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      "title": "Allegory of the cave; A first person perspective and modern (crypto) reference",
      "body": "Here is my take on a first person perspective of Plato's 'Allegory of the cave'. For reference, in case you haven't read Republic, here is a picture of the layout of Plato's cave; \n![platoscave.gif](https://steemitimages.com/DQmZP63kezEKM2qkNuof7CPQu1bD9xkmPdudXXRfAshDcGs/platoscave.gif)\nI like to think that the ideals of this philosophical text are directly related not only to education, but the crypto community also. In Plato's original text, the allegory is referencing the manipulated minds of society and how we refuse to accept truths, preferring to grasp what is familiar rather than what is reality. In a sense, this can be depicted as people who solely rely on fiat currency and refuse to accept crypto-currency as the future of monetary value. Nevertheless, here is my first person perspective of Plato's literature. If you haven't read the original I would highly recommend it, it is a very insightful piece. I added a quote at the end since it seemed relevant ;)   \n\nThe envisagement of beginnings, and that of ends is much the same. One could say that darkness is a mutual standpoint between life and death. And this beginning, this life, is not an exception. This life starts as any other life, it starts with sensation, with air meeting supple flesh for the first time. It starts with shrill cries and confusion. It starts with darkness. But this life is not just any other life. There is darkness and hardness and coldness, there is no living. \nA life grows with the time. The time that smells of cold and dark is changing shape. The time is familiar and the time is home. \nNought but a wall stares back. A head that cannot move, that is restricted, and yet it knows no movement. Legs that cannot walk, that don’t know how to walk and useless arms that slump from bony frames. They are not needed. The eyes and the ears, the nose and the mouth is enough to survive, it is enough for life. \nSometimes the walls talk. The sound seems to bounce upon every inch of the ear canal, a too loud sound that is not loud enough. It is meaningless, simply white noise that is frustratingly comforting. It is the noise of the gods. \nThe walls that talk, can also dance. The darkness dances and speaks and makes shapes with its body. And we give the shapes names, so as to identify them. A long shape with arms we call a ‘tree’. A square shape we call ‘book’, and a shape with many legs we call ‘dog’. The shapes speak to us, they teach us what is right and what is wrong, they teach us to live. \nTime and life move together, constantly fighting but never leaving the other behind. They do not exist without one or the other, they are a pair. \nThe calmness that accompanies familiarity is a blessing. The walls do not move, only the darkness which dances upon its willing surface. The walls are familiar, the shapes are familiar, and the sound is familiar. This is the life that is known, and none else is needed or desired. But a life is not truly living. \nThe life is untimely ripped from the grasp of familiarity and thrust into the unknown, the undesired.  A wall that talked and sang and danced is not living at all. A violent shape that holds pigment and texture and depth is moving like no other shape could. They call it fire. The familiar shapes on the wall mimic unfamiliar shapes that have eyes and ears and mouths and noses. The unfamiliar shapes produce noise, the noise of the gods. But they are not gods. They are the same as this life. \nThe scene that has been set for this life is a dark encampment which only the light of fire can touch. This life is the embodiment of lies and deceit. This life does not know how to live. The puppeteers, as this life has declared them, mould the dark shapes against their will. Their bodies move as this life never has, and they make trees and books and dogs. They make reality, the living. \nThe life is horrified as reality crushes all that was known and desired.  Consent was not given. The life wants to flee. The reality is unwanted, it is hated. This life wishes to return to the sweet, innocuous realm of ignorance. Instead, further torment ensues. \nThe life’s realm, the imaginary bliss, is named a ‘cave’. The passage from security and serenity opens to blindingly immense shapes. One from above that could truly be named God, is overwhelmingly powerful. The life ought but shiver under the rule of true purity and realness. That which far outweighs the magnitude of any imagination. For once, there is no hardness and coldness and darkness, only light and warmth and a gentle touch. But it is too much.\nThe life desperately releases itself from the grasp of its perpetrator. It staggers down a narrow path, away from the metaphysical holiness, and delves into darkness once more. It descends past the roaring flame and the barbarous puppeteers. It doesn’t cease its descent until it has reached rock bottom. There is nowhere else to run. All that is left is large sparse of dark wall with shapes dancing in every corner. This is all the life needs, all the life wants. The hardness, coldness, darkness of descent is a comfort that the life indulges. The life stays, gazing upon the wall and rejecting the falsehood of reality from its mind. The life never questions, never leaves, ever again. \n\n“facilis descensus Averno;\nnoctes atque dies patet atri ianua Ditis;\nsed revocare gradum superasque evadere ad auras,\nhoc opus, hic labor est.” – Vergil, Aeneid",
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      "body": "I'll keep that in mind. A real world example of this is light beams hitting a paraboloid surface and reflecting off on an angle equal to the angle between the focal point and the incident beam relative to the parabola.",
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      "body": "Good stuff cabbage, would be beneficial if the explanation was  in 'laymans' terms as well as scientific. Also an example of how we interact and experience in day to day life.",
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      "permlink": "proof-the-reflective-property-of-a-parabola",
      "title": "PROOF: The reflective property of a parabola",
      "body": "For this proof, the parabola y^2=4ax will be used, where point P = P(at^2, 2at)\nTo prove the reflective property of the parabola, it must be proven that RF= FP as this would form an isosceles triangle, meaning angles PRF and RPF are equal. Consequently, angle RPF must be equal to the angle between the reflected beam and the tangent at P as they are vertically opposite angles. The angle between the reflected beam and the tangent at P is equal to the angle of the line through P parallel to the x axis because the angle of the line through P and angle PRF are alternate angles, and it is known that angle PRF is equal to angle RPF, which is also equal to the angle between the reflected beam and the tangent at P. Hence, reflected rays will appear to emanate from the focus because FP directly corresponds to the outwards appearance of the beam due to the equality of the angles. If point P were to shift along the parabola, the line FP would also move, altering the angle RPF and consequently the angle between the reflected beam and the tangent at P. \n![Diagram 1.jpg](https://steemitimages.com/DQmYh9jnSa6Syz9SNKXHLY4HDccz5uUHWc4sY5mEfiqR1bG/Diagram%201.jpg)\nPA is the distance perpendicular to the x axis, the y value of P, = 2at\nOA is the x value of P= at^2\nOF = a, therefore FA = (at^2-a)\nFP= FP (unknown) \nAngle PAF = 90 degrees, FP = hypotenuse \nc^2 = a^2+b^2\na = PA = 2at , b = FA = (at^2-a)\nc^2 = (2at)^2+(at^2-a)^2\nc^2 = 4a^2t^2 + a^2t^4 - a^2t^2 - a^2t^2 + a^2\nc^2 = 2a^2t^2 + a^2t^4 + a^2\nc^2= a^2(t^4+t^2+t^2+1)\nc^2= a^2(t^2+1)^2\nc= √ a^2(t^2+1)^2\nc= a(t^2+1)\nc= at^2+a , FP = (at^2+a)\n\nProve that FP= RF\nRF= gradient of tangent at point P\n\nm= (y2-y1)/ (x2-x1)\nm= the derivative of the equation of the parabola which equals the slope of the line (change in y over change in x), hence m=y’ \n\nUsing implicit differentiation; \ny^2= 4ax \ny’= d(y^2)/d(y)\nd(y^2)= 4a \nd(y)= 2y \ny’= 4a/2y\ny’= 2a/y\n\nRF= x1 + a\n\n(2a/y)= (y2-y1)/ (x2-x1)\ny2= 2at , y1= 0 , x2= at^2 , x1= x1\n(2at-0)= (2a/y)(at^2-x1)\ny= (2a/y)(at^2-x1)\ny.y= 2a(at^2-x1)\ny^2= 2a(at^2-x1)\n4a^2t^2= 2a(at^2-x1)\n(4a^2t^2/2a) = (2a(at^2-x1))/2a\n2at^2= at^2-x1\nx1= at^2-2at^2 \nx1= -at^2\nRF (magnitude) = (at^2+a)\nRF= FP\n(at^2+a)= (at^2+a)\n\nTherefore RF= FP and the triangle RPF must be isosceles. This indicates that angles PRF and RPF are equal, as an isosceles triangle has two equal sides and two equal angles. \nLight appears to emanate from the focal radius because angle RPF and theta x are vertically opposite and equal as a result. Angle Q and PRF are also equal as they are alternate angles. Angle PRF is known to be equal to angle RPF which is equal to theta x, hence angle Q is equal to theta x and angle RPF as a result. This means that the angle of the incident beams at Q will be equal to the angle of the reflected beams at theta x. Although the beam will appear to emanate from the focus because angle RPF also directly corresponds to these angles and as FP moves, the angle RPF will change, resulting in an equal change of angles Q and theta x. Essentially, the reflected beam could also be seen as an extension of FP outside of the parabola.",
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