Geometry/physics mysteries

[Big Bad Voodoo]

I notice that we have some realy mysterious things in geometry and physics which I dont understand and I try to connect.

To not taking your precious time I will start in second.

How is it possible that sphere can be turned inside out? I realy cant grasp that. Is that some quantum/math trick?

Geometry is quite interesting when we know that Einstein believed that we can explain all trough geometry, Poincare conjecture and that often stumble upon geometry in math where we dont excepct it, like in prime numbers etc etc.

In searching for answer I stumble to even more mysterious things. Atleast 20 mysteries which even scientists have problems to explain. Because time is factor to me I just picked up few which I try to connect although they dont have nothing with above. Or they have.(?)

a) Banach Tarski paradox said that when we cut ball into finite number of pieces then reassemble it we get two balls equal size as first ball. Interestingly Paul von Nuemann also conclude on 2D analogy. (He is interesting to me because Game theory). Anyway is hard to me to grasp that. So can we conclude that surface is some quantum trampoline thing or does it replicate or perhaps infinite?

b.)Why I think is infinite? Because we have thing called Gabriel horn where we have finite volume and infinite surface object. Obviously we have objects with infinite surface.

c) There is paradox called coastline paradox where in a sense coastline is infinite.

d) Holographic paradox say that information that can be stored in volume isnt proportional to volume but to area that bounds that volume.

Obviously surface is REALY mysterious to us. Or is it just to me.

Thanks in advance.

Edited June 23, 2013 by the L

[sepulchrave]

How is it possible that sphere can be turned inside out? I realy cant grasp that. Is that some quantum/math trick?

There is no quantum physics involved in any of these examples. It is all geometry, and usually connected to set theory.

I assume you are talking about ``Smale's Paradox'' here?

It isn't actually possible to physically do this. The sphere used in this context can freely pass through itself; the act of turning inside out (eversion) just requires that the sphere not tear or crease.

You can see videos of how it is done (again, of course, you can't do it in real life because you can't make a sphere of something that is infinitely stretchy and deformable and also able to freely pass through itself).

Geometry is quite interesting when we know that Einstein believed that we can explain all trough geometry, Poincare conjecture and that often stumble upon geometry in math where we dont excepct it, like in prime numbers etc etc.

Geometry is basically a different ``language'' for expressing mathematics, just like algebra.

Sir Isaac Newton originally developed calculus purely from geometric proofs.

And, of course, the most famous equation in mathematics provides a direct link to convert geometry into algebra and vice versa.

a) Banach Tarski paradox said that when we cut ball into finite number of pieces then reassemble it we get two balls equal size as first ball. Interestingly Paul von Nuemann also conclude on 2D analogy. (He is interesting to me because Game theory). Anyway is hard to me to grasp that. So can we conclude that surface is some quantum trampoline thing or does it replicate or perhaps infinite?

Again, this has nothing to do with quantum mechanics, and it is impossible to do with a real sphere. This is just a ``mathematical trick'' involving group theory.

In the Banach-Tarski paradox, the surface of the sphere is divided into a finite set of ``orbits''; that is the collection of points which can be reached with repeated applications of a single group element (in this case a 3D rotation).

These ``orbits'' have finite surface area, but they are not physically discrete pieces (each may consist of a scattering of points and infinitesimally thing strands around the surface of the sphere).

A real sphere is made from a finite collection of atoms, not the result of an infinite application of generators to some initial element. The Banach-Tarski paradox doesn't exist in real life.

In fact, the use of a sphere is - in some sense - only to help visualize the problem. The Banach-Tarski paradox doesn't say so much about geometry as it does about reconstructing a full set of generators from a discrete subset.

b.)Why I think is infinite? Because we have thing called Gabriel horn where we have finite volume and infinite surface object. Obviously we have objects with infinite surface.

Gabriel's horn isn't a real object. It is impossible to make. We don't really have physical objects with infinite surfaces.

Again, this isn't so much an issue with geometry as it is with calculus. Some infinite series converge, others do not.

Obviously surface is REALY mysterious to us. Or is it just to me.

I don't think it is that mysterious.

It seems mysterious because, as I said above, we can ``translate'' any abstract mathematical problems into geometry - and then it seems more real (because obviously we directly interact with geometry every day in the real world).

In my opinion, this all boils down to different manifestations of the ``cardinality of the continuum''; that every uncountably infinite set is equivalent to every other set.

In other words, there is an infinite number of infinitesimally small points that make up the volume of an object, and there are an infinite number of infinitesimally small points that make up the surface of an object, and that these two infinities are exactly the same.

In real life this is not the case because objects are not made up of an infinite number of infinitesimally small points, they are made up of a large, but finite, number of small, but finite, atoms.

Of course the holographic principle differs from this argument somewhat because it relates to the accessible information of an object - i.e. what is needed to determine how an object interacts with others.

However it is also similar because it is a rather general application of large-scale space-time; it doesn't necessary apply to every-day life.

[Big Bad Voodoo]

For the second I understand all.

Thanks Sepul, anyway, on your good will.