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History of Non-Euclidean Geometry - Lies - Extra History - #6 (Patreon)

Published:
2018-06-29 17:31:00
Imported:
2022-01

Content

You asked for so long, and here it is! SUGGESTED READING for this Extra History series: 

--Euclid's The Elements 

--Newton's Philosophiæ Naturalis Principia Mathematica 

--Lobachevsky's Geometrical Investigations on the Theory of Parallel Lines 

Happy reading! 

Files

History of Non-Euclidean Geometry - Lies - Extra History - #6

You gently corrected out our math mistakes and artistic slip-ups, and we're here to tell you it was all part of Bismarck's plans--er, it's Euclid's fault. Time for another episode of Lies! Support us on Patreon! http://bit.ly/EHPatreon (--More below) Grab your Extra Credits gear at the store! http://bit.ly/ExtraStore Subscribe for new episodes every Saturday! http://bit.ly/SubToEC Talk to us on Twitter (@ExtraCreditz): http://bit.ly/ECTweet Follow us on Facebook: http://bit.ly/ECFBPage Get our list of recommended games on Steam: http://bit.ly/ECCurator ____________ ♪ Get the intro music here! http://bit.ly/1EQA5N7 *Music by Demetori: http://bit.ly/1AaJG4H ♪ Get the outro music here! *Music by Sean and Dean Kiner: http://bit.ly/1WdBhnm

Comments

Anonymous

Keeping it easy with those recommendations, eh? I recall Newton's book being written in a form "to avoid being baited by little Smatterers in Mathematicks". For those interested, I'd recommend Jeffrey Weeks' "The Shape of Space". It can be a bit tough going at times, but doesn't require much formal math and does a good job at explaining this material as well as introducing the field of topology.

Anonymous

So, is the Flu Pandemic Series going to be the first one with Matt as narrator?

Anonymous

should not there be a one off ?

Anonymous

when is the next topic vote ?

Anonymous

what took them so long to figure out the 5th postulate was independent of the first 4??? in the axioms there really isn't anything that says a "line" is a line as we understand and not say a great circle on a sphere or that points cannot be lines and lines planes (projective geometry) Or what if we let the "points" of our geometry to be everything except a point p in the plane along with a point at infinity "lines" in our geometry all the circles and lines that go through p and the angle of two circles C_1 ,C_2 is the angle made by the lines f(C_1) f(C_2) where f(x)=p+ (x-p)/|x-p|^2 is the inversion fcn which one of the 4 postulates says a line is not a curve?

Anonymous
>>13532477

I understand to some people I sound a bit ridiculous but just from the 4 axioms we only have that there are elements of a set called points and a subset of the power set L that we call "lines" with the property that for any two points a,b there is exactly one line going through them where in the 4 axioms have we used the fact that our "lines" are straight or even continuous? Where have we used the fact that our points are all in a plane?

Bill Lemmond
>>13532477

o_O I've been told I have a 160 IQ, and I tried to follow these two comments, but they were too dense for me, with too any unfamiliar terms. Any chance for pointers to illustrated discussions of these subjects?

Anonymous
>>13532477

Perhaps projective geometry, pole to polar transformations, inversion mapping of the plane, geometry of Z_p (integers modulo p) algebraic geometry, elementary set theory other college lvl mathematics are good for intuition However you may not to read that much for an example try to rigorously define distance using only the 5 postulates and reword Euclids 5 postulates in set theoretic terms as rigorously as possible to show that your definition of distance works

Anonymous
>>13532477

I'm not aware of Euclid formally defining distance as far as I'm aware he just kinda assumes the notion of distance is obvious and further that given A, B, C the distances btw A,B,C are all finite and there is a point B' on AC which is kind of his circle axiom although to be fully rigorous he should say that any line drawn through the center of a circle intersects the circle at 2 points Here I'm assuming as few axioms as possible that is there are these objects P called points and 2 sets of objects S_L, S_C called lines and circles respectively such that given a,b in P not equal there is a unique L in S_L that contains a,b ... etc as an exercise using this set theory framework define distance and prove (introducing new axioms if necessary ) the triangle inequality