Mathematics, Physics. an irregular geometric structure that cannot be described by classical geometry because magnification of the structure reveals repeated patterns of similarly irregular, but progressively smaller, dimensions: fractals are especially apparent in natural forms and phenomena because the geometric properties of the physical world are largely abstract, as with clouds, crystals, tree bark, or the path of lightning.
Architecture, Decorative Art.a design or construction that uses the concept and mechanics of fractal geometry.
The English word "fractal" comes from Latin word "fractus", which means "fractured"
Fractals in Nature:
Fractals in Nature: Fractal geometry is stunning and exists throughout the natural world. Phenomena such as lightning bolts, river networks and even galaxies contain fractals. Trees are one of the best examples of a fractal, branching out in the most delightful ways.
Examples of fractals in nature:
Fractals in Math: Fractals are a rough geometric shape that can be split into parts, each part being roughly a reduced size version of the whole, this property is known as self-similarity.
Examples of fractals in mathematics include:
De Rham curve
“Bottomless wonders spring from simple rules which are repeated without end.” – Benoît Mandelbröt (1924-2010)
The Mandelbrot Set - Benoit Mandelbrot is often cited as the “father of fractal geometry” for the pioneering use of computers for research. In 1982 he released his book “The Fractal Geometry of Nature” altering the field of applied mathematics. The Mandelbrot set is the most famous example of a fractal in mathematics.
The complex shape of the Mandelbrot set was only revealed through the use of computer calculations, allowing for millions of iterations that were unfeasible to plot by hand.
Julia Set Fractal
Julia sets are closely related to the Mandelbrot sets. The Mandelbrot set can be considered a "map" of all the Julia sets (each point on the Mandelbrot set corresponds to its own unique Julia set.) Julia sets are some of the most beautiful fractals that have been discovered. They are named after the French mathematician Gaston Julia.
The Phoenix fractal is a modification of the Mandelbrot and Julia sets. It was discovered by Shigehiro Ushiki in 1998 in a paper published in the 'IEEE Transactions on Circuits and Systems' journal.
The Newton fractal is a boundary set in the complex plane. The fractal is characterized by Newton's method (a well known procedure for finding roots of functions) applied to a fixed transcendental function.
Most people learn Newton's method using real functions, however it can be applied to complex functions, such as in the Newton Fractal.
Koch Fractal (Koch Snowflake)
The Koch Snowflake fractal (also known as Koch curve or Koch star) is a fractal curve and one of the earliest known fractals to have been discovered. It was first described in 1904 by a Swedish mathematician Helge von Koch.
The Koch Snowflake can be built up iteratively. The above illustration is a Koch snowflake with 5 iterations. Iteration 1 is a simple equilateral triangle. This iterative process can go on forever, creating an infinite fractal curve. This creates a fractal that encloses a finite area, but has an infinite perimeter.
In geometry the golden spiral is a type of logarithmic spiral whose growth factor is the golden ratio. The Fibonacci spiral is an approximation of the golden spiral and gets closer as the numbers increase toward infinity.
Approximations of logarithmic spirals can occur in nature at various scales. A prime example being spiral galaxies.
Other types of fractals include:
Barnsley Fern Fractal:
The Barnsley fern is a fractal named after British mathematician Michael Barnsley, who first described it in his book 'Fractals Everywhere'. The book was based on courses Barnsley taught at the Georgia Institute of Technology. This is an example of using an iterated function system (IFS) to develop a fractal.
Sierpinski Carpet Fractal:
The Sierpinski carpet is a fractal that was first described in 1916 by a Polish mathematician named Wacław Sierpiński. The same method used to create the Sierpinski carpet also works with other shapes such as the equilateral triangle or a pentagon.
Sierpinski Triangle Fractal:
Sierpinski Pentagon Fractal:
Harter-Heighway Dragon Fractal:
The Heighway dragon fractal was first investigated by NASA physicist John Heighway. Details of this fractal were first published in 1957 in the Scientific American magazine.
Fractal Artist: Pardesco
Our best fractal art and one of a kind wood wall sculptures
The following is a series of one of a kind fractal wall art. Each piece is intricately carved into hardwood and hand finished. All of our fractal carvings are original custom designs made by artist and parametric designer: Pardesco
Growth Spiral I, 2022
Crucible, Fractal Art, 2022 (sold)
Brinicle, Fractal Art, 2021 (sold)
Ripple I, Fractal Art, 2022 (sold)
Hyperstar I, Fractal Art, 2022
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“Chaos, present in everything from a drop of water to the galaxies in our universe, has long fascinated people from cultures across the world. The natural disorder present in the branches of trees, lightning, and coastlines, to name a few examples, may seem to be completely chaotic; however, self-similarity within these phenomena is much more organized than it appears. Repeating patterns in many natural objects and processes are known as fractals, figures that show self-similarity at different levels. The study of fractals, both in nature and pure mathematics, has advanced numerous branches of science, such as computing, telecommunications, fluid dynamics, biology and medicine, and can offer innovative new perspectives on human science and technology.
Almost everything in the universe is constantly changing, and while fractals have been described as patterns of chaos, they can also be described as patterns of change in the context of natural fractals. In many cases, fractal patterns are the most efficient way to deliver nutrients, harvest sunlight, or form support structures for plants or animals. For example, the nautilus shell, a naturally occurring object that results from the growth of a nautilus, forms in a very precise logarithmic spiral, meaning every compartment of the shell is larger than the preceding one by the same factor regardless of its position in the shell. The veins of many leaves are also arranged in a fractal pattern, making it easy for nutrients to flow into and out of the leaf. Romanesco broccoli, perhaps one of the most well-known fractal patterns in nature, grows in branches or buds; each large bud has several smaller buds, which themselves have smaller buds, and so on. While these natural fractals are not infinite like their mathematical counterparts, their self-similarity is well-modeled by fractals.
Many non-biological processes or objects demonstrate repeating, self-similar fractal features. Snowflakes have a fractal structure due to the organization of water molecules within them, which is often reflected in the six branches of snowflakes or the frost seen on windows. Many metals have branching microstructural features close in appearance to trees, which are known as dendrites, resulting from specific crystallization parameters. Flowing water often spreads out in fractal patterns, such as the alluvial fans of rivers that terminate near mountains. The flow of water, wind or other sources of erosion can form fractal patterns in features such as coastlines and mountains, as the driving forces of change often act more strongly on areas which have already been acted upon in a recursive loop. On a larger scale, solar systems and galaxies have been observed to organize in self-similar patterns, something that defies previous theories.
At the most basic level, fractals occur in nature because of the repetition of a process or force acting on something, such as the erosion of land by water or the growth of leaves on a fern plant. The persistent action of water on a surface can introduce cracks, which expand and open up more surface area for new cracks to form, thus creating self-similar features. The force of evolution drives living things to be as simple as possible while still being able to survive; the shortening of an organism’s genome can be beneficial, as it would require less resources to reproduce. In this respect, fractals offer a marvelous method for living structures to simplify, as it is easier to genetically store the self-similar information of a fractal than the details for every small structure, as is seen in the fractal patterns of fern leaves. For some natural fractals, it is still unclear why they organize into self-similar patterns, although the driving principle is likely related. Humans have noticed the patterns in natural objects for thousands of years, and many human structures have been inspired by fractal patterns. For example, the organizations of towns and villages historically have often followed fractal distributions. The logarithmic spiral is a prevalent feature of much of Renaissance art. Humans seemingly followed the self-similar patterns seen in nature subconsciously, and modern research has shown that humans prefer art and designs that show fractal geometry over those that don’t. Many scientists investigated the patterns of natural structures such as the nautilus shell, ice crystals, or the veins in the human body.
Human models of these patterns were simple, such as the logarithmic function or repeating geometrical features, but advanced several areas of mathematics, science and technology. Many scientists continued to investigate the mathematics of recursion and self-similarity, but the study of fractals was reinvigorated with the advent of computing in the late 1900s. In the late 1970s, Polish-French mathematician Benoit Mandelbrot began studying fractals, specifically the Julia sets, created by French mathematician Gaston Julia six decades prior. At the time, Mandelbrot was among the research staff at IBM, and used the computers available to him to plot the Julia sets, specifically highlighting the points in the set that did not go to infinity, but stayed between certain bounds, now known as the Mandelbrot set. Building on the work of Georg Cantor, Gaston Julia, Felix Hausdorff, and Lewis Fry Richardson, Benoit Mandelbrot wrote many papers and books on the new field of study - fractals, a name he derived from the Latin word for fragmented. Mandelbrot demonstrated the application of fractals to natural phenomena such as coastlines, as well as unnatural phenomena such as the stock market, and popularized fractals. In addition to applications in mathematics, fractals have been used to model or design novel systems in fields such as medicine, computer and electrical engineering, biology, chemistry, environmental science, cosmology, and many more. Unlike the familiar 1, 2, and 3-dimensional figures, fractals do not have integer topological dimensions, and instead can have non-integer Hausdorff dimensions, which allow for closer approximation of natural features. The simplicity and high surface area of fractals make them ideal models for simplifying and improving existing or theoretical designs. Fractals have found ingenious applications in biology and medicine due to the multitude of natural systems that follow recursive patterns in at least one respect. Some bacteria’s growth has been well-approximated by fractal models, and these models can also help predict the locations of bacterial colonies. The growth of blood vessels in the human body follows fractal division, a phenomenon that can be disturbed in cancerous tissue; fractal analysis of blood vessel growth in tumor tissue can offer new insights into cancer. The organization of neurons follows fractal patterns, a feature that may help understand how the human brain functions. Fractal distributions have been used to determine the distributions of plants in managed forests, analyze the distributions of alveoli in lungs, and even research the genetic systems of cells.Fractal models have been successfully applied to physical processes such as turbulent flow and aggregation of molecules due to the similarity of the physical laws that govern how such things occur. Fluid dynamics, as chaotic and potentially recursive phenomena, have been reproduced numerically by fractal approximations, which has helped model such things as fluid flow around airplanes and ships, the kinetics of mixing in industrial processes, and many other engineering problems. Some porous structures are fractal to a degree, and fractal models have been used to model porous media for petroleum engineering. The randomness seen in fractals can be similar to the Brownian motion of molecules in liquid or gas, and semi-fractal models have been used for approximation of diffusion, aggregation, and electrodeposition of molecules in materials engineering.Due to the high surface area to volume ratio of fractal patterns, designs that incorporate fractals have been implemented to minimize the mass or size of applications that require high surface areas, such as antennae or cooling units. This approach has shown great promise, achieving equal or higher specifications with significantly less material used. Using the principles of self-similarity at any scale, fractal patterns have been used in a new method of image compression, which offers clear resolution at any scale.The natural world has an abundance of chaotic processes that shape the world we live in, and these processes often create the self-similar things we see around us. Learning from these natural fractals has given us the ability to advance our knowledge of the physical world and to apply this knowledge to novel creations. In the future, the study of fractals will lead to greater inventions and may help answer our lingering questions about the universe.”