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The Ontological Beauty of the Mandelbrot Explored and Explained

Updated on November 24, 2015
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The author, Mario Buildreps, is a graduated engineer, and specialized in showing you the same topic from a different angle.

Details in One and the Same Mandelbrot

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This beautiful tornado-like shape, that is part of the Mandelbrot set, is located in the complex plane. The tornado continues to go on forever, with an infinite amount of other tornado's around it. It's a whole universe on its own.This curl, that looks somewhat like a tentacle of an octopus, keeps on curling inward forever. You can keep on zooming in for eternity, while the shape becomes more complex. With every step the required computing power increases.This part of the main shape sits on top, near +1 on the Imaginary Axis. It can be considered as an anti-matter shape.An infinite amount of Black Holes ordered around one huge Black Hole in the middle?This shape turns up in the 'anti-matter universe', with an infinite amount of basic Mandelbrot shapes in it (the black spots).This basic Mandelbrot shape is everywhere to be found. This one is on the negative Real Axis near -2. It contains infinite amount of detail. This part of the Mandelbrot deeply zoomed in shows something that looks very similar like a neural network. This deeply zoomed in part of the original Mandelbrot shows again the original shape but now somewhat distorted. The amount of iterations is set on 5,000.The same picture as before but now after 100,000 iterations, showing a huge amount of detail.This detail shows the basic shape turned under a very specific angle. Around it there are candlestick-like structures with 3 arms. There are also other candlesticks with 4, 5, 6, 7 and even 8 arms. Every structure has infinite depth.
This beautiful tornado-like shape, that is part of the Mandelbrot set, is located in the complex plane. The tornado continues to go on forever, with an infinite amount of other tornado's around it. It's a whole universe on its own.
This beautiful tornado-like shape, that is part of the Mandelbrot set, is located in the complex plane. The tornado continues to go on forever, with an infinite amount of other tornado's around it. It's a whole universe on its own. | Source
This curl, that looks somewhat like a tentacle of an octopus, keeps on curling inward forever. You can keep on zooming in for eternity, while the shape becomes more complex. With every step the required computing power increases.
This curl, that looks somewhat like a tentacle of an octopus, keeps on curling inward forever. You can keep on zooming in for eternity, while the shape becomes more complex. With every step the required computing power increases. | Source
This part of the main shape sits on top, near +1 on the Imaginary Axis. It can be considered as an anti-matter shape.
This part of the main shape sits on top, near +1 on the Imaginary Axis. It can be considered as an anti-matter shape. | Source
An infinite amount of Black Holes ordered around one huge Black Hole in the middle?
An infinite amount of Black Holes ordered around one huge Black Hole in the middle? | Source
This shape turns up in the 'anti-matter universe', with an infinite amount of basic Mandelbrot shapes in it (the black spots).
This shape turns up in the 'anti-matter universe', with an infinite amount of basic Mandelbrot shapes in it (the black spots). | Source
This basic Mandelbrot shape is everywhere to be found. This one is on the negative Real Axis near -2. It contains infinite amount of detail.
This basic Mandelbrot shape is everywhere to be found. This one is on the negative Real Axis near -2. It contains infinite amount of detail. | Source
This part of the Mandelbrot deeply zoomed in shows something that looks very similar like a neural network.
This part of the Mandelbrot deeply zoomed in shows something that looks very similar like a neural network. | Source
This deeply zoomed in part of the original Mandelbrot shows again the original shape but now somewhat distorted. The amount of iterations is set on 5,000.
This deeply zoomed in part of the original Mandelbrot shows again the original shape but now somewhat distorted. The amount of iterations is set on 5,000. | Source
The same picture as before but now after 100,000 iterations, showing a huge amount of detail.
The same picture as before but now after 100,000 iterations, showing a huge amount of detail. | Source
This detail shows the basic shape turned under a very specific angle. Around it there are candlestick-like structures with 3 arms. There are also other candlesticks with 4, 5, 6, 7 and even 8 arms. Every structure has infinite depth.
This detail shows the basic shape turned under a very specific angle. Around it there are candlestick-like structures with 3 arms. There are also other candlesticks with 4, 5, 6, 7 and even 8 arms. Every structure has infinite depth. | Source

Ever Seen a Mandelbrot Set Before?

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Basics of the Mandelbrot Fractal

Fractals, and especially the Mandelbrot set, are recursive structures in the complex plane.

Every mathematician marvels at the amazing beauty and complexity of the Mandelbrot, and of course of fractals in general.

Ain't she beautiful? Just enjoy them. This article will get too difficult for many of us.

Don't try to wrap (or warp) your head around this mathematical discovery of the Polish mathematician Benoît Mandelbrot. Not one mathematician really understands this, besides as being just another abstraction.

These shapes are insanely complex, while they are derived from a ridiculous simple looking formula: Zn+1 = Zn2 + C. Where Z is a complex number, or a point in the complex coordinate system. The first recursive shapes were drawn on the early PC's of the late 70s.

The Mandelbrot Set is shaped from an infinite amount of so called Julia's. The offset from the origin changes its shape and forms. That's why it's officially called the Mandelbrot Set. It's a collection of Julia's. Every Julia has infinite depth on its own. It is real mad science.

By the way: Ontology is the study of the nature of being and the nature of true reality.

Fractals in Nature

What makes fractals even more interesting is that they are the best existing mathematical descriptions of many natural forms, such as coastlines, mountains, ice crystals, parts of living organisms, and even complete organisms. How is that even possible? Is the universe mathematical in its true nature?

The most significant features are:

  • infinite intricacy,
  • zoom symmetry,
  • complexity from simplicity,
  • fractional dimensions.

It seems as if the artificial and perfect symmetrical fractals in real life become asymmetrical by countless interferences from the natural surrounding, making them even more complex and more seemingly chaotic.

The question which is asked too less: how is it possible that mathematics describes nature so well? Is it because everything is ultimately mathematical? Is it even thinkable that it would be otherwise?

Modern cities are madhouses, aimed at obscuring the true reality of existence. Being in pure nature helps to understand the Truth of existence.

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This image shows how after countless iterations from a simple asymmetrical branch a complex treelike shape is formed.The complex branching behaviour of lighting can be found in the Mandelbrot Set.The Romanesco Broccoli has a typical fractal shape. The patterns on the large cones return on the smaller cones. The Barnsley Fern is a fully mathematical object created by matrix iterations. Snowflakes are fractal objects as well.
This image shows how after countless iterations from a simple asymmetrical branch a complex treelike shape is formed.
This image shows how after countless iterations from a simple asymmetrical branch a complex treelike shape is formed. | Source
The complex branching behaviour of lighting can be found in the Mandelbrot Set.
The complex branching behaviour of lighting can be found in the Mandelbrot Set. | Source
The Romanesco Broccoli has a typical fractal shape. The patterns on the large cones return on the smaller cones.
The Romanesco Broccoli has a typical fractal shape. The patterns on the large cones return on the smaller cones. | Source
The Barnsley Fern is a fully mathematical object created by matrix iterations.
The Barnsley Fern is a fully mathematical object created by matrix iterations. | Source
Snowflakes are fractal objects as well.
Snowflakes are fractal objects as well. | Source

Recursion and Iteration

Recursion or Iteration are the core engine of fractals. This article won't zoom in on the differences between the two. Both engines are focussed on finding a solution within a certain amount of loops. The moment that the loop is allowed to stop is set by the programmer or user of a program.

There is a distinction to make between converging and diverging iterators. The former runs in circles towards a certain point, while the latter runs to infinity, and is considered as a 'bad' iterator.

A cancer cell can be regarded as a 'bad' iterator. A part of the formula (the DNA) has mutated, turning the cell into the wrong iteration mode - a diverging mode.

In the table is a simple example used to show what iteration is. Fractals are based on the same principle.

Xn+1 = Xn / (Xn + 1) ---> Let's make Xn = 1.

Example of an Iteration Cycle

Iteration cycle
Xn
Xn+1
1
1
0.500
2
0.5
0.330
3
0.33
0.248
....
 
 
0
0

The result of this iterator is that it goes to zero in an infinite amount of steps. Iterators are often used in all sorts of software to find a solution for a specific problem.

Nature uses iterators all the time to find a solution. With our modern equipment we use iterators many times a day as well without knowing it.

So, they appear to be very useful. They are the drivers of existence.

A Walk Through Mandelbrot

The Complex Plane

The difficult part of the Mandelbrot set is that the plane covered by this formula is called the complex plane. An important factor in complex numbers is the definition that i2 = -1. In the world of the real numbers is taking the square root of -1 regarded as impossible. In complex numbers is the square root of -1 simply defined as i. Complex numbers, this i, allow all polynomial equations to have solutions, and gives a powerful tool for working with rotations.

What no one ever tells you, is that i is nothing else than time. Many mathematicians have no clue what i actually is. They regard it as an abstraction, while in fact everything is real in mathematics.

Complex numbers made Quantum Mechanics possible and sky rocketed physics, creating a new world of possibilities. Even many advanced prediction models for weather and stock markets use complex numbers. That they are not accurate enough is caused by a lack of understanding of the real underlying mechanisms, and by a lack of computing power and grid.

Nevertheless are imaginary numbers still regarded as 'unreal' by most scientists. They cannot be detected by the senses. This shows a deep misunderstanding of the real thing.

Physics abounds in formulas with complex numbers and is unreasonable successful in explaining our world. The question that never has been answered how this comes, besides the idea that imaginary numbers are just unreal. All formulas deliver miraculous results in the 'real' world. Are these results then imaginations too?

Is there a world above and beyond that governs matter? Yes, there is.

Everything is energy and that's all there is to it. Match the frequency of the reality you want and you cannot help and get that reality. It can be no other way. This is not philosophy. This is physics.

— Albert Einstein

Zero, Infinity and Insanity

Anyone who knows nothing about higher mathematics can still quit now. Or maybe I should recommend to lean back and just to view some of the beautiful pictures, and stop reading.

Browsing through the Mandelbrot makes you fall from one amazement into the other. You can keep zooming in until your computer crashes, because the more you zoom in, the more computing power is required. If your CPU has no sufficient cooling it will shut down or even partially melt. So be careful.

Just to imagine how complex this structure is. If everyone in the world would explore on their computer, a tiny, but different section of the Mandelbrot, they could be doing this for eternity. Even if every computer would be infinitely fast in computing power. And every day everyone could discover a whole new realm full of new shapes and new possibilities.

An infinitesimal section has still an infinite complexity. Isn't that amazing?

Loosing yourself in this shape could turn people into insanity, so a note of caution is at its place here as well.

The Best Kept Secret

As if that isn't enough. Did you know that zero is the most important number of the whole universe? It is the core of existence. We all learned that zero is nothing. But every mathematician knows that 00 = 1. It's not even imaginary. Just tap it in your calculator (0^0). It will verify this. If you don't believe your calculator, there is extensive proof. You can just believe it, since the proof is even harder than this article.

So 00 + 00 = 2. Exactly as 1 + 1 = 2. The operation (00 / 0) delivers infinity (∞). There are even no real numbers. Only just zero exists, and nothing else. Isn't that amazing?

Did you know this is one of the best kept secrets of the most secret societies?

Zero is the number of the soul, the singularity. This is regarded as information that is not suitable for the masses. Isn't that a shame?

The graphical representation of the Casorati–Weierstrass theorem.
The graphical representation of the Casorati–Weierstrass theorem. | Source

Reality is Found Inside

It has even a much deeper meaning. Turning inward (meditation) leads you straightforward to reality.

Turning outward is the illusion. The proof is in the numbers.

Did you know that mainstream science is probing an illusion? Just like in the Matrix.

The knowledge goes even further by saying there are no real numbers. There is only Zero. That is what Patañjali actually meant by saying the world is an illusion.

This illusory world can be confirmed by the Casorati-Weierstrass theorem, that states there is an essential Singularity at Zero. By using this formula, any singularity can transform itself from the origin to any point in the complex plane (the 'real world'), proving that the universe is nothing else than a Singularity. We are singularities in the Singularity.

By turning inward you can do anything you want. The Taoists, Greeks, Egyptians and Yogi's knew this already long time ago.

The last step you need to take is to understand what Pythagoras meant with "all things are numbers". What he actually meant is all things are Zero. A number stands for a frequency from which our world emerges. All from just Zero.

Did you get this part? Meditation is crucial, and to understand why you do this is crucial as well.

Julia Offset - the Building Blocks of the Mandelbrot

This simple illustration shows what happens when a Julia is shifted over the complex plane. Every little shift causes the Julia to change. The amount of possible shapes are infinite.
This simple illustration shows what happens when a Julia is shifted over the complex plane. Every little shift causes the Julia to change. The amount of possible shapes are infinite. | Source

The Complex Plane

The complex plane where these shapes occur.
The complex plane where these shapes occur. | Source

The Mandelbrot Set in the Complex Plane.

The Mandelbrot Set in the complex plane. The colours assigned outside the Mandelbrot set show how many iterations were required to determine that they are outside the Mandelbrot set. The closer to a black area, the more iterations are required.
The Mandelbrot Set in the complex plane. The colours assigned outside the Mandelbrot set show how many iterations were required to determine that they are outside the Mandelbrot set. The closer to a black area, the more iterations are required. | Source
Euler's formula results in a perfect circle in the complex plane.
Euler's formula results in a perfect circle in the complex plane. | Source

Introduction to the 'God Equation'

This article was probably already hard for many people, but when for the sake of simplicity this part would be skipped, it would even be harder to understand how this theory got at the finish.

Most people have already heard of the idea of modern science that everything is vibration.

Who doesn't know this these days?

But matter is so enormously persistent and real, that it is hard to bypass. When you bump your little toe, you're mercilessly confronted with the reality of matter. And still, the pain exists only in your head. It is a very persistent illusion.

Despite all the physical pleasures and horrors that can happen to us, on microscopic level there's nothing else than vibrations. An infinite amount of them.

Euler's formula, also called the 'God Equation', is an important factor in explaining the deeper aspects of our world where everything is ultimately vibration.

e = cos(φ) + i·sin(φ), which is often famously written as e + 1 = 0 for φ=π. This gives the astounding possibility to state that e = i2. And this makes deep connections possible to all underlying, hidden realities. While we already knew that all numbers are derived from Zero. The 'holy' number.

Can you still keep up? We are going even deeper down, into the best kept secrets.

The Queen of Physics

Quantum Mechanics contains in fact nothing else than formulas of vibration and spin. Vibration is like sound that can be written as a function of sines and cosines, and so is Euler's formula sitting at the core of this Queen of Physics, the most successful scientific theory ever produced by mankind.

And that in a time that the Relativity Theory of Albert Einstein has been proven to be completely false. Relativity doesn't exist, which Einstein as a genius realized at old age. Everything is based on the Absolute, and that is Zero.

There is even no limit to speed. You can travel to the other side of the galaxy in an instant. Why? Because we are ultimately singularities. You only have to know how to tap into that infinite source.

Materialize, Transform or Teleportation

A small hint. Look at the Mandelbrot and try to understand why it can be so hard to reach this singularity. There are an infinite amount of realms, channels, and even complete galaxies to get lost in.

The more matter you gather, the deeper you get lost. The more desires you have, the deeper you are warped into these infinite realms of oblivion. The less balanced you are, the more you are pushed away from your own zero point.

You must find the zero point of your own fractal equation, which is basically the same for most people. Zero is the Omega point. The point of your own divinity.

There happens something very interesting when we throw the God Equation, the perfect circle in the complex plane, into the ultra centrifuge of the Julia iterator. We can create an infinite depth in any shape, from nothing more than just a circle.

The only thing we need is the correct formula for the iterator. Circles, or vibrations, are the basics of all existence. We don't need to parrot what others tell us. It is not impossible to find out yourself, and even more important, to understand it.

It is even possible from this understanding to materialize, transform, decompose, teleport, or create anything you want.

Fractal Fourier analysis provides a method to break complicated signals up into a set of well understood building blocks, in a similar way to how conventional Fourier analysis breaks signals up into a set of smooth sine waves.

— Professor Barnsley

Our Universe is Divided into Four Quadrants

The universe as described by the current state of science is just an infinitesimal part of the real universe. Re=1 is the scientific size of the universe. Im=1 is the speed of light.
The universe as described by the current state of science is just an infinitesimal part of the real universe. Re=1 is the scientific size of the universe. Im=1 is the speed of light. | Source

How Our Universe Evolves

Our universe is one big problem solving computer, iterating at an infinite speed, solving its own formula. The basis is Zero, that raises itself to the power of itself and becomes One.

This is how the Big Bang started. All secret knowledge.

The iterative universe has the consequence that the subjective perception of the inhibitors, which is us, animals, and plants, perceive a world that is 'probed and explored' by senses.

This dualistic arena is necessary to make a constant new synthesis from a thesis and anti-thesis. If there would be nothing to push you off, you would free float for eternity in nothingness. There would be no evolving consciousness. And therefore no life possible.

The free float is only possible for the divine mind, or the God consciousness. It needs no duality anymore to develop its consciousness, and is able to vibrate infinitely close around Zero or go at will wherever it chooses to go.

The probing and data collection through our senses we constantly do, is necessary to make evolution possible. It enables us to evolve from our current state of consciousness into a gradual higher state of consciousness. That is to say, if we don't depend too much on foolish gadgets. Otherwise we will degrade, and finally become extinct.

Can you still keep up?

You might ask what this has to do with this Mandelbrot? Everything. The Mandelbrot shows how our universe gets bended and distorted by endless iterations of perfect Eulerian circles. Einstein called it bending of spacetime by gravity, which is despite his geniality, an outdated understanding of quantum reality.

How Our Universe Curves and Evolves

Like explained in former paragraphs, there's no need for other numbers than Zero to create all other numbers. The Singularity creates a realm on its own by simply raising itself to its own power. This is the first step in consciousness, creating the number One.

After an infinite amount of iterations since the Big Bang, our Universe has become infinitely complex. It is still becoming even more infinitely complex. The meaning of: "the one is in the whole and the whole is in the one" might become more clear now. Remember the key hole (the basic shape of the Mandelbrot) that returns everywhere you look? This is the same principle as the holographic universe. It's just a more clever and clear way of expressing this.

To understand how infinite complexity can become even more complex, there's a simple way to imagine this.

Imagine there is a Hotel Infinity somewhere in the universe. The hotel has an infinite amount of rooms, that is occupied by an infinite amount of guests. So, Hotel Infinity is full. At a certain moment there arrive an infinite amount of new guests at the desk asking for a room. There is a problem, which is brilliantly solved by the management. All guests that currently occupy a room are shifted from their current room to a room with an odd number. All even numbers are now available for the newly arrived guests. That means there are now an infinite amount of even rooms available. This switching trick of the management can continue for eternity.

The same with our universe. It can become every time complexer, while it already was infinitely complex.

A Final Word

What is your place in this infinite complex constellation? Zero is the point to go to, which is seated in your heart. But to make this possible you must be perfectly balanced around your heart. When you reach this point, divinity is at your doorstep.

The trinity religion, science and politics despise these kind of dissidents, who challenge the idiocy of their doctrines, policies, or institutions. That attitude has a price tag you must be willing to pay. The rewards will increase gradually.

In our modern society our development is mainly material, and much less spiritual.

To put it in the mathematical way.

  • The three lower chakra's can be projected on the Real axis (matter)
  • The three highest chakra's on the Imaginary axis (time, dynamics)
  • The heart is in the middle at Zero.

There are techniques that belong to the highest secret teachings to find this Zero point in your heart, which warps you into the divine mode. But these teachings are not published online. They are only passed on personally, together with an extensive training.


© 2015 by Buildreps

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    • Nell Rose profile image

      Nell Rose 19 months ago from England

      Cool! lol! now all I need to do is remember all that, for at least ten minutes before it disappears into my cerebal cortex! lol! seriously cool stuff and I love the patterns in science!

    • Jodah profile image

      John Hansen 19 months ago from Queensland Australia

      Wow, Buildreps, this is totally beyond my mathematical comprehension (anything outside basic addition, subtraction, division and multiplication is :) but that doesn't stop me from marvelling at how spectacular the Mandelbrot Fractal is. That video and pics are amazing. Good work.

    • Buildreps profile image
      Author

      Buildreps 19 months ago from Europe

      Haha, Nell. The fun of these fractals is that the right brain absorbs them like a sponge. The brain loves these patterns. Thanks for your nice comment.

    • Buildreps profile image
      Author

      Buildreps 19 months ago from Europe

      I think almost everyone is dazzled by this almost incomprehensible math, but what David Eagleman (you can watch it on TED talk or on Youtube: https://www.youtube.com/watch?v=4c1lqFXHvqI) proved with his latest invention is that any brain is able to figure out complicated patterns unconsciously. He showed how to make deaf people 'hear'. The brain loves to figure out math patterns, and any brain is able to do this.

    • billybuc profile image

      Bill Holland 19 months ago from Olympia, WA

      I know just enough math to stumble through this article, but the pictures sure are pretty. :) As always, I learned something, and it wasn't painful, so it's all good. :)

    • Buildreps profile image
      Author

      Buildreps 19 months ago from Europe

      I'm all happy it wasn't as painful as bumping a little toe. :) Thanks a lot for reading and commenting, dear billybuc. Have a great day!

    • Carb Diva profile image

      Linda Lum 19 months ago from Washington State, USA

      And I thought this was going to be an article on Jewish almond cookies. You didn't lose me, but my last math course was over 40 years ago so I'm a tad bit rusty. Fascinating.

    • Buildreps profile image
      Author

      Buildreps 19 months ago from Europe

      Haha, that could be easy the case :) Mandelbrot's (grand)mother probably baked many cookies for him. It must have inspired him. Thanks for the comment.

    • Larry Rankin profile image

      Larry Rankin 19 months ago from Oklahoma

      Wonderfully analyzed.

    • AliciaC profile image

      Linda Crampton 19 months ago from British Columbia, Canada

      This is fascinating, Buildreps! I'm going to have to read the hub several times to absorb all the information, but I'll enjoy the process. I've found the idea that the universe is based on math very interesting for some time and want to explore the idea further. Thank you for creating this very thought provoking hub.

    • Buildreps profile image
      Author

      Buildreps 19 months ago from Europe

      Thanks for the compliment, Larry. I'm a lucky man who is able to make much time free for this kind of work.

    • Buildreps profile image
      Author

      Buildreps 19 months ago from Europe

      Your compliment means much to me, Alicia. I feel honoured by your comment. Thank you very much. There is among the many gifted physicists one who entirely agrees with this view, and that is Max Tegmark. If you're interested in this you might want to watch this: https://www.youtube.com/watch?v=_3UxvycpqYo

    • avorodisa profile image

      Anna Sidorova 19 months ago from Russia

      This hub shows how science and religion are interconnected. The statement about reality and illusion, for example. And these Mandelbrot pictures are beautiful as well as the video is great for meditation ptactice. Nature is a wonderfl teacher about the truth, well-being and real beauty. The creator lives in each particle. Great job and thumbs up!

    • Buildreps profile image
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      Buildreps 19 months ago from Europe

      Watching the Mandelbrot videos are indeed great before starting your meditation practice. Just a few minutes at a time. The mind (spirit) loves to process these images much more than anything else. It increases consciousness, intuition, feelings and intelligence. So, that's quite a lot :) Thanks for reading and commenting, Anna.

    • Jason Ironstone profile image

      Jason Ironstone 19 months ago from Canada

      Fascinating. This did make sense to me overall; I've studied related subjects in college so I have context for some of it. I've also found my higher self in my heart, which may correspond to this "zero point". It's like a still small voice.

      I don't fully understand what the Zero point is about, or what I can do with it ... limitless potential, really? I'm curious and wondering what else I might learn.

      Thanks for posting this.

    • Buildreps profile image
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      Buildreps 19 months ago from Europe

      The zero point is the singularity, which means everything and nothing. Zero and Infinity are two sides of the same coin. Finding the perfect balance in your heart is not that difficult as it seems. A good starting point is the practice of Hermes Trismegistus (Thoth). You can find it here: http://hub.me/agFpT. This practice takes at least a year to master, and balances you perfectly around the heart.

      Kriya yoga and the Thoth practice are the basics of the secret teachings to master yourself completely, which unlocks unlimited creative powers. Unlimited creative powers come with an unlimited responsibility as well :) There's not really much space to create unnecessary things like sport cars or huge villa's. That's out of the question. :) These creations warp you back into the Halls of Amenti, so to speak.

    • just helen profile image

      just helen 6 months ago from Dartmoor UK

      I've seen a few Mandelbrots but not one that goes in so far! We were listening to Craig Pruess's Sacred Chants of Shiva whilst watching it, which was very spacey...

      I agree reality is found within, to the extent time and space and the outward manifestation of the universe is an illusion. A Rumi said, the Universe is within you.

    • Buildreps profile image
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      Buildreps 6 months ago from Europe

      Beautiful! Thank you for your comment, just helen. A wonderful choice to listen to this chant while watching the Mandelbrots.

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