Since the advent of the modern age we have been riddled by the question of how closely technology and consciousness are related, and how they interact. Various literature and movies explore this question and have taken it to fictional heights of possibilities.

Many publicly available studies have thoroughly investigated the workings of our brain, including how brain activity and states of consciousness are connected. This article will have a look at how much science has caught up with science fiction.

In research labs, scientists have monitored monks and Yogis in an attempt to understand the correlations between their brainwave activity and meditative states. Results of this research have enabled us to understand and categorize how certain states of consciousness are directly linked to brainwave frequency ranges.

These results, however, posed the question of whether we can also vice versa influence the state of consciousness by inducing certain brainwave frequencies. And, if so, how can this be done?

The idea behind this is to influence brainwaves using an external stimulus with a certain frequency. If a person receives a stimulus of a frequency within the range of brainwaves, the predominant brainwave frequency is likely to adapt to or at least move towards the frequency of the external stimulus. This process is called entrainment and can be explained by the law of resonance.

Studies have found that certain frequencies are predominant when certain activities, such as being active, relaxed, concentrating or sleeping, are performed. So the idea is to support and create the desired outcome of consciousness states by inducing the according frequencies with external influences.

Electrical charges in the brain are created by billions of neurons. Neurons are electrically charged and they pump positively and negatively charged ions across their membranes to their surrounding neighbor neurons. Ions are constantly being exchanged and, when many ions are released from groups of neurons simultaneously, they trigger their neighbouring groups of neurons to do likewise, creating a wave of so called “volume conduction”.

These waves of electrical activity can be measured by instruments such as a voltmeter or an EEG (Electro Encephalo Gram) device. The rate at which this volume conduction occurs can be measured in cycles per second or Hertz (Hz).

Here is a short story from my first experience with Brainwave Entrainment.

I used to be in a group involved in shamanic trance journeys. Our tool was the shaman drum. We would sit in a circle of seven to eight people each drumming on a handheld drum. After being immersed in that sound for five to ten minutes my consciousness would independently start to slip into a different state. It’s very hard to resist this process for long. It is rather like falling asleep. I had different experiences of varying degrees of emotional and spiritual intensity during these shamanic trance journeys.

What I never understood, though, was the fast pace of our drumming which I had initially expected to be rather slow. Since it worked well, I didn’t question it any further… until I came across binaural beats and isochronic tones. It took me quite a while to put two and two together. The shaman drumming was at the pace of about 4-5 Hz – which is exactly in the same brainwave frequency band as the so called “Theta State”.

Scientific studies have shown that theta brainwaves are associated with deep relaxation and trance states. This technique has been used by shamanic cultures since time immemorial to induce trance states. The theta state has only recently been discovered and studied in modern technological terms.

Now, with today’s technology, we can easily create and precisely engineer sinus waves, rhythms and beats according to our needs. There are three common methods of inducing particular frequencies using sound.

**1. Binaural Beats warrant some extra attention, because the actual beat is created in the brain.**

When two different tones with a slight difference in frequency are separately introduced to each ear, the difference is perceived in the brain as a third tone or beat. When headphones are used, each ear can only hear one steady sinus tone without the beat. For example a 400Hz tone and a 410Hz tone will produce a subsonic beat of 10Hz in the brain. The “carrier frequency” is then at 405Hz.

**2. Monaural Beats are pretty much binaural beats but the third tone is created before the sounds enter the ears.**

This basically means that, when playing binaural beats using two boxes next to each other, the sum of the waveforms interfere with each other and keep oscillating between louder and quieter – becoming monaural beats. The third tone or beat is created in open air before entering the ears.

**3. Isochronic tones are simply rhythmic tones or sounds that are turned on and off at a certain frequency. They do not rely on the combination of two tones.**

Shamanic drumming is a good example of isochronic tones. Studies have shown that isochronic tones produce a stronger response in the brain, making them more effective than monaural or binaural beats.

**Gamma Waves (30-100Hz)** Gamma waves are measured when different senses are processed at the same time, implicating the unity of conscious perception. This state is associated with the high neural activity necessary for carrying out cognitive and physical actions.

**Beta Waves (13-30Hz)** Beta waves are known to be the normal awake state. When we are active, busy, thinking, concentrating or feeling anxious, our brainwaves are in the beta frequency range.

**Alpha Waves (8-13Hz)** Alpha waves are known to be the state of relaxation or daydreaming. They are also associated with the meditative state including being relaxed or before and after going into deep sleep. This state is very good for working with the sub-consciousness.

**Theta Waves (4-8Hz)** Theta waves are known as the deep relaxation state or sleep. They can be measured during deep meditation and trance states, and are also associated with states of high creative inspiration. Interestingly, they occur normally in young children, but ongoing theta states in adults could indicate disorders.

**Delta Waves (0.5 – 4Hz)** Delta waves are associated with deep, dreamless sleep or unconsciousness. When this state is reached through deep meditation, it is associated with out-of-body experiences or astral traveling.

**Sub Delta Waves (0.4-0.1Hz)** Sub Delta waves are associated with a brain state, in which the brain produces more neurochemicals that are beneficial for the regeneration of the body and the psyche.

**Epsilon Waves (0.1-0.001Hz)** Epsilon waves are experimental in use, for those who want to go as deep as possible in meditation. I am only aware of one company that provides the Epsilon Waves in their program. Read about “iAwake Technologies” below.

**iAwake Technologies
**

iAwake produces Brainwave Entrainment programs of a high standard. Their quality, content, effectiveness and taste are unmatched. This is the only site I know of that offers the “Epsilon” frequencies (that go as low as 0.1 – 0.001 Hz) in their programs. These sub-delta frequencies are unique and especially suitable for mystical experiences.

Another special trait of iAwake is that, next to simple brainwave entrainment, they have included what is called “Biofield Entrainment” where, based on research, other biorhythms of our organism are taken into account. They also have a programm based on technology called “Fractal Entrainment”. This is based on cycles and proportion instead of on fixed frequencies.

What iAwake does not have, however, is a great amount of voice guided meditations. They do have a few, like their “Profound Release”, but most of their programs are in the form of soundscapes.

Another great advantage of iAwake is that all their products come with user manuals that guide you in the appropriate use of the technology. (Thanks Steve for the reminder.)

You can download a free audio package for test drive on their site:

When you are busy or working, then brainwave entrainment can’t get you into the alpha state. And when you are driving or doing something that needs your full attention, you sure don’t want to go into alpha or theta states. For meditating, relaxing or calming down, audio entrainment can be a perfect tool and support. It can be most helpful in sinking into meditation practices, especially when you are just starting out.

It does not, however, take the work or practice off of you. There are no magic tunes that will automatically do any kind of development for you. Audio entrainment is an aid and a supporting tool that can catalyze and enhance your progress and experience, but it cannot replace your personal input and effort. More and more beats and isochronic tones based on specific carrier frequencies are being studied and released. Some of these have been verified by different institutions while others still need to be confirmed.

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]]>This section is for those of you who want to know more about the mathematical and geometrical background of fractals. I am no mathematician, but I am fascinated by the underlying constructions, because they give us a better idea of how these beautiful and intriguing images are created.

There are many different types and classifications of fractals. After my own studies I came to the conclusion that a simple basic classification is useful of:

**1. fractals based on algebraic iteration (iteration = repetition), meaning formulas and numbers and **

**2. fractals based on geometric iteration.
**

These two classes produce such different looks that a distinction is useful.

This category of fractals includes all that are based on formulas and numbers. The classifications I would make within this category are the different sets of numbers that are used. Fractals like the mandelbrot set, julia set and the burning ship are made from the set of complex numbers, whereas 3d fractals like the quaternions, tessarines, octonions, better known as the mandelbulb and mandelbox fractals are based on the set of hypercomplex numbers. So what are these sets of numbers?

At the beginning we have the set of natural numbers. That is 1, 2, 3, 4…. and so on. Within this set we can do all kind of mathematical operations such as addition, subtraction. But when we get to this formula “2-3=x” then we cannot solve it within the set of natural numbers. So to solve that we need to extend the boundaries to the set of whole numbers. This set also includes negative whole numbers like -1, -2, -3, -4… and so on.

This is a very simple example, but it illustrates very well the method and concept of expanding the area of operation to a larger set of numbers in order to solve an equation. Now within this set of whole numbers we can solve the formula above where “x=-1”, but still no equation like “3/2=x”. To solve this, we have to further extend the boundaries to the set of rational numbers including non-whole numbers.

This set includes all numbers that can be expressed through fractions, so now the solution is “x=0.5”. Then we still have the set of irrational numbers, that includes all numbers that cannot be expressed as fractions, such as Pi, Phi or other number that do not have repetitions after the comma. All these sets of numbers together make up the set of real numbers, which are basically all numbers between minus infinity and infinity:

Now we have a horizontal axis with all numbers we know – the set of real numbers. And we add a vertical axis that represents the imaginary numbers. The name “imaginary” has developed over time, since it lies outside of the “real” numbers, but there is actually nothing imaginary or mystical about it. Simply put, it is another axis of “real” numbers. A complex number on the 2D-plain has a part real value and and part imaginary value and is written as a pair of values, for example (3,2i) and is hence a pair of two “real” values like on a normal coordinate system.

The only difference is a special multiplication rule that allows “i” to be “i=?-1”. “i” is the so called imaginary unit and is on the vertical axis equivalent to 1 on the horizontal axis. This imaginary value i is not “countable”, but we can use it and do mathematical operations just like any other number or any other variable. Now any number can be put as a dot across this 2-dimensional plain. This is the set of complex numbers which includes all real numbers and the imaginary numbers that lies outside of the set of real numbers.

Now the magic happens, when we put complex numbers into the most simple formula like “z=z²+C”, where of course this is already an iterative function. That means we start out with a value for z and C and whatever comes out is being used in the same formula for z again and so forth. That way we create for every number we start with a chain of values.

Now when iterating a function this way up to infinite times, the numbers have two possibilities: either they tend towards 0 or they tend towards ?. Now when putting all this into a graphic, for those numbers going towards 0 we give them a black color and for those numbers going towards ? we assign them a color. This is how the mandelbrot image is created. Where as the fractal structure is exactly at the border between those numbers going to 0 and those going to infinity.

Now the concept of extending the boundaries or sets of numbers can be continued. One might ask what numbers can be outside of the set of complex numbers? To solve the equation of “x=?i” we need to extend the 2D-plain into a 4-dimensional hypersapce.

This is how sets like the “quaternions” (4 dimensions), “tessarines” or “octonions” (8 dimensions) are made. Now to avoid confusions, we do not talk about the 3 dimensions as space and the 4th as time. We talk about space-dimensions only. So how to imagine what a 4-dimensional space is? The way of operation to go from a 1-dimensional line to a 2 dimensional plain to a 3 dimensional space is simply repeated. A 4th axis is added and the operation is repeated onto the 4th space dimension. This principle can be repeated to create x-dimensional hyperspaces.

Now if putting a dot anywhere into this 4-dimensional space, we get a number that has 4 values, for example (2,i,j,k), whereas “i,j,k” each represents one value from each of the 4 axis. For mathmatical operations such as mulitplications, matrices are used. The image below shows a 4-dimensional hypercube.

Images creative commons from wikipedia.com

Now one might ask, what this strange math is good for and this is a good question. I am no mathematician, but I know that for example complex numbers are used for calculations in astronomy and for finding patterns in chaos theory. And hypercomplex numbers are used for example in mechanics where dynamics of machines with 3 joints can be predicted.

The most extreme idea that I know of is the E8-model related to quantum physics. This mandala like shape consist of 248 space dimensions. In quantum physics scientists have discovered that sub-atomic particles often appear in groups of 3 which was unexplainable. When the E8 model came out, they found out that by simply spinning it, the intersections would reveal the constellations of 3 in a very natural way, describing the behavior of the sub-atomic particles. So all this shows us that for some solutions the boundaries need to be extended and the solutions can be found in a larger context – outside the box.

And of course hypercomplex numbers are good for art!

A while ago I had the chance to talk to a math professor and I asked him a lot of questions about this topic. He gave me some very interesting insights on how the complex and hypercomplex numbers work. Soon it became clear that the extension to the additional axis was actually a man made definition to solve certain math problems. So at the end I asked him whether the complex and hypercomplex numbers, hence the fractals are actually a man-made construction or rather a discovery of something natural or universal that is already there.

His answer was really intriguing to me – he said, this is one of the oldest debates in the experts world of math where representatives of both sides argue against each other. So there is no clear answer to that. He himself believes, when looking at the images, that we have truly discovered something universal.

This category of fractals includes all that are made based on geometric operations. That means that this kind of fractals can be created without numbers and formulas for example with a pen on a sheet of paper. Of course when generating them on a computer the geometric operations need to be translated into numbers and formulas for the computer to operate. The so called iteration function systems (IFS) is a standard class of software that was developed in the 1970-80s. Some of the most appealing fractal programs for artistic use have emerged out of this development.

One of the most popular fractals is the Sierpinski triangle. Using this example we can easily demonstrate a geometrically created fractal. So obviously for this operation we need no more than a pencil and a sheet of paper and no math or formula is needed. The Sierpinski triangle can be constructed in many ways. I have come across at least 4 different methods that lead to the same result.

Another good example for geometrically iterated fractal would be the “Koch snowflake”. The operation is also very simple and self-explaining.

A specific class are fractals that are dot or pixel iterated. The smallest building block here is a dot. The iteration functions systems and also contemporary software like Apophysis or J-Wildfire are based on that principle. Due to their striking look that is quite different from the formula based fractals these images are categorized into an extra class of the so called “flame fractals” or “fractal flames”. To understand how such complex images can be created, I would like to show it using an analog example with a sheet of paper, a pen and a dice.

Creating the Sierpinski triangle on a sheet of paper using the IFS principle. Everyone can do it!

So first of all we point out the corners of a triangle on a sheet of paper. We give the first point the name “1,2”, the second point “3,4” and the third point “5,6”. Now we start off with placing a dot anywhere between the 3 corner points – that is our starting point.

Now we toss the dice and if for example we get a “2” then measure the distance between our starting point and the corner point named “1,2” and place a new dot at exactly half of the distance. This new dot is our new starting point and we repeat that operation, whereas we always use the corner point that matches the tossed dice. We can also add color by using a different color for every corner point.

Now by slightly changing the operators by adding a rotation to 2 attractors, the Sierpinski triangle transforms and we get organic looking structures.

Besides “linear” there is an increasing amount of non-linear operators. Now an unlimited number of attractors can be added and each can have a free combination of transformers and values. The possibilities are literally infinite, though it takes 2 attractors only to create obvious fractal structures.

The kind of images these flame fractals produce are more dreamy than the ones created using hypcercomplex numbers. In both we find similarities to countless structures known from nature.

Though most of the shapes that these fractals produce are not to be found in nature and we may ask whether they visualize something invisible, but real, existing beyond the manifestations of nature as we know.

Want to see more of Flame Fractals? Visit my Gallery here:

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