[AntonioLao]

The key to background independence or dependence is singularity. In one sense, it is equivalent to a singular square symmetric Hadamard matrix. That is a matrix without an inverse matrix. In another sense, it is a rational function of two integer variables X and Y such that X=Y/(1-Y) and Y=X/(1+Y). Their product is simply XY=X-Y. If an inverse function is defined as symmetric switching of X and Y, X«Y then the resulting graph is equivalent to a point reflection about the origin of a coordinate system: (X,Y)®(-X,-Y). This is also equivalent to orthogonal reflection about each axis or a 180° rotation about the origin.

In a polar-trigonometric coordinate system of (R,f), where X=Rcos(f) and Y=Rsin(f) then tan(f)+Rsin(f)=cot(f)-Rcos(f)=1 and Rsin(f)cos(f)=cos(f)-sin(f) or ½Rsin(2f)=cos(f)-sin(f) or R=csc(f)-sec(f). This appears as a dimensionless quantity of pure numbers: real, complex, imaginary, or hypercomplex depending on its usage in a particular situation. So, it is possible for R to be a singular square symmetric Hadamard matrix.

[Profpat]

Well if it's a singular square it would have 4 corners and 4 lines.

[AntonioLao]

Originally Posted by Profpat 4 corners and 4 lines

This is true for the plane geometric figure but in topology a plane circle is topologically equivalent to a square and for that matter any polygon, concave or convex, is topologically equivalent to a square. In abstract sense, the corners and lines are the dimensional boundary and dimensional connectivity between two topologies.

[Profpat]

OK but our singularity would have complementary opposites. i.e concave vs convex, opposing sides or lines, etc.

[AntonioLao]

Originally Posted by Profpat have complementary opposites

In quantum mechanics it is the principle of complementarity that is playing a key role but not in general relativity. See http://en.wikipedia.org/wiki/Complementarity_(physics)

Originally Posted by Profpat concave vs convex, opposing sides or lines

Concavity and convexity and lines lose their meaning in higher dimensional descriptions which is described as compactness or connectivity of topological manifolds.

[Profpat]

Thanks for the link Antonio. I would have thought concave or convex being a definition as to the curvature would be the same regardless of the complexities of the dimensions.

[AntonioLao]

Originally Posted by Profpat being a definition as to the curvature would be the same regardless of the complexities of the dimensions

Works well in a finite structure but in superstring there are the infinitely extended multidimensional branes.

[Profpat]

Is science sure of infinitely extended? It seems like nature puts limits on everything, whether it's the speed of light or the number of neutrons and protons that can bind together. It seems like nature doesn't like infinity. A concept for mathematicians.

[mkirkpatrick]

Originally Posted by AntonioLao The key to background independence or dependence is singularity. In one sense, it is equivalent to a singular square symmetric Hadamard matrix. That is a matrix without an inverse matrix. In another sense, it is a rational function of two integer variables X and Y such that X=Y/(1-Y) and Y=X/(1+Y). Their product is simply XY=X-Y. If an inverse function is defined as symmetric switching of X and Y, X«Y then the resulting graph is equivalent to a point reflection about the origin of a coordinate system: (X,Y)®(-X,-Y). This is also equivalent to orthogonal reflection about each axis or a 180° rotation about the origin. In a polar-trigonometric coordinate system of (R,f), where X=Rcos(f) and Y=Rsin(f) then tan(f)+Rsin(f)=cot(f)-Rcos(f)=1 and Rsin(f)cos(f)=cos(f)-sin(f) or ½Rsin(2f)=cos(f)-sin(f) or R=csc(f)-sec(f). This appears as a dimensionless quantity of pure numbers: real, complex, imaginary, or hypercomplex depending on its usage in a particular situation. So, it is possible for R to be a singular square symmetric Hadamard matrix.

What really goes on behind the scene?what links all this together,is the singularity the
provider of the "field"?

regards michael.

[AntonioLao]

Originally Posted by Profpat Is science sure of infinitely extended?

Theoretically not physically and mathematically infinity makes sense only in differential and integral calculus at the foundation of analysis.

Originally Posted by mkirkpatrick What really goes on behind the scene?

Your guess is as good as mine but please read the post on quantum bullies entitled: rollie pollie QUOM collie.