Nature's patterns: Golden spirals and branching fractals (2024)

FAQs

What is the golden spiral in fractals? ›

In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes. Golden spirals are self-similar. The shape is infinitely repeated when magnified.

Spiral galaxy

The golden spiral is based on the golden ratio. Symbolised by the character φ (Phi), it's found when a line is split in such a way that the larger part divided by the smaller part is equal to the whole part divided by the larger part -- a ratio of (rounded) 1.618.

Examples of the Golden Ratio in nature include the spiral patterns found in sunflowers, pinecones, and seashells. Additionally, the growth patterns of certain plants, like the arrangement of leaves around a stem, exhibit the Golden Ratio.

The golden ratio, also known as the golden number, golden proportion or the divine proportion, is a ratio between two numbers that equals approximately 1.618. Usually written as the Greek letter phi, it is strongly associated with the Fibonacci sequence, a series of numbers wherein each number is added to the last.

Separate from Euclidean geometry, fractal geometry addresses the more non-uniform shapes found in nature, such as mountains, clouds and trees. Fractals provide a systematic method to capture the “roughness” of some objects.

A fractal is a kind of pattern that we observe often in nature and in art. As Ben Weiss explains, “whenever you observe a series of patterns repeating over and over again, at many different scales, and where any small part resembles the whole, that's a fractal.”

According to Culture-Cross the Fibonacci spiral embodies profound symbolism and purpose. It represents the interconnectedness and harmony found in nature and the universe. This sacred geometrical pattern signifies growth, evolution, balance, and the divine order.

Spirals appear in nature due to radial growth or the shape of an organism such as a chameleon's tail or a fiddlehead fern. Spirals are a natural pattern produced as the organism develops or a hurricane is formed depending upon the dynamics of growth and formation.

In trees, the Fibonacci begins in the growth of the trunk and then spirals outward as the tree gets larger and taller. We also see the golden ratio in their branches as they start off with one trunk which splits into 2, then one of the new branches stems into 2, and this pattern continues.

Sunflowers provide a great example of these spiraling patterns. Snail shells and nautilus shells follow the logarithmic spiral, as does the cochlea of the inner ear. It can also be seen in the horns of certain goats, and the shape of certain spider's webs. Spiral galaxies are the most common galaxy shape.

Where can you find the golden spiral in real life? ›

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!

There are many variations on what the spiral represents. The spiral motif is a link to nature, representing the ever changing seasons. It represents the cycle of life; birth, growth, death, and re-incarnation.

The essential part is that as the numbers get larger, the quotient between each successive pair of Fibonacci numbers approximates 1.618, or its inverse 0.618. This proportion is known by many names: the golden ratio, the golden mean, ϕ, and the divine proportion, among others.

The Fibonacci sequence, also known as the golden ratio, is utilized in architectural designs, creating aesthetically pleasing structures. In engineering and technology, Fibonacci numbers play a significant role, appearing in population growth models, software engineering, task management, and data structure analysis.

While the curvature of the golden spiral changes continuously, that of the Fibonacci spiral varies discontinuously. Because discontinuities in curvature are usually not liked, it was expected that the Fibonacci spiral is perceived as less beautiful than the golden spiral.

The Golden Spiral has the special property such that for every 1/4 turn (90° or π/2 in radians), the distance from the center of the spiral increases by the golden ratio φ = 1.6180. For this to occur, cot b must take the value (which comes from solving our function):

The basics of sacred geometry

The golden ratio is renowned for producing some of the most beautiful shapes. From it, we get the golden spiral (a logarithmic spiral that gets wider by a growth factor of the golden ratio), and the golden rectangle (a rectangle whose side lengths are in the golden ratio).

The Fibonacci Spiral, which is my key aesthetic focus of this project, is a simple logarithmic spiral based upon Fibonacci numbers, and the golden ratio, Φ. Because this spiral is logarithmic, the curve appears the same at every scale, and can thus be considered fractal.