It 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.
This section is about telecommunication systems that are based on top of telephone lines.
Telephone lines were ubiquitous from early on, and many technologies used them to send data, including much after regular phone calls became obsolete with VoIP.
These market forces tended to eventually crush non-telephone-based systems such as telex. Maybe in that case it was just that the name sounded like a thing of the 50's. But still. Dead.
Long Distance by AT&T (1941)
Source. youtu.be/aRvFA1uqzVQ?t=219 is perhaps the best moment, which attempts to correlate the exploration of the United States with the founding of the U.S. states. Television series that Ciro Santilli watched all episodes of Updated 2025-06-17 +Created 1970-01-01
This classification is too restrictive, and too South-centered. But if is worth knowing.
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. A different and more elegant way to express Maxwell's equations by using the:instead of the:
Crystallography determination with a transmission electron microscopy instead of the more classical X-ray crystallography.
Breakdown:
Zen master and the little boy scene from Charlie Wilson's War
. Source. This is how Ciro Santilli first heard the old man lost his horse! The scene happens after the Americans seem to have had sucess in arming the Taliban in the 1980s to fight the Soviets, and then they decide to stop all monetary help to rebuild the country, which then leads to a terrorism threat many years later.The Fourier series of an function (i.e. the function generated from the infinite sum of weighted sines) converges to the function pointwise almost everywhere.
The theorem also seems to hold (maybe trivially given the transform result) for the Fourier series (TODO if trivially, why trivially).
Only proved in 1966, and known to be a hard result without any known simple proof.
This theorem of course implies that Fourier basis is complete for , as it explicitly constructs a decomposition into the Fourier basis for every single function.
TODO vs Riesz-Fischer theorem. Is this just a stronger pointwise result, while Riesz-Fischer is about norms only?
One of the many fourier inversion theorems.
There are unlisted articles, also show them or only show them.