Hidden numbers and basic mathematics in the Mandelbrot-Set

To the untrained eye, the Mandelbrot-Set might look chaotic and the arrangement of the patterns seem random.
But they are the opposite, they are arranged to very precise rules and you can find all kind of mathematics just by observing the Mandelbrot-Set.

Basic Calculus: 

  • Counting: The tips of the smaller and smaller bulbs to the right always add one.
  • Multiplication: A period 3 bulb will have smaller bulbs on it that are 3*x period
  • Exponential: Use a bulb of period x as base and find the attached bulb that is the y-largest to calculate x^y, repeat y-2 times. Example 3^3=27 is shown in image below.
    simpler example is the tip to the left, each next bulb is 2^(y+1)
  • Addition: http://math.bu.edu/DYSYS/FRACGEOM2/node6.html#SECTION00060000000000000000

Look at this fullscreen and try for yourself.

Fibonacci Sequence & Golden Ratio:
The Main Cardiod has a Period of 1, the next largest bulb on it has a period of 2, the thirdlargest has a period of three, now find the next largest between 2 and 3, it will have 5 arms, so a period5. Now the next largest between 3 and 5 will have 8 arms and so on..
Pi Image

Power Law & Logarithmic Map:

Spirals with all possible angles:

Let us know in the comments if you know more! Pretty sure there are!

Fractal electrode prototype could revolutionize energy storage

A study published in Nature showed that embracing the structure of fractal Fern leaves, is a possible new and better way to store Solar energy. Today, solar energy is stored in battery banks. Researchers are working on using supercapacitors to do the same thing, but the energy density is restricted which limits the storage.

By using a fractal fern leave structure, the ratio of the relative active area compared to the volume can be increased, which reduces the electrolyte ionic path. This results in a 30 times higher energy density within the supercapacitators compared to the current way of using supercapacitors.

According to the researchers, energy storage with these fractal electrodes opens the prospects of efficient self-powered and solar-powered wearable, flexible and portable applications. Read the full article here

Cytoskeleton Actin discovered to have fractal structure

Until recently, scientists couldn’t understand why a broad range of membrane proteins display anomalous diffusion on the cell surface within human cells. Due to several technical, and experimental restrictions, scientists could never visualize this odd behavior off cells. In this new study, scientists were finally able to visualize this process by imaging cortical actin, and by tracking individual membrane proteins in live mammalian cells. This study found that cortical actin is organized into a self-similar fractal structure. This is quite a huge deal because actin participates in many important cellular processes, including muscle contraction, cell motility, cell division and cytokinesis, vesicle and organelle movement, cell signaling, and the establishment and maintenance of cell junctions and cell shape. Furthermore, this study showed that the the plasma membrane is compartmentalized in a hierarchical fashion by a dynamic cortical actin fractal. According to the researchers. this finding has many applications for better understanding of cell communication. Nevertheless, we are wondering what the applications are for the better understanding of life in general.

Original Paper


Rohrschach Tests: Fractals determine the imagery you perceive

Rorschach inkblots (like the ones shown in the picture) have had a striking impact on the worlds of art and science because of the remarkable variety of associations with recognizable and nameable objects they induce. Originally, these inkblots were adopted as a projective psychological tool to probe mental health, but now psychologists and artists have interpreted the variety of induced images simply as a signature of the observers’ creativity.

In this study, the researchers analyzed the relationship between the spatial scaling

Number of percepts, n, induced by the five black blots plotted as a function of the D value of their boundaries.

From the result, the researchers suggest that the perceived images are induced by the fractal characteristics of the blot edges. They also note, that humans have a remarkable sensitivity to fractals. Nevertheless, this fact is perhaps not surprising given that many of the physical fractals that define our daily visual environment exhibit fractal characteristics, but commonly overseen.
parameters of the inkblot patterns and the number of induced associations. A traditional technique was used for measuring a boundary’s dimension fractal D, referred to as the box-counting method. The boundary’s D value describes how the patterns occurring at different magnifications combine to build the resulting fractal shape. A smooth line (containing no fractal structure) has a D-value of 1, while for a completely filled area (again containing no fractal structure) its D-value is 2.

According to the researchers, their analysis of Rorschach inkblots provides an appealing framework for understanding the rich variety of visual associations induced by fractal patterns spanning psychology, art and nature. The fractal properties of the inkblots induce of a more ‘organic’ visual character, which might enhance the pattern’s ability to induce associations with nameable objects and/or recognizable imagery. Within this fractal model, low D-fractals provide the optimal distribution of spatial frequencies to induce the images. In contrast, the dominance of fine structure in high D-fractals appears to reduce the perception of recognizable images. Moreover, it is intriguing to note that self-reported creative people prefer to look at fractals with higher D-values.

Read the full study here:

Fractal analysis reveals early signs of cognitive decline in paintings

Psychologist Dr Alex Forsythe from the University of Liverpool has examined 2092 paintings of seven famous artists for changes in the fractal dimension in ther paintings as they age.
The analyzed artists were:

  • Salvador Dali and Norval Morrisseau (who suffered from Parkinson’s disease)
  • James Brooks and Willem de Kooning (who suffered from Alzheimers disease)
  • Marc Chagall, Pablo Picasse and Claude Monet (no known cognitive disease)

According to Dr Forsythe changes in the fractal dimension became noticeable when the artists were in their 40ies, long before other signs of the disease could be diagnosed.

Woman 5 – Willem De Kooning (1952)

Though the findings are unlikely to lead to a early test for dementia, it might lead to new approaches in investigating the desease.

Untitled 2 – Willam Kooning (1981)

Chaos based microchips could help to keep up with Moore’s Law

Just as more and more articles proclaim the end of Moore’s Law and applicable quantum computers still seem far away, new developments using Chaos Theory could help to keep up the doubling of computation power every 2 years, proclaimed by Gordon Moore 50 years ago.

Researchers at North Caroline State university developed new non-linear, chaos-based integrated circuits that are able to perform multifple functions with fewer transistors.

Reconfigurable chaotic integrated circuit. Credit: Behnam Kia

The lead author of the paper Behnam Kia says:

“We propose utilizing chaos theory – the system’s own nonlinearity – to enable transistor circuits to be programmed to perform different tasks. A very simple nonlinear transistor circuit contains very rich patterns. Different patterns that represent different functions coexist within the nonlinear dynamics of the system, and they are selectable. We utilize these dynamics-level behaviors to perform different processing tasks using the same circuit. As a result we can get more out of less.”

Usually each transistor-based circuit performs only one task. Processors operate by directing every instruction to the proper transistor circuit on the integrated circuit that implements that particular instruction.

In Behnam Kia’s design, the transistor circuit can be programmed to execute distinctive instructions by morphing between various operations and functions.

“In current processors you don’t utilize all the circuitry on the processor all the time, which is wasteful,” Kia says. “Our design allows the circuit to be rapidly morphed and reconfigured to perform a desired digital function in each clock cycle. The heart of the design is an analog nonlinear circuit, but the interface is fully digital, enabling the circuit to operate as a fully morphable digital circuit that can be easily connected to the other digital systems.”

The researchers say:

“The potential of 100 morphable nonlinear chaos-based circuits doing work equivalent to 100 thousand circuits, or of 100 million transistors doing work equivalent to three billion transistors holds promise for extending Moore’s law – not through doubling the number of transistors every two years but through increasing what transistors are capable of when combined in nonlinear and chaotic circuits.”

The new design is compatible with existing technology and uses the same fabrication process as existing computer chips. This could help significantly with commercial adoption.

We are nearing commercial size and power and ease of programming in our evolving designs that could well be of significant commercial relevance within a few months with our three month design/fabrication cycle of improvements and implementations

The full paper can be viewed here

Juno arrives – time to explore Jupiter’s fractals

Today, after a 5 year journey the Juno-probe arrives at Jupiter.
If you ever wondered why the gas-giant’s surface looks the way it does, with all its whirls and twirls and the famous big red spot – fractals are the answer.

It’s quite well known that the turbulences in our atmosphere, responsible for our weather and climate show fractal patterns. But this phenomenon is not limited to earth.

Here’s original footage of Jupiter from NASA:

And here’s a simple simulation of fluid dynamics, showing fractal patterns on many scales, nearly identical with the turbulences we observe on jupiter:

In this interview Professor Philippe Lavoie (UTIAS) explains how turbulence is basically a fractal process:

About 10 years ago, researchers started examining something called fractal turbulence. They would pass a fluid flow through a fractal object, such as a grid, so that it forces the turbulence at different scales. They can then see how it’s affected.

The results of these experiments did not behave as we would have expected…. it questioned some of the basic principles of turbulence theory.
We conclusively showed that fractal turbulence was behaving the same way as classical turbulence, which has not been shown before.

Although fractal turbulence has some different features, it’s not fundamentally different. This evidence settled a long debate, as it demonstrated that our understanding of turbulence does not need to be fundamentally altered.

If creating turbulences with the help of a fractal object doesn’t change the actual turbulence we can conclude that it also is a fractal process when ‘regular’ objects cause the turbulence.

Also, the magnetic field of Jupter can be described using multifractal laws.

More resources about fractal turbulences, also in earths atmosphere.