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Fractals: The beauty of order within the Chaos

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From the veins of a single leaf to a whole of a rainforest, at first glance everything that’s around seemed random and chaotic. When scientists and mathematicians looked at it no one was particularly intrigued by this unexplainable chaos. This led to everyone curling back to the world they created and could explain. Circles, lines, triangles etc were simple, easy and something that we could easily put into mathematical words as equations. Our artificial world was built around it. But as we went into explaining the true shapes of nature, most of it won’t fit into this simple geometry.

Then in 1980 a mathematician named Benoit Mandelbrot, then working in IBM discovered the Mandelbrot set. His equation looked something like this:

z->z ² + c

 

The equation took an initial number and gave an output and again used the output as initial number. The equation was iterated millions of times. The result was plotted and the figure above is what he got from the plot.

 

This psychedelic looking image when zoomed in on, repeats itself and never ends. The randomness of the simple equation when first plotted with few numbers, after thousands of iteration repeats itself and produces such complex pattern. He named it the “Fractals”, as the geometry was fractured but the shape had order and can be produced by simple equations. This opened up a whole new world of natural mathematics.

From graphically designing mountains, flowing lava in movies to studying the natural phenomenon like weather patterns, building compacted antennas; Fractals geometry had numerous implication. When we search around our real natural world, the pattern of delta formed by rivers, the branching of trees, the inside of our lungs, the beating of our heart and most of the nature is found to be following this fractal geometry.

After watching the world around us through the fractals, this chaotic and random nature has a much simpler order and beauty within the seeming chaos. Pleasing to eyes and satisfying to the soul these fractal patterns, irrespective of naturally occurring or computer generated resonates perfectly with every human to some degree. This leads some to believe that maybe even our mind itself is a fractal, as infinite as it is and well compacted with a skull.

Lichtenberg figure is fractal and closely resembles the thunder-lightning.

 

The shell shows a fractal pattern

 

Veins in leaves are fractals

We are all just an arbitrary constant looking for a value that fits the equation of our life. Guess I will find one someday.

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Arts & Literature

Philosophical Enlightenment: Plato’s Allegory of the Cave.

This article talks about Plato’s allegory of the cave which heavily influenced the thinkers in later centuries and became one of the causes of European enlightenment or, the Age of Enlightenment.

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Enlightenment in western philosophy was given a platform by European thinkers such as Voltaire, Diderot, Montesquieu, Hume, Adam Smith, Thomas Reid, Immanuel Kant, and Christian Wolff among others. Philosophical enlightenment during the 18th Century Europe was thought to be an answer to dramatically improve Human life. This enlightenment was born out of the scientific revolution of the 16th and 17th century. The scientific revolution broke age old thinking patterns and gave birth to realism.

Philosophical enlightenment as a concept urges its followers to aspire for intellectual progress. Such progress could improve individual lives and also the society as a whole. Thinkers such as Hobbes, Descartes, Spinoza, Locke and others stressed on “logical arguments” which could be achieved through research.

This article talks about Plato’s allegory of the cave which heavily influenced the thinkers in later centuries and became one of the causes of European enlightenment or, the Age of Enlightenment.

Plato’s cave is a chapter in The Republic (514a-520a) which explores the effect of education and the lack of it on our nature. Allegory of the Cave is written as a conversation between Socrates and Glaucon (son of Ariston and the major conversant with Socrates in the Republic).

Plato writes the allegory to show the importance of philosophical education. In this allegory of the cave, Socrates tells Glaucon to imagine a cave where there are a group of prisoners. These prisoners were born there and are chained since they were children (the hands, feet and neck have all been chained). For the entirety of their lives, these prisoners can only see what is in front of them – which is the back wall of the cave.

Behind them at some height, there is a blazing fire. Between the fire and the prisoners is a low wall, behind which, there are men who carry various objects around. The fire casts the shadows of these objects directly on the back wall of the cave. These shadows are the only things the prisoners ever see and thus they think that the shadows are themselves the objects.

When the people who carried the objects spoke or engaged in conversation with each other, the prisoners could have thought that the sound came from the shadows – making the prisoners strongly think that the shadows were real.

“Then in every way such prisoners would deem reality to be nothing else than the shadows of the artificial objects.” (Socrates to Glaucon, Book VII, The Republic by Plato)

The prisoners talk to each other regarding the shadows and their meaning of them. They feel that the shadows give them wisdom to learn about life. They form dialogues and are happy that they have found meanings in life.

When one of the prisoners is freed from the shackles, the first thing he finds is that the shadows are similar to the objects that were being carried. But the objects that were forming the shadows had so much more to them.

The prisoner is now taken outside of the cave and is in front of the great sun. He sees the sun and is temporarily blinded by the light and thus suffers pain in his eyes.

“…, what do you suppose would be his answer if someone told him that what he had seen before was all a cheat and an illusion, but that now, being nearer to reality and turned toward more real things, he saw more truly? And if also one should point out to him each of the passing objects and constrain him by questions to say what it is, do you not think that he would be at loss and that he would regard what he formerly saw as more real than the things now pointed out to him?” (Socrates to Glaucon, Book VII, The Republic by Plato)

At first, he is only able to see the shadows that have been formed on the ground, because the shadows are familiar to him and the sun has caused pain in his eyes. He is slowly able to see reflections and finally shifts his focus from the shadows of trees and animals on the rocks, the reflections in the water – to the “real” objects that are creating them.

He sees the colors of the birds and the flowers, the patterns on rocks and the intricacies of tree barks.

He sees the objects as they are – in reality. This makes him confused, for he was habitual to the shadows. He denies the reality of the objects at first, but then questions if the fire that is burning so bright behind the objects actually created the shadows. He now starts questioning more. But this time his questions are more guided by the complexities of things “as they are”.

“…there would be need of habituation…to enable him to see the things higher up. And so, finally, I suppose, he would be able to look upon the sun itself and see its true nature, not by reflections in water or phantasms of it in an alien setting, but in and by itself in its own place.” (Socrates to Glaucon, Book VII, The Republic by Plato)

The prisoner finally understands that the sun is the cause of everything. He figures out that the sun causes light, sight and then reveals the true nature of the objects.

He then sees the world grow darker – the sun has gone and the moon is being seen. He sees the patterns of stars in the sky. He starts questioning about the vastness of the universe that surrounds him.

He sees objects in their own place and not just as reflections.

He finally starts contemplating about the sun as it is. He starts seeing objects first as they are and then starts to reason about them.

He sees that the shadows are nothing but illusions created due to the direct contact between realistic objects and the sun.

He can see clearly now that the world is not finite. He sees that knowledge is not finite.

He is astonished by this new found knowledge and searches more for truth. He becomes realistic and is now unresponsive towards the shadows which previously governed his thought patterns.

The prisoner has now risen above the wisdom of the cave.

He is excited to share his knowledge with the prisoners he has lived with.

As the prisoner returns back to the cave, he finds out that his visions have been changed. He can now see less in the dark and is less responsive to the shadows that had governed his life. It takes some time to adjust to the darkness.

The other prisoners laugh at him for having his visions destroyed by going out of the cave. They laugh at him because – he now no longer shares the same belief. They then start becoming angry at him for going outside the cave – he has become blasphemous. The prisoners think he should be executed.

He is now a stranger in a pack that he was a part of. They despise him – for he is an offender.

For Plato and for many philosophers after him, the allegory of the cave is the story of enlightenment, a story of the time before philosophy.

We as humans are born in the shadow and many of us live the rest of our lives in it, without ever questioning the actual truth. Many of us never question the reason for shadows and blindly follow what is shown to us.

We are forced to experience things as the desires of the society (the cave) we live in. We derive our understanding of cultures, traditions, religions, romance, fame, power, politics and other concepts (shadows) as per the desires of the group (the men carrying the objects behind the wall).  Humans are pleasured by the false conception of these things. They have been created by the society and reflect as shadows on the walls of our mind. These phantoms or shadows are what keeps us from being enlightened or “Conscious”.

When we get out of the cave (that is our society) and see the light (the world of ideas, thoughts and reality), and finally start questioning, we are primarily mocked for being cynical. This could discourage many who have seen the light and thus send them back to the cave that they were liberated from. But some of the humans revolt against the stronghold of the shadow.

But Plato discourages us to share our “new found” knowledge, bluntly with the others. This could anger others and surely get us into troubles.

Socrates was executed by the Athenian government under accusations that he was corrupting the youth with false ideologies and that he was guilty of impiety (not believing in the gods of the state). He was killed by the decision made by the majority of fools. Plato hated Athenian democracy because he saw the possibility of cave dwellers having majority power to take false decisions. These decisions were based upon the concept of the shadows and did not include higher consciousness.

But Plato still insisted that willingness to step outside of the cave is not enlightenment alone – and that there must be a willingness to return back and educate those who are yet to see the sun. However, this is to be done carefully.

This process of carefully educating the masses is known as the “Socratic method”. Those who have seen the sun should not be indulged into forcing people to see “what is”, but rather give their arguments and stimulate dialogue from the listeners. This would generate critical thinking and eliminate hypotheses by eradicating those ideas that lead to multiple contradictions. This is Plato’s ideal method of dialectics.

Plato insisted that “wisdom starts with owning up to ignorance”. We should confess that we actually know nothing. This would make us open to understanding things and listen to different views of people regarding these things. For, not knowing, according to Plato, was paradoxically a kind of knowledge in itself. This has been referred to as the “Socratic ignorance” which paradoxically is also known as “Socratic Wisdom”.

We need to understand that we indeed do not recognize how life is supposed to be. We should understand that we do not comprehend the ways some concepts like politics, government, relationships, and others work. We need to lay out our ideas and welcome others’ to eliminate contradictions. One way of eliminating contradictions could be through logic. Logical interpretations of our ideas, and, citing resources and existing patterns of behavior of humans when faced with similar situations should be done.

The allegory of the cave describes the metaphor of life and shows us the relation between education and truth. For Plato, the major function of education is to dispose us towards the truth and not just show them to us. Education should be able to make us face the truth.

For Plato, education was awareness of “being”, along with, the awareness of other objects. Education is the source of enlightenment.

The stages of enlightenment in the allegory of the cave are:

1. Owning up to ignorance.

2. Reaching knowledge.

3. Building the capacity to understand “the good” (the sun).   And, finally,

4. Raising questions and fostering dialogues.

Those who have been educated are the teachers (Socrates) who do not themselves bring the truth out, but rather by means of questioning create a platform for dialectics and then cause the learner to rationally speak out and finally figure out the truth.

Plato further says that one has to have a desire to learn. Those who do not desire cannot be forced to learn. Thus, acquiring knowledge is a transformative process that requires painful movement. In order to learn, one has to be willing to turn their chained bodies towards the light, one has to be willing to turn around their soul and be willing to change their ideas.

We are all prisoners of the ideologies fed to us, by our society, by our state. Many of us do not dare to seek knowledge outside the cave. Many of us remain prisoners for the rest of our lives, but at times there are those who seek to learn more than what has been offered. These individuals who are willing to learn are open to new dimensions of truth that they had never before encountered.

An enlightened person for Plato means that this person is willing to learn and stops clinging to ordinariness and pleasures and is willing to rise above and look upwards.

In conclusion

Enlightenment is not something that opens portals to secretive universal knowledge, but is rather what gives rise to inquisitiveness. This inquisitiveness to learn about “the good” (the source of reality) fosters individual development. Enlightenment is thus, a process of understanding the difference between the illusions that create a facade and the reality behind those facades.

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Word Of The Day

Word Of The Day: Tump

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Curated by: Srisha Poudel

Definition 1

a small rounded hill or mound

(eg: There is a little tump over the pond.)

Definition 2

to tip or turn over especially accidentally

(eg: He tumped over.)

Synonyms

accident, sudden, bush

Antonyms

intentional, huge

Nepali Meaning

उल्टिनु


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Word Of The Day

Word Of The Day: Tousle

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Curated by: Srisha Poudel

Definition 1

to make untidy

(eg: Nathan’s tousled head appeared in the hatchway.)

Definition 2

to mess up

(eg: He always had time to tousle a young boy’s hair .)

Synonyms

untidy, messy, unkempt, disordered, disarranged

Antonyms

arrange, unsnarl, untwist, disentangle, tidy

Nepali Meaning

सफा सुग्घर नभएको


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