Quote from: username18333 on February 05, 2015, 08:57:55 PM

Quote from: username18333 on February 04, 2015, 02:11:33 AM

Quote from: coric on February 03, 2015, 11:37:04 AM

So, you're trying to reinvent limits? Nice, but you're some centuries late in that.

Yes, the singularity of 1/x can be "removed", even on the complex plane if you add an infinity point to make it a sphere, it becomes a simple reflection
Now Riemann's function is a wee bit more difficult Wink

Code:

( ∀𝑥 𝑥 ∈ (−0⁺, 0⁺) ) ⇒ ( −0 ± 𝑥 = {−𝑥, 𝑥} ) ∧ ( 0 ± 𝑥 = {0⁻ + 𝑥, 0⁺ − 𝑥} )

It is not an infinity point (coric), for such a point would not accomodate conventional mathematics hyperreal numbers. Instead, it is an originone that has been missed sorely.

Earths set of all real numbers is, essentially, a Möbius strip fashioned from a line where one surface extends from −0⁻ to 0⁻, the other from −0⁺ to 0⁺, and all edges are retained.

An interesting fact is that the Möbius strip and the Torus are topologically related.

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Topologically, the Möbius strip can be defined as the square [0, 1] × [0, 1] with its top and bottom sides identified by the relation (x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1, as in the diagram on the right.

A less used presentation of the Möbius strip is as the topological quotient of a torus.[7] A torus can be constructed as the square [0, 1] × [0, 1] with the edges identified as (0, y) ~ (1, y) (glue left to right) and (x, 0) ~ (x, 1) (glue bottom to top). If one then also identified (x, y) ~ (y, x), then one obtains the Möbius strip. The diagonal of the square (the points (x, x) where both coordinates agree) becomes the boundary of the Möbius strip, and carries an orbifold structure, which geometrically corresponds to "reflection" geodesics (straight lines) in the Möbius strip reflect off the edge back into the strip.

The Möbius ladder might help demonstrate this a bit better.

http://upload.wikimedia.org/wikipedia/commons/7/71/Moebius-ladder-16-animated.svg

Which brings us to the Hopf fibration, that's where the fundamental existential "twist" is hidden.

http://upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Hopfkeyrings.jpg/250px-Hopfkeyrings.jpg

A couple more pages and we will finally be getting to the point of making PoW useful! Grin