About fractals. A pattern confirmed for the first time
Many people are fascinated by the beautiful images called fractals.
Expanding beyond the typical perception of mathematics as a body of complicated, boring formulas, fractal geometry blends art with mathematics to demonstrate that equations are more than just a collection of numbers. What makes fractals even more interesting is that they are the best existing mathematical descriptions of many natural forms.
The concept of “fractals” has become increasingly popular, although it first appeared more than two centuries ago, with a prominent and prolific mathematician and philosopher Gottfried Wilhelm Leibnitz.
Leibnitz is believed to have first addressed the notion of self-symmetry or self-similarity, but it was not until 1960 that the concept was formally stabilized both theoretically and practically by Benoit Mandelbrot, through mathematical development and computer visualizations, when the name “fractals” was also established.
But let’s see what a fractal is. A fractal is an infinite pattern that repeats itself at different scales. This property is called, as mentioned above, “self-similarity”. The entire pattern is contained in each of its parts.
Fractals are defined mainly by three characteristics:
- Self-similarity: identical or very similar shapes at all scales
- Repetition: a recursive relationship limited only by the capacity of the computer. With high enough performance, the iterations could be infinite. They are highly detailed shapes at each scale, which change in relation to the first iteration, manifesting the same original shape at different iteration levels.
- Fractal dimension or fractional dimensions: a measured length changes depending on the length (scale) of the measuring instrument used.
Fractals in nature
The universe has a similar and dynamic structure on all scales, from the infinitely large to the infinitely small, from the macrocosm to the microcosm, and within each scalar boundary condition there is a holographic representation of the whole: an infinitely connected holofractographic universe.
We find fractals, the same patterns over and over again, from the tiny branching of our blood vessels or neurons, to the branching of trees, lightning and river networks or snowflakes, and even further to the branching of galaxies.
The spiral is another extremely common fractal in nature and is found widely throughout the universe. Biological spirals are found in the plant and animal kingdoms, while non-living spirals are found in the turbulent swirl of fluids and the pattern of star clusters in galaxies.
All fractals are formed by simple repetition. Combining iteration with expansion and rotation gives rise to the ubiquitous spiral.
An even more amazing view of the universe is described by Carl Sagan. He offers the speculative idea that our universe may be just an electron in a much larger universe. Similarly, the electrons in our own universe may themselves contain entire universes.
This is similar to a fractal theory of the multiverse. There are multiple universes on different scales.
Instead of there being elementary particles and nothing smaller (or a fundamental building block of matter) there are countless universes contained within the elementary particles of each universe – the fractal multiverse. And it could even be infinite – we could magnify an object and reach from one universe to another forever.
Fractals in geometry and algebra
Purely geometric fractals can be made by repeating a simple process. The Sierpinski triangle is made by repeatedly removing the middle triangle from the previous generation. The number of colored triangles increases by a factor of 3 at each step: 1,3,9,27,81,243,729, etc.
We can also create fractals by repeatedly calculating a simple equation over and over again. For example, Z=Z³ + C.
Because equations need to be calculated thousands or millions of times, we need computers to explore them. Not coincidentally, the Mandelbrot Set was discovered in 1980, shortly after the invention of the personal computer. Exploring fractals is fun, and we can play with the equations to see what happens.
Applications in the world of science
An important question arises: what are fractals useful for?
Nature has used fractal patterns since the beginning. Only recently, however, have human engineers begun to copy natural patterns to build successful devices. Below are just a few examples of fractals used in engineering and medicine:
- A computer chip cooling circuit etched in a fractal branching pattern. Developed by researchers at Oregon State University, the device channels liquid nitrogen over the surface to keep the chip cool.
- Fractal antennas developed by Fractenna in the US and Fractus in Europe have paved the way for mobile phones and other devices. Because of their fractal shape, these antennas can be very compact while receiving radio signals over a wide range of frequencies.
- Researchers at Harvard Medical School and elsewhere are using fractal analysis to assess the health of blood vessels in cancerous tumors. Fractal analysis of CT scans can also quantify the health of the lungs of patients suffering from emphysema and other lung diseases.
- Amalgamated Research Inc (ARI) is creating space-filling fractal devices with high precision in mixing fluids. Used in many industries, these devices allow fluids such as epoxy resins to be mixed carefully and precisely without the need for turbulent agitation.
Why are fractals so important? If you apply these principles to yourself, you can begin to visualize, to imagine, to experience that the finite structure that you are contains the possibility of infinite division and the possibility of infinite information; that you have within you the information of the entire universe.
And this obviously has consequences in science. Why? Here’s an example.
When we discovered biological cells, we used microscopes whose magnification power, resolution, was the maximum at that time. So we thought, “These things are the smallest particles produced by the universe.”
Then we discovered the atom. There are billions of atoms in each cell. And we said again, “This must be the smallest thing produced by the universe.”
And then we discovered protons and neutrons, in the center of the atom. The nucleus of an atom was discovered, and we were still thinking, “This is so small, this is the smallest thing in the universe.” Then came quarks, antiquarks, gluons, etc.
Fractal pattern confirmed in a quantum material for the first time!
For the first time, MIT physicists have discovered fractal patterns in a quantum material.
The material is NdNiO3, and it either conducts electricity or acts as an insulator, depending on its temperature. It also exhibits inhomogeneous magnetism: regions with a specific magnetic orientation that vary in size and shape throughout the material.
The material exhibits this unique electronic and magnetic behavior as a result of quantum effects at the atomic scale, and for this reason it is called a quantum material.
The researchers used X-rays to map the size, shape, and orientation of the magnetic regions point by point at different temperatures. It was confirmed that, above a certain critical temperature, the regions disappear, erasing the magnetic order.
However, if they cooled the sample below the critical temperature, the magnetic regions reappeared in almost the same place as before! This means that the system has memory, which was very unexpected. We could have a system that is robust against external disturbances, so that the information is not lost.
Secondly, after mapping the magnetic regions of the material and measuring the size of each one, the researchers found the same pattern over and over again, and that these magnetic patterns are fractal in nature!
Because the material acts as an insulator or conductor depending on temperature, scientists are exploring NdNiO3 for neuromorphic devices – devices that mimic biological neurons.
Another potential application – magnetically resistant data storage devices.
“There are fractal codes that contain the laws of creation, and the more we understand these laws, the more we can bring harmony into our own lives. There is truth and purity in natural things, and our contact with them nourishes the soul and enlightens the mind.” – Jonathan Quintin
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Source: fractalfoundation.org, resonancescience.org
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