A Miraculous Proof (Ptolemy's Theorem) - Numberphile
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. .. I ask, because inversion is certainly an elegant method, but if its use is essentially limited to this proof, it doesn't seem worthwhile to go through all the steps introducing inversion just for one proof. There are other, simpler ways to proof Ptolemy's theorem.
Date: 2022-04-09
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Comments and reviews: 9
Cran
Very cool! If you make a plot of velocity space, the circle of inversion corresponds to the Light-like region, inside the circle corresponds to the Time-like region and outside the circle corresponds to the Space-like region. The inverted points' distance from the origin corresponds to velocities of equal energies for 'real' particles and tachyon particles.
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Very cool! If you make a plot of velocity space, the circle of inversion corresponds to the Light-like region, inside the circle corresponds to the Time-like region and outside the circle corresponds to the Space-like region. The inverted points' distance from the origin corresponds to velocities of equal energies for 'real' particles and tachyon particles.
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LT42
Couldn't Ptolemy 's theorem simply just be taken just be stated that the average of one set of parallels squared added to the average of the other set squared must be equal to the average of the diagonals squared? This seems to automatically tie Ptolemy and Pythagoras both under one catch-all, easily geometrically proven axiom.
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Couldn't Ptolemy 's theorem simply just be taken just be stated that the average of one set of parallels squared added to the average of the other set squared must be equal to the average of the diagonals squared? This seems to automatically tie Ptolemy and Pythagoras both under one catch-all, easily geometrically proven axiom.
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Jesper
It is exceedingly interesting how dependent we (animals) are upon some of these 'geometries' to judge social cues and 'read' things, so that e. g heavy glasses may in fact distort reality, make us less able to utilise some otherwise easy to read things (there are afterall no straight lines in nature)
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It is exceedingly interesting how dependent we (animals) are upon some of these 'geometries' to judge social cues and 'read' things, so that e. g heavy glasses may in fact distort reality, make us less able to utilise some otherwise easy to read things (there are afterall no straight lines in nature)
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SchrodingerBraCat
excellent exposition!
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if we DEFINE the radius of the inversion circle as 1 unit, then -inversion- is exactly the same as 1/z transformation in complex analysis. we can patch a -point of infinity- with the plane and link it to a sphere via sterographic projection in the usual way.
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excellent exposition!
.
if we DEFINE the radius of the inversion circle as 1 unit, then -inversion- is exactly the same as 1/z transformation in complex analysis. we can patch a -point of infinity- with the plane and link it to a sphere via sterographic projection in the usual way.
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Venkat
The relationship between lines and surface is Pi. PTheorem is a reverse relations. Right angle is a square relation. Just because lines form a closed loop. Waves can be formed only using closed loop. The relationship between 1 and 0. Otherwise numbers cannot be formed. Let alone structures.
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The relationship between lines and surface is Pi. PTheorem is a reverse relations. Right angle is a square relation. Just because lines form a closed loop. Waves can be formed only using closed loop. The relationship between 1 and 0. Otherwise numbers cannot be formed. Let alone structures.
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Mona
Oh, have nothing to say other than admitting that this video is highly enjoyable, wholly to that I shared it with my classmates.
)unworthy of mentioning sidenote: cyclic q. are actually included in 9th grade curriculum, so I will see how to demonstrate (problems ) using the P. th. )
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Oh, have nothing to say other than admitting that this video is highly enjoyable, wholly to that I shared it with my classmates.
)unworthy of mentioning sidenote: cyclic q. are actually included in 9th grade curriculum, so I will see how to demonstrate (problems ) using the P. th. )
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RCB
-Stronger- is not really the same as -more general- -- Pythagoras is a more specific case but it's still hugely useful in a wide variety of applications and the relationship it describes is far more succinct and efficient that Ptolemy. It's by far the more useful tool in my toolbox.
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-Stronger- is not really the same as -more general- -- Pythagoras is a more specific case but it's still hugely useful in a wide variety of applications and the relationship it describes is far more succinct and efficient that Ptolemy. It's by far the more useful tool in my toolbox.
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Thinking
I really enjoyed this not simply for the classical geometry, a subject that is so delightful if taught well but for the sheer enjoyment conveyed by Zvezdelina Stankova. People are so lucky to be able to see this in their formative years, I wish I was seeing this 50 years ago.
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I really enjoyed this not simply for the classical geometry, a subject that is so delightful if taught well but for the sheer enjoyment conveyed by Zvezdelina Stankova. People are so lucky to be able to see this in their formative years, I wish I was seeing this 50 years ago.
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Oscar
Laborious but worth the watch.
Interesting to me how a circle inside the conversion, maps to either a line or a circle depending on whether the circle itself touches the center of inversion.
Is there anything that behaves that way in physics?
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Laborious but worth the watch.
Interesting to me how a circle inside the conversion, maps to either a line or a circle depending on whether the circle itself touches the center of inversion.
Is there anything that behaves that way in physics?
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