Fibonacci Sequence

A sequence of numbers in which each number is the sum of the preceding two.

A Fibonacci sequence is a sequence of numbers in which each number is the sum of the two preceding numbers (e.g., 1, 1, 2, 3, 5, 8, 13). Patterns exhibiting the sequence are commonly found in natural forms, such as the petals of flowers, spirals of galaxies, and bones in the human hand. The ubiquity of the sequence in nature has led many to conclude that patterns based on the Fibonacci sequence are intrinsically aesthetic and, therefore, worthy of consideration in design. id="footnote84a"> class="nounder totri-footnote" href="https://www.safaribooksonline.com/library/view/universal-principles-of/9781592535873/xhtml/ch41_fn.html#footnote84">1

Fibonacci patterns are found in many classic works, including classic poetry, art, music, and architecture. For example, it has been argued that Virgil used Fibonacci sequences to structure the poetry in the Aeneid. Fibonacci sequences are found in the musical compositions of Mozart’s sonatas and Beethoven’s Fifth Symphony. Le Corbusier meshed key measures of the human body and Fibonacci sequences to develop the Modulor, a classic system of architectural propotions and measurements to aid designers in achieving practical and harmonious designs. id="footnote85a"> class="nounder totri-footnote" href="https://www.safaribooksonline.com/library/view/universal-principles-of/9781592535873/xhtml/ch41_fn.html#footnote85">2

Fibonacci sequences are generally used in concert with the golden ratio, a principle to which it is closely related. For example, the division of any two adjacent numbers in a Fibonacci sequence yields an approximation of the golden ratio. Approximations are rough for early numbers in the sequence but increasingly accurate as the sequence progresses. As with the golden ratio, debate continues as to the aesthetic value of Fibonacci patterns. Are such patterns considered aesthetic because people find them to be more aesthetic or because people have been taught to believe they are aesthetic? Research on the aesthetics of the golden ratio tends to favor the former, but little empirical research exists on the aesthetics of non-golden Fibonacci patterns. id="footnote86a"> class="nounder totri-footnote" href="https://www.safaribooksonline.com/library/view/universal-principles-of/9781592535873/xhtml/ch41_fn.html#footnote86">3

The Fibonacci sequence continues to be one of the most influential patterns in mathematics and design. Consider Fibonacci sequences when developing interesting compositions, geometric patterns, and organic motifs and contexts, especially when they involve rhythms and harmonies among multiple elements. Do not contrive designs to incorporate Fibonacci sequences, but also do not forego opportunities to explore Fibonacci relationships when other aspects of the design are not compromised.

src="https://www.safaribooksonline.com/library/view/universal-principles-of/9781592535873/images/f0095-01.jpg" alt="image" width="550" height="458" data-mfp-src="/library/view/universal-principles-of/9781592535873/images/f0095-01.jpg" />Le Corbusier derived two Fibonacci sequences based on key features of the human form to create the Modulor. The sequences purportedly represent a set of ideal measurements to aid designers in achieving practical and harmonious proportions in design. Golden ratios were calculated by dividing each number in the sequence by its preceding number (indicated by horizontal lines).