Generalizations of n-dimensional Inner Product Spaces

[benb]

I liked increasing the quantity of orthants of a given space by increasing its dimension but I wanted to increase the denominator in the 2^n expression indicating the quantity of orthants of an n-dimensional space to any natural number. I think I may have found a way to make this computable and the results describe spaces and objects with unusual properties, including achirality. I have been writing up a draft of the constructions and findings and I think at this stage it may be of interest and benefit from feedback and perhaps discussion.

2020-03-28MultipolarEquations.pdf

(260.9 KiB) Downloaded 234 times

[mr_e_man]

Do I understand this correctly?

The multipolar set M_G(p) can be thought of as p rays sharing an endpoint. (This is shown in the first picture, with p=5.) A multipolar number is specified by two pieces of information: which ray it's on (an integer from 1 to p), and the distance from the endpoint (a positive real number). The rays correspond to elements of some algebraic group {g₁, g₂, ... , g_p}.

Multipolar multiplication is defined by (a g_j) * (b g_k) = (a*b) (g_j*g_k), where a,b are positive real numbers, and g_j,g_k are elements of the group.

Multipolar addition ("consolidation") is defined piecewise:

(a g_j) + (b g_k) = {

(a + b) g_j, if j = k;
(a - b) g_j, if j =/= k, and a > b;
(b - a) g_k, if j =/= k, and a < b.

Geometrically, you just add the numbers using the vector addition "parallelogram rule", then travel "diagonally" along the grid formed by the two rays until you hit one of them:

multipolarSet.png (19.56 KiB) Viewed 2255 times

The "multipolar-to-real product" is defined by

mrp(a g_j, b g_k) = {

a*b, if j = k;
-a*b, if j =/= k.

All of this describes a "confield", which is considered one-dimensional; then we can construct an n-dimensional "convector space" as n-tuples of these numbers.

[mr_e_man]

I suppose we could also define a "multipolar conjugate", analogous to the complex conjugate: (a g_j)^~ = a (g_j^-1).

More generally, any automorphism F of the group (that is, an invertible function that preserves the group structure: F(g_j*g_k) = F(g_j)*F(g_k) ) can be extended to the multipolar set, by defining F(a g_j) = a F(g_j). Then F is also an automorphism of the multipolar set: F(x*y) = F(x)*F(y), and F(x + y) = F(x) + F(y), where x,y are multipolar numbers.

But the multipolar conjugate has the special property that x^~*x = |x|² g₁, where g₁ is the group's identity. We could write g₁ = 1, so that x^~*x is a positive real number.

[benb]

Yes, "confield" is the algebraic structure and a multipolar set M_G(p) would be an instance of that structure whose dimension may be considered in geometric terms as you described.

The visualizations indicate what I was getting at and map to my own early intuitions. That every coaxial pair of points on the continua they represent is separated by pi radians adds to the challenge of depiction.

You are correct regarding the possibility of defining a multipolar conjugate and defining it as 1. When that possibility is played out, it means that in some multipolar systems all numbers have square roots and in others 1 and | x |= 1 may have more than one square root.