When looking at x^y as hypercube in dimension y with edge x, I have made implicit assumption that the ANGLE in which each edge of such hypercube creates with another is 90 degrees in any dimension y, just by analogy, or Pi/2, as in general definitions of hypercube.
However, this angle may be another,may be negative, may be imaginary.
This leads to interpretation of real etc. extensions of hyperoperation number z in x[z]y as angle between such edges x of a hypercube in dimension y either directly, or via some exponential relation like log (pi/2)^n/log (pi/2) = n for integer z, and z*log pi/2/log (pi/2) =z for all other. , or n= log (e^n*I*pi/2)/log I , z= log (e^z*I*pi/2)/log I.
As for imaginary, negative angles and other imaginary things like points, edges etc. it seems obvious that such interpretation of hyperoperations and its extension for real/imaginary/negative number of operations leads to/is connected with projective geometry in its original complex form, as developed by Lambert, Gauss, Stoudt, Cayley, Klein. It seems to me that real numbers as foundations of geometry fail here(?) (in interpretation of hyperoperations to real , imaginary, negative numbers) , which is of no big surprise since their foundation are based on physical convenience but not abstract proofs (since it is unprovable so far) , so alternative models are possible as well.
But since I only conjectured this yesterday, there are many things to read, particularly about origins of non-euclidian geometry . Just to summarize:
In expression :
x[z]y
x- edge of hypervolume in y dimensions
y- dimensions of hypervolume with edge x
z- is related to varying angle between edges x of hypervolume in y dimensions, perhaps via logarithm or some trigonometric function. A meaningful definition should recover usual angles pi/2 in case of hypercube in integer dimensions.
The geometric interpretation of hyperoperation in general and its real, complex extensions is related to projective geometry in imaginary form extended to non-integer, negative and imaginary dimensions.
To give a more concrete test example:
I[I]I is a hypervolume with edge I in I dimensions with angle between imaginary edges (e^(I*I*pi/2) = e^(-pi/2)
I[-I]I is a hypervolume with edge I in I dimensions with angle between imaginary edges e^pi/2. In I dimensional space, such angle might have a meaning.
It also seems that his hyperangle conveyed via z is a composite one, so related to twisting of hypervolume x[z]y of dimension y. This is probably possible to check if some volumes in ordinary integer spaces of twisted hypercubes are known.
Ivars
However, this angle may be another,may be negative, may be imaginary.
This leads to interpretation of real etc. extensions of hyperoperation number z in x[z]y as angle between such edges x of a hypercube in dimension y either directly, or via some exponential relation like log (pi/2)^n/log (pi/2) = n for integer z, and z*log pi/2/log (pi/2) =z for all other. , or n= log (e^n*I*pi/2)/log I , z= log (e^z*I*pi/2)/log I.
As for imaginary, negative angles and other imaginary things like points, edges etc. it seems obvious that such interpretation of hyperoperations and its extension for real/imaginary/negative number of operations leads to/is connected with projective geometry in its original complex form, as developed by Lambert, Gauss, Stoudt, Cayley, Klein. It seems to me that real numbers as foundations of geometry fail here(?) (in interpretation of hyperoperations to real , imaginary, negative numbers) , which is of no big surprise since their foundation are based on physical convenience but not abstract proofs (since it is unprovable so far) , so alternative models are possible as well.
But since I only conjectured this yesterday, there are many things to read, particularly about origins of non-euclidian geometry . Just to summarize:
In expression :
x[z]y
x- edge of hypervolume in y dimensions
y- dimensions of hypervolume with edge x
z- is related to varying angle between edges x of hypervolume in y dimensions, perhaps via logarithm or some trigonometric function. A meaningful definition should recover usual angles pi/2 in case of hypercube in integer dimensions.
The geometric interpretation of hyperoperation in general and its real, complex extensions is related to projective geometry in imaginary form extended to non-integer, negative and imaginary dimensions.
To give a more concrete test example:
I[I]I is a hypervolume with edge I in I dimensions with angle between imaginary edges (e^(I*I*pi/2) = e^(-pi/2)
I[-I]I is a hypervolume with edge I in I dimensions with angle between imaginary edges e^pi/2. In I dimensional space, such angle might have a meaning.
It also seems that his hyperangle conveyed via z is a composite one, so related to twisting of hypervolume x[z]y of dimension y. This is probably possible to check if some volumes in ordinary integer spaces of twisted hypercubes are known.
Ivars