Natalie has taken a stunning image of a dahlia.  When we look at this image, our human brain responds with fondness and warmth.   What is it exactly that we are responding to?

In 1202 CE, Leonardo Bonacci from the Republic of Pisa, wrote an incredible mathematical text.  In Liber Abaci, he introduced arithmetic with Hindu-Arabic numerals to the masses in Europe.  This is the arithmetic that you learned in primary school and that he had learned throughout his travels in North Africa.  Like all good texts, it provided example problems and exercises for the reader.  One such problem asks the reader to model the growth of a rabbit population over time, and the solution to the problem is a sequence in which, after two initial terms, each term is the sum of the previous two terms.   If we start with the terms 1 and 1 then the sequence begins

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, …

Leonardo’s fame was such that, probably after his death, he earned “one-name nickname status.” Filius Bonacci (son of Bonacci) became Fibonacci, and the sequence we see above is known as the Fibonacci Sequence.  

The Fibonacci Sequence holds many delights, and one is a gateway to understanding beautiful things like Natalie’s dahlia.   Here is a way that we can sketch a particular type of spiral called a “Fibonacci Spiral.”  Starting with a grid, we can colour squares in a way that corresponds to the Fibonacci Sequence, and use our squares to draw a spiral, as follows.   Start with a square with side-length 1, then add another square of side length 1 that shares an edge with the first square.   Now we have a rectangle, and the long side has length two (1+1).  Now we add a square of side-length two (1+1) so that you make a rectangle with a long side of length 3 (1+2).  Now we add a square with side-length 3 so that you make a rectangle with a long side of length 5 (2+3).  Squares are added in a clockwise direction.  In each square, identify opposite corners in a way that spirals out from the original square.  Join up the corners you identified with a spiral curve, and you have a Fibonacci Spiral.

The petals of the dahlia are arranged like a Fibonacci spiral.  Dahlia’s have evolved to grow this way because it has a benefit—it provides the maximum amount of exposed petal area with the fewest resources spent growing petals.  Evolution strives for optimality, and so Nature is a wonderful mathematician.

Have humans evolved to appreciate the aesthetic qualities of the Fibonacci Spiral because we subconsciously recognise its efficiency? Or do we simply recognise it as being natural?  For the artist, Natalie, this flower evokes fond memories of her family as well as a human response to natural beauty.  I wonder if the image also makes her think of a well-travelled renaissance intellectual with a profound legacy.

Associate Professor Adam Piggott

Mathematical Sciences Institute