Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Infinite tetration and superroot of infinitesimal
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- 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.


Messages In This Thread
RE: Infinite tetration and superroot of infinitesimal - by Ivars - 07/05/2008, 08:12 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
  [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 1,231 03/20/2018, 12:16 AM
Last Post: tommy1729
  [MO] Is there a tetration for infinite cardinalities? (Question in MO) Gottfried 10 12,283 12/28/2014, 10:22 PM
Last Post: MphLee
  Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 5,267 05/06/2014, 09:47 PM
Last Post: tommy1729
  Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 7,023 07/17/2013, 09:46 AM
Last Post: Gottfried
  Wonderful new form of infinite series; easy solve tetration JmsNxn 1 4,453 09/06/2012, 02:01 AM
Last Post: JmsNxn
  the infinite operator, is there any research into this? JmsNxn 2 5,555 07/15/2011, 02:23 AM
Last Post: JmsNxn
  Infinite tetration of the imaginary unit GFR 40 55,973 06/26/2011, 08:06 AM
Last Post: bo198214
  Infinite Pentation (and x-srt-x) andydude 20 26,596 05/31/2011, 10:29 PM
Last Post: bo198214
  Infinite tetration fractal pictures bo198214 15 23,400 07/02/2010, 07:22 AM
Last Post: bo198214
  Infinite towers & solutions to Lambert W-function brangelito 1 3,870 06/16/2010, 02:50 PM
Last Post: bo198214

Users browsing this thread: 1 Guest(s)