How the Golden Ratio Manifests in Nature

Researchers have also found evidence of the golden spiral and golden ratio is many other plants, including fiddleheads — the the curled up fronds of a young fern — daisies and spiral aloe vera. In plant biology, the golden ratio and Fibonacci numbers have fascinated botanists for centuries. Phi controls the distribution and growth of leaves and other structures in many species — while others grow at a growth constant that is astonishingly close to this magic number. The terms Fibonacci spiral and golden spiral are often used synonymously, but these two spirals are slightly different. A Fibonacci spiral is made by creating a spiral of squares that increase in size by the numbers of the Fibonacci sequence. When the golden ratio is applied as a growth factor constant to a spiral (meaning the spiral gets wider — or further from its origin — by a factor of the golden ratio (1.618) for every quarter turn it makes) we get the golden spiral.

  1. Phi can be defined by taking a stick and breaking it into two portions.
  2. Then the trunk and the first branch produce two more growth points, bringing the total to five.
  3. Sunflowers provide a great example of these spiraling patterns.
  4. The atomic radius of hydrogen in methane is the Bohr radius over the golden ratio.

Objects designed to reflect the golden ratio in their structure and design are more pleasing and give an aesthetic feel to the eyes. It can be noticed in the spiral arrangement of flowers and leaves. Emergence theory is based deeply on the notion of language. A powerful language is one which has the ability to express the maximum amount of meaning with the least number of choices, since each choice requires resources. A resource in this sense can be a unit of electricity spent for a logic gate to be opened to activate a binary choice in a computer language, or a calorie or two of energy needed to make a mental choice of what shirt to wear.

Fibonacci Numbers and How Rabbits Reproduce

Shown is a colour diagram of a rectangle divided into a pale purple square and a smaller pink rectangle. The top and left edges of the square are each labelled with a blue, lower case, italic a. The top edge of the smaller rectangle is labelled with a red, lower case, italic b. The golden ratio in nature entire bottom edge of the larger rectangle is labelled with a + b, in green italics. Look at the array of seeds in the center of a sunflower and you’ll notice they look like a golden spiral pattern. Amazingly, if you count these spirals, your total will be a Fibonacci number.

Golden Ratio Equation

You’ll notice that most of your body parts follow the numbers one, two, three and five. You have one nose, two eyes, three segments to each limb and five fingers on each hand. The proportions and measurements of the human body can also be divided up in terms of the golden ratio.

Practice Questions on Golden Ratio

Note that we are not considering the negative value, as \(\phi\) is the ratio of lengths and it cannot be negative. Since both the terms are not equal, we will repeat this process again using the assumed value equal to term 2. The value of the golden ratio can be calculated using different methods. Why don’t you go into the garden or park right now, and start counting leaves and petals, and measuring rotations to see what you find.

DNA molecules follow this sequence, measuring 34 angstroms long and 21 angstroms wide for each full cycle of the double helix. Some plants express the Fibonacci sequence in their growth points, the places where tree branches form or split. One trunk grows until it produces a branch, resulting in two growth points. The main trunk then produces another branch, resulting in three growth points. Then the trunk and the first branch produce two more growth points, bringing the total to five. This pattern continues, following the Fibonacci numbers.

Shown is a black and white illustration of a rectangle divided into smaller squares and rectangles that get smaller as they move around the page, towards a spot in the bottom right quadrant. The smallest square is not labelled, but it looks like this pattern could continue, becoming smaller and smaller with each iteration. In 1868, Wilhelm Hofmeister suggested that new cells destined to develop into leaves, petals, etc. (primordia) “always form in the least crowded spot” on the meristem (growing tip of a plant). Each successive primordium of a continuously growing plant “forms at one point along the meristem and then moves radially outward at a rate proportional to the stem’s growth” (Seewald).

Misconceptions About the Golden Ratio

We celebrate Fibonacci Day Nov. 23rd not just to honor the forgotten mathematical genius Leonardo Fibonacci, but also because when the date is written as 11/23, the four numbers form a Fibonacci sequence. Leonardo Fibonacci is also commonly credited with contributing to the shift from Roman numerals to the Arabic numerals we use today. The famous Fibonacci sequence has captivated mathematicians, artists, designers, and scientists for centuries. Also known as the Golden Ratio, its ubiquity and astounding functionality in nature suggests its importance as a fundamental characteristic of the Universe.

And while phi is said to be common in nature, its significance is overblown. Flower petals often come in Fibonacci numbers, such as five or eight, and pine cones grow their seeds outward in spirals of Fibonacci numbers. But there are just as many plants that don’t follow this rule as those that do, Keith Devlin, a mathematician at Stanford University, told Live Science. Have you ever wondered why flower petals grow the way they do? Why they often are symmetrical or follow a radial pattern. The unique properties of the Golden Rectangle provides another example.

The golden ratio appears fundamentally in quantum mechanics and in black holes. The atomic radius of hydrogen in methane is the Bohr radius over the golden ratio. In 1993 Lucien Hardy of the Perimeter Institute discovered that the probability of entanglement for two particles projected in tandem is the golden ratio over negative 5. This phenomenon is the ubiquity of the golden ratio in nature from the micro (including the Planck scale) to the macro scale. Upon learning of a golden ratio related fact, most scientists will often treat it as a coincidence. However, the statistical probability of the golden ratio’s unrelenting prevalence to such high accuracy is practically zero.

For example, lilies and irises have three petals, buttercups and wild roses have five, delphiniums have eight petals and so on. Other polyhedra are related to the dodecahedron and icosahedron or their symmetries, and therefore have corresponding relations to the golden ratio. In four dimensions, the dodecahedron and icosahedron appear as faces of the 120-cell and 600-cell, which again have dimensions related to the golden ratio.

Palms are ideal specimens of the plants that display spiral phyllotaxis because their large leaves are prominently arranged (and therefore easily observed) on the trunk. Phi can be defined by taking a stick and breaking it into two portions. If the ratio between these two portions is the same as the ratio between the overall stick and the larger segment, the portions are said to be in the golden ratio. This was first described by the Greek mathematician Euclid, though he called it “the division in extreme and mean ratio,” according to mathematician George Markowsky of the University of Maine.

Its centre consists of tiny, pointed, deep yellow structures, densely packed into a circle. The spiral demonstrates that the tiny pointed structures are laid out in a spiral pattern. We can grow this pattern by adding a new, larger square to the long side (a + b) of the rectangle. This square, combined with the previous shapes, results in a new, larger rectangle. Do this again and again, and you can create a growing pattern, like the diagram below.

There are many examples of the golden ratio in nature — yet many people have no idea what it is or how to appreciate the planet’s stunning geometry. This might be because the US as a nation, does not appear to excel in the subject of mathematics. As Hart explains, examples of approximate golden spirals can be found throughout nature, most prominently in seashells, ocean waves, spider webs and even chameleon tails! Continue below to see just a few of the ways these spirals manifest in nature.