What fractals, Fibonacci, and the golden ratio have to do with cauliflower

This vegetable’s tendency to produce buds at an accelerating rate causes this spiral pattern and resulting conical shape. It has long been observed that many plants produce leaves, shoots, or flowers in spiral patterns. Cauliflower provides a unique example of this phenomenon, because those spirals repeat at several different size scales—a hallmark of fractal geometry. This self-similarity is particularly notable in the Romanesco variety because of the distinctive conical shape of its florets.

The golden spiral and the Fibonacci spiral are very similar in shape, and many use them interchangeably, but they’re not the same. Everything can be explained by mathematical calculations, and they won’t have the same exact pattern when measured. In mathematics, the golden ratio is a special number that’s approximately equal to 1.618 and represented by the Greek letter Φ (Phi). You may wonder where this golden spiral comes from—and the answer to that lies within the golden rectangle. In geometry, the golden spiral can be drawn from a golden rectangle whose sides are proportioned according to the golden ratio.

Rectangle Method

And finally, I would like to thank Ryan Butler, Auckland University of Technology, for his help with photography to document this experiment. In other situations, the ratio exists because that particular growth pattern evolved as the most effective. In plants, this may mean maximum exposure for light-hungry leaves or maximized seed arrangement.

• The golden spiral plays a role in the geometry of fractals, a complex pattern that repeats forever.
• Rivers branch out into streams because water flows from the main artery downhill, but why always in meandering S shapes?
• In fact, if the angle between the appearance of each seed is a portion of a turn which corresponds to a simple fraction, 1/3, 1/4, 3/4, 2/5, 3/7, etc (that is a simple rational number), one always obtains a series of straight lines.
• (It is obtained by multiplying the non-whole part of the golden mean by 360 degrees and, since one obtains an angle greater than 180 degrees, by taking its complement).
• The explanation is linked to another famous number, the golden mean, itself intimately linked to the spiral form of certain types of shell.

Forces acting in opposing directions causes skin and plant tissue to buckle inward as it grows, he said. When the golden ratio is applied as a growth factor (as seen below), you get a type of logarithmic spiral known as a golden spiral. The spirals of sunflowers and pinecones are pretty, but they’re not golden spirals. In fact, their spirals don’t even wrap around the center, as opposed to the golden spiral. While some flowers have number of petals that correspond with the Fibonacci numbers, there are several exceptions found. Built between 447 and 438 BCE, the Parthenon in Athens, Greece is one of the most aesthetically pleasing structures ever made.

Since the ratio between consecutive Fibonacci numbers approaches the golden ratio as the Fibonacci numbers approach infinity, so too does this spiral get more similar to the previous approximation the more squares are added, as illustrated by the image. Contrary to popular belief, the nautilus shell isn’t a golden spiral. When measured, the two wouldn’t match no matter how they were aligned or scaled. Also, not every nautilus shell is created equal, as each has variations and imperfections in shapes. The golden spiral is thought to hold a fascinating influence over the motion of the human hand. According to an anatomist, the movement of human fingers follows the pattern of the golden spiral.

Mathematics is the science of patterns, so it’s not surprising that spirals have inspired mathematicians for centuries. One of these spirals is the golden spiral, thought to be a sort of code that governs the architecture of the universe. The golden spiral is a broad, fascinating subject that has played a prominent role in history and works of art. The golden spiral is a pattern created based on the concept of the golden ratio—a universal law that represents the “ideal” in all forms of life and matter. In fact, it’s often cited as an example of the connection between the laws of mathematics and the structure of living things.

Now, a team of French scientists from the CNRS has identified the underlying mechanism that gives rise to this unusual pattern, according to a new paper published in Science. Before one can study skin bio-dynamics or surgical closure techniques, one needs a good understanding of the directions of skin tension lines of normal skin. It is quite amazing that the Fibonacci number patterns occur so frequently in nature
( flowers, shells, plants, leaves, to name a few) that this phenomenon appears to be one of the principal “laws of nature”. Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition, numerous claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves  The Fibonacci numbers are also found in the family tree of honeybees. From hurricanes to flowers and pinecones, spiral patterns are abundant in nature.

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Improving observation skills is the first step to enjoying nature and building our relationship to it. The lists below will give you tips for where to look, but why look for shapes in nature? The golden ratio symbol is a Greek work known as the phi spiral.

1.Every spiral found in the earth follows the golden spiral rule. But this is not true because many spirals do not follow the same rule. Later, Leonardo da Vinci painted Mona Lisa’s face to fit perfectly into a golden rectangle, and structured the rest of the painting around similar rectangles. In order to optimize the filling, it is necessary to choose the most irrational number there is, that is to say, the one the least well approximated by a fraction. (It is obtained by multiplying the non-whole part of the golden mean by 360 degrees and, since one obtains an angle greater than 180 degrees, by taking its complement). With this angle, one obtains the optimal filling, that is, the same spacing between all the seeds (figure 3).

You can actually see the Fibonacci sequence in action in some natural spirals. In visual art like painting and photography, the golden ratio is used in composition because it is considered aesthetically pleasing. The universe may be chaotic and unpredictable, but it’s also a highly organized physical realm bound by the laws of mathematics.

The Experiment

Of course the Cabbage display’s the spiral common in most growth forms of leaf and petal rotation which again comes back to the growing tips forming at most efficient angle. Below is a fairly comprehensive list of where to look to easily find naturally occurring spirals. I have also compiled more info on the maths in spirals or particularly on how Fibonacci numbers manifest in nature, and how to spot the interconnected spirals in a sunflower so be sure have a good look around the Smart Happy Project website. Golden spiral tattoos are famous among those people who love maths, geometry, and symmetry. If you are one of them don’t hesitate to get the golden spiral tattoo.

The video below gives you an idea of how all the different shells we see all conform to the same factors that determine their shape. Also at the other end of the flower cycle the seed formation often clearly presents a multitude of interconnected spirals. The golden spiral gave the proper symmetry to things as well as humans and made them look more aesthetic.

Rivers branch out into streams because water flows from the main artery downhill, but why always in meandering S shapes? While streams can sometimes be established in a straight line, they quickly become bendy as they adapt to disturbances like wildlife dens. Just one disturbance can throw off the flow of a river and cause it to curve throughout. As they grow, branches develop from the trunks, and each of these branches is like a smaller tree in itself, developing its own branches and their own branches.

How the Golden Ratio Manifests in Nature

Its a simple nature craft that encourages the observations skills and involves the hands and the eyes. What the tourist industry along the UK’s Dorset coastline is built on – Fossils. There are many fossil types but the Ammonite fossil is the spiral we are looking for and easily found along this stretch of this Jurassic coast in the UK both in calcite and pyrite forms its definitely worth a day trip. Looking is the key to finding Spirals in Nature (no way Lisa, you’re not kidding?) I know, obvious, but you’d be surprised how little we see things sometimes.

In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to j, the golden ratio. Specifically, a golden spiral gets wider (or further from its origin) by a factor of j for every quarter turn it makes. Apparently, this is the result of self-selected mutations during the process of domestication, which over time drastically changed the shapes of these plants. The authors found that, while the meristems fail to form flowers, the meristems do experience a transient period where they’re in a flower-like state, and that influences later steps in development. In the case of Romanesco cauliflower, the curd adopts a more conical shape instead of a round morphology. The end result is those fractal-like forms at several different size scales.

We create a compelling visual brand and messaging platform by focusing on your buyers and communicating your unique solutions to their problems. We operate at the intersection of brand and growth marketing to empower companies with buyer-centric strategies that drive revenue. Although few of us ever see it, the inner ear is spiral shaped and the DNA cell structure is a double helix. The Pineapple demonstrates the same pattern as the pinecone we painted.

Continue below to see just a few of the ways these spirals manifest in nature. Our universe is filled with spirals, so it’s not surprising that many have become interested with the math behind them and their golden spiral in nature meanings. Artists have long recognized the golden spiral as the most pleasing to the eyes. It’s indeed one of the most inspiring patterns in nature that can be translated to creative artistic expressions.

You’ll even find images of a clenched fist with the spiral symbol as overlay. The Nautilus shell is the closest thing you will find that resembles what the creatures were that created the fossil we mentioned earlier. The Nautilus spiral is hailed as being a perfect Fibonacci spiral, reflecting the growth in the relationship between the sequence of numbers, sometimes also called a Golden Spiral. You can clearly see the spiral as a flat curve in ferns, but also as a 3D spiral in petals as they unfurl around the flower bud.

Given this this is an interesting and exciting new discovery, and may lead to further research in other fields of scientific endeavor, I am presenting this to the wider scientific community. As discussed earlier, spiral formations abound in nature and in tissues; they may represent a feature of rapid expansion along an advancing front. After all, the opening of flowers is often rapid –for example, Hedera helix, the English Ivy, opens in about 5 minutes [24], and many flowers demonstrate spiral patterns around their inflorescence axis. Indeed studies in monozygotic twins have shown individual variations of scalp whorls, and therefore both skin stretch and the subsequent gene expression can be considered responsible for the formation of spirals [27]. As an animal biologist and skin cancer academic, the author’s fascination has been with the fact that humans are the only animals to have scalp whorls on the top of their heads that follow spiral patterns seen in nature. Others have noted that of all mammals, only humans have hair whorls on the vertex of the scalp, and that each human individual must have a hair whorl [9].

Some other examples are pinecones, shells, seed heads, hurricanes, shells, insects’ bodies, etc. It is a ratio that is put forward in case of a line splitting into two parts. Even the pyramids of Giza are also designed with the help of this spiral ratio.