# The Mathematical Beauty in Flowers

Art by Jake Upton

By Alice Blake

Most people believe nature to be beautiful. The beauty that we see in nature is what drives us to buy flowers, travel to national parks, take photographs and create paintings of natural landscapes, and maintain gardens. There is beauty in both the symmetry and the asymmetry of nature. There is beauty in the placement of flower petals and leaves. Proceeding with the assumption that nature is beautiful, we can then explore why we find it beautiful. Mostly, this discussion will center around the beauty of rose-like flowers.

Flowers evolved significantly before humans, so their evolution into something we find beautiful has nothing to do with our existence. Flowers evolved approximately 140-130 million years ago, while homo sapiens began their evolution 0.2 million years ago (Ulrich et al). Thus, flowers and nature evolved to be beautiful because their structure is useful to their existence. Flowers have evolved such that they capture the maximum amount of sunlight and can hold the maximum number of seeds. We happen to find that beautiful and are able to describe those systems with mathematical angles and series of numbers. The flower petal placement, the leaf placement, and the location of seeds can all be explained with the golden ratio.

We find flowers beautiful because of the golden number/golden section/golden ratio. The golden number is (1+√5)/2, which is approximately equal to 1.618. The number is irrational, which literally means it is not a ratio. While there can be close decimal or fractional approximation so the golden number, the most exact way to express it is as (1+√5)/2. The number √5 is the most irrational number, according to number theory.

Related to the golden number is the golden angle. The golden angle is approximately 137°30’. This angle is found when the circumference of a circle is divided into two sections based upon the golden ratio. The golden angle determines the leaf and petal placement on flowers because it allows for the optimal light capture efficiency. Imagine you are holding a clipped flower and place your finger at a leaf growing off the stem. Now, turn the stem in your hand a little over 137°, or the golden angle. This is likely where the next leaf is placed.

While the leaf position on flowers often follows the golden angle, it also is related to the Fibonacci Series. The Fibonacci series is a list of numbers where you add the previous two numbers to get the next number. It goes like this: 1, 1, 2, 3, 5, 8, 13, etc. There is a variation of the Fibonacci Series called the Schimper-Braun principle series which has the Fibonacci Series in both the numerator and the denominator of the series. The Schimper-Braun principle series goes like this: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21 … As you can see, the numerator (the top number in the fraction) follows exactly with the Fibonacci Series of 1, 1, 2, 3, 5, 8… and the denominator (bottom number in the fraction) skips ahead in the Fibonacci Series to the 2 and then picks up with the series, going 2, 3, 5, 8, 13, 21…. The Schimper-Braun principle series gets closer and closer to the golden angle. Thus, this series, as well as the golden angle, can accurately describe the leaf placement on a plant stem and sometimes the petal placement on flowers.

Flower and nature mostly follow the golden angle and the Schimper-Braun principle series. However, due to a wide range of factors, including less than ideal environmental surroundings, flowers and other plants rarely grow ideally. This is why mathematics can closely approximate it but will never exactly describe the structure of the plant. The mathematical beauty of plants is part of what makes them beautiful, because we find the golden ratio to be beautiful. There is no concrete consensus as to why we find the golden ratio beautiful. Adrian Bejan, a mechanical engineering professor at Duke University argues that the golden ratio is beautiful because the human eye interprets images containing the golden ratio faster than other images. However, we also find the asymmetry and variation in plants to be beautiful. It would be boring and rather strange if all flowers had the exact same shape. We find nature beautiful because of the mathematical symmetry and because of the asymmetry in growth variation.

Carlson, Benjamin, Why Do People Like the Golden Ratio. The Atlantic. January 2010. __https://www.theatlantic.com/technology/archive/2010/01/why-do-people-like-the-golden-ratio/341546/__

Ulrich Lüttge, Gustavo M. Souza, The Golden Section and beauty in nature: The perfection of symmetry and the charm of asymmetry, Progress in Biophysics and Molecular Biology, Volume 146, 2019, Pages 98-103, ISSN 0079-6107, https://doi.org/10.1016/j.pbiomolbio.2018.12.008.