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From: Archimedes Plutonium on 29 Jul 2010 15:01
It is a shame, perhaps in math history that the Recalculus was not
borne first and only later
would the Calculus be borne, but if we look at technology we
understand instantly why Recalculus is borne later, because strip
making technology is sophisticated. To make wine
barrels out of strips of oak wood; to make baskets out of strips of
reeds.

In ancient Greek times it was known: "the surface of the sphere is
four times the area of
its greatest circle"

Or in other words: sphere surface area = 4(pi) r^2

We can sort of picture this as taking the globe with its continents
and countries and
seeing the 4 semihemispheres, or 4 compartments. Much like a orange
sliced into 4
sections. And if we take one of those 4 sections and turned the
corners into a circle.

But this leads to a more important feature. Consider that 2 of those 4
sections forms
2 funnel shaped objects and thus forms a pseudosphere, but a
pseudosphere *not* having the same diameter as the sphere. What
mathematical relationship does that pseudosphere have
to the sphere?

Now consider 2D geometry of the circle inside a square. Let us cut out
the circle and
we have remaining 4 hyperbolic triangles. Now let us position the 4
hyperbolic triangles
to remove the straight line sides and we have a figure that can be
called a 4 pointed star.
We can nest this 4 pointed star inside the original circle. We can
call this 4 pointed star
the square minus circle residue. And we know for sure that the area of
square equals
the area of circle plus residue area (hyperbolic star). Now this
hyperbolic star is in fact
a pseudosphere analog of 2D and call it a pseudocircle.

In the case of the pseudosphere it is 1/2 the area of the sphere of
3D, but in the case of
the pseudocircle we have something different.

Now we repeat this experiment by placing the sphere inside a cube and
removing the sphere
and then cutting the residue into, not 4 compartments but making 3
cuts into forming 8 equal
compartments of the residue. Now we assemble this residue parts into a
pseudosphere. And
we ask what is the mathematical relationship of this 3D pseudosphere
to the sphere itself?

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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