Synthetic geometry of the real projective plane by 
Ciro Santilli 37 Updated 2025-07-16 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-07-16 +Created 1970-01-01It good to think about how Euclid's postulates look like in the real projective plane:
- Since there is one point of infinity for each direction, there is one such point for every direction the two parallel lines might be at. The parallel postulate does not hold, and is replaced with a simpler more elegant version: every two lines meet at exactly one point.One thing to note however is that ther real projective plane does not have angles defined on it by definition. Those can be defined, forming elliptic geometry through the projective model of elliptic geometry, but we can interpret the "parallel lines" as "two lines that meet at a point at infinity"
- points in the real projective plane are lines in
- lines in the real projective plane are planes in .For every two projective points there is a single projective line that passes through them.Note however that not all lines in the real plane correspond to a projective line: only lines tangent to a circle at zero do.
Unlike the real projective line which is homotopic to the circle, the real projective plane is not homotopic to the sphere.
The topological difference bewteen the sphere and the real projective space is that for the sphere all those points in the x-y circle are identified to a single point.
One more generalized argument of this is the classification of closed surfaces, in which the real projective plane is a sphere with a hole cut and one Möbius strip glued in.
Ultimate explanation: math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426#3732426
Only normal subgroups can be used to form quotient groups: their key definition is that they plus their cosets form a group.
One key intuition is that "a normal subgroup is the kernel" of a group homomorphism, and the normal subgroup plus cosets are isomorphic to the image of the isomorphism, which is what the fundamental theorem on homomorphisms says.
Therefore "there aren't that many group homomorphism", and a normal subgroup it is a concrete and natural way to uniquely represent that homomorphism.
The best way to think about the, is to always think first: what is the homomorphism? And then work out everything else from there.
History of quantum mechanics bibliography by Ciro Santilli 37 Updated 2025-07-16 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-07-16 +Created 1970-01-01Quantum is getting hot in 2019, and even Ciro Santilli got a bit excited: quantum computing could be the next big thing.
No useful algorithm has been economically accelerated by quantum yet as of 2019, only useless ones, but the bets are on, big time.
To get a feeling of this, just have a look at the insane number of startups that are already developing quantum algorithms for hardware that doesn't/barely exists! quantumcomputingreport.com/players/privatestartup (archive). Some feared we might be in a bubble: Are we in a quantum computing bubble?
To get a basic idea of what programming a quantum computer looks like start by reading: Section "Quantum computing is just matrix multiplication".
Some people have their doubts, and that is not unreasonable, it might truly not work out. We could be on the verge of an AI winter of quantum computing. But Ciro Santilli feels that it is genuinely impossible to tell as of 2020 if something will work out or not. We really just have to try it out and see. There must have been skeptics before every single next big thing.
Very practical, low-cost experiments.
Alan Watts controlled dream of life talk by Ciro Santilli 37 Updated 2025-07-16 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-07-16 +Created 1970-01-01Goes along: if you could control your life multiple times to be perfect, you would eventually get tired of paradise, and you would go further and further into creating uncertain worlds with some suffering, until you would reach the current real world.
Very similar to The Matrix (1999) when Agent Smith talks about the failed Paradise Matrix shown at www.youtube.com/watch?v=9Qs3GlNZMhY:
Did you know that the first Matrix was designed to be a perfect human world where none suffered, where everyone would be happy? It was a disaster. No one would accept the program. Entire crops were lost. Some believed that we lacked the programming language to describe your "perfect world". But I believe that, as a species, human beings define their reality through misery and suffering. So the perfect world was a dream that your primitive cerebrum kept trying to wake up from.
E.g. monotremes laying eggs did not evolve separately after function loss, it comes directly from reptiles.
2011 PHYS 485 lecture videos by Roger Moore from the University of Alberta by Ciro Santilli 37 Updated 2025-07-16 +Created 1970-01-01
Ciro Santilli 37 Updated 2025-07-16 +Created 1970-01-01 Partial label partial derivative notation by Ciro Santilli 37 Updated 2025-07-16 +Created 1970-01-01
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