitum). The first 21 numbers in the sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, and 6765.

Since the numbers 2,3, and 5 are part of the Fibonacci sequence, and since limericks are poetic verse based on the numbers 2,3, and 5 (five lines with an AABBA rhyming structure and a 33223 beats per line structure), then the following is a Fibonacci poem about Fibonacci sequences:

Zero, one! One, two, three! Five and eight!
Then thirteen, twenty-one! At this rate
Fibonacci appears;
The man's sequence for years
Has kept math students studying late.
From "The Omnificent English Dictionary In Limerick Form"

The relationship of φ to Fibonacci sequence, as we have seen above, is approximate. It turns out that dividing a number in the Fibonacci sequence by its immediate predecessor will give the approximate value of φ. As we divide each number in the sequence by the preceding number, these approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase. Dividing the 25,001 number in the Fibonacci sequence by the 25,000 number yields a result that is accurate for φ out to at least 10,000 significant digits!

And honestly, even though this image has elements that match up to lines from five golden rectangles, I think the strength of the composition is probably more due to the two diagonal lines and curves that intersect at the face of the locomotive.

Notice that we were able to place elements of the scene along each of these five successive compositional lines. Sometimes the element is shorter than the compositional line, sometimes vice-versa. But each line has a corresponding element in the scene approximately along at least part of its length. We also have a very strong diagonal and a strong curve traversing the largest square that also lead the viewer's eye to the locomotive that occupies the fifth redactive square.

Notice that we were able to place elements of the scene along each of these five successive compositional lines. Sometimes the element is shorter than the compositional line, sometimes vice-versa. But each line has a corresponding element in the scene approximately along at least part of its length. We also have a very strong diagonal and a strong curve traversing the largest square that also lead the viewer's eye to the locomotive that occupies the fifth redactive square. If one were to draw the tangential arcs in each square to create a near-Fibonacci spiral, the fifth arc would cross the nose of the locomotive from lower right to upper left, the sixth would arc above the train and then the seventh and all successive ones would fall in the space occupied by the freight cars being pulled by the locomotive.