I was sure is not so simple as it has been portrayed. Below is text from Wikipedia on quaternions, but my idea is related to the "point" at origin of pure imaginary 2-sphere, whose projection on unit sphere in that space is equivalent to of a complex plane, and its antipode to . One unit sphere in imaginary 3D space contain infinity of complex planes , each corresponding to axis going via 2 points on the surface of imaginary 2 sphere.

The question is about the nature of the point at origin, which is not normal 0, since it is present in pure imaginary space.

I still have and always have had a gut feeling that the infinity of such axis going via imaginary 0 and tetration of real numbers to imaginary values are linked. That is why I have been trying to use dI since change from one complex plane defined by one of the axis in purely imaginary 3D space to another defined by another infinitesimaly close axis would lead to ANOTHER in that space, dI away from previous. dI would be than a vector on the surface of the Unit sphere in space, probably orthogonal to , and dI would be possible to decompose into dj and dk, or have spherical imaginary angle representation.

So when we get as a result of infinite tetration e.g. we do not know do we speak about one of these complex planes or all of them. We do not know are there quaternions or even higher dimensional imaginary spaces involved or not; finally, if x^2+1=0 has infinitely many solutions in such space, what about x^x+1=0 in infinitedimensional imaginary space?

Of course, I am not sure if the point of origin in 3D imaginary space is not represented as some hyper surface in e.g. 7D pure octonion space which in turn is a hyper surface in 15D pure sedenion space etc.

This all can be found in Wikipedia under Quaternions.

Ivars

The question is about the nature of the point at origin, which is not normal 0, since it is present in pure imaginary space.

I still have and always have had a gut feeling that the infinity of such axis going via imaginary 0 and tetration of real numbers to imaginary values are linked. That is why I have been trying to use dI since change from one complex plane defined by one of the axis in purely imaginary 3D space to another defined by another infinitesimaly close axis would lead to ANOTHER in that space, dI away from previous. dI would be than a vector on the surface of the Unit sphere in space, probably orthogonal to , and dI would be possible to decompose into dj and dk, or have spherical imaginary angle representation.

So when we get as a result of infinite tetration e.g. we do not know do we speak about one of these complex planes or all of them. We do not know are there quaternions or even higher dimensional imaginary spaces involved or not; finally, if x^2+1=0 has infinitely many solutions in such space, what about x^x+1=0 in infinitedimensional imaginary space?

Of course, I am not sure if the point of origin in 3D imaginary space is not represented as some hyper surface in e.g. 7D pure octonion space which in turn is a hyper surface in 15D pure sedenion space etc.

Quote:H as a union of complex planes

Informal Introduction

There exists an intriguing way of understanding H that links its structure closely to the surface of an ordinary sphere of radius 1. In mathematics such a sphere is called a unit 2-sphere to emphasize that only its two-dimensional surface is being considered.

The first step is to translate the XYZ coordinates of the unit 2-sphere into the ijk coordinate system of quaternions, keeping the scalar (first) value of the quaternions set to zero. For example, the XYZ point <1,0,0> becomes the quaternion 0 + 1i + 0j + 0k. Since quaternion absolute lengths are calculated in the same way as XYZ radii, the resulting unit 2-sphere quaternions also all have absolute lengths (radii) of 1.

A less intuitive property of unit 2-sphere quaternions is that their squares all equal -1. This is true by definition for the three main axes of i, j, and k, but it can also be verified easily by trial for any arbitrary unit 2-sphere quaternion.

Since a length of 1 and a square of -1 are the defining properties of i, these unit 2-sphere quaternions look suspiciously like mathematical analogs to i. Furthermore, since each such quaternion has an "unused" scalar value associated with it, a fascinating conjecture becomes possible:

For any given ijk-only point on the quaternion unit 2-sphere, does the set of all quaternions that can be expressed as the sum of a real number and a multiple of that ijk-only point behave like a complex plane?

Somewhat unsurprisingly, the answer is yes.

That is, H can be partitioned in such a way that it looks like an infinite set of complex planes. Each such plane has its own unique version of i, although they all share the same real (scalar) axis. Furthermore, each unique i value corresponds to and is fully defined by a point on the surface of an ordinary unit-radius sphere, thus providing a strong connection between the geometry of ordinary spheres and the far less intuitive four-dimensional properties of H. Once a point on the unit 2-sphere has been selected, there is no mathematical difference in the behavior of the resulting subset of H and the more traditional concept of a single abstract complex plane.

Thus quaternions do not just extend the concept of i just to the two new axes j and k. They generalize i to an infinite set of points that happen to be the same ones found on the surface of an ordinary unit-radius sphere!

A more precise mathematical profile of how H can be interpreted as a union of complex planes is provided below.

Detailed Specification

Isomorphisms to the imaginary unit

The set of quaternions of absolute length (radius) 1 has the form of a 3-sphere or hypersphere, which is also called S³. Within this hypersphere there exists a subset of quaternions with the additional property that their squares are equal to −1. This subset has the geometric form of an ordinary sphere, or 2-sphere (S²). It can be understood as a three-dimensional "slice" of the larger hypersphere in much the same way that a circle is a two-dimensional "slice" of an ordinary sphere. For reasons explained below, this sphere-like subset of H is referred to here as Hi, where the i subscript refers to the imaginary unit.

This all can be found in Wikipedia under Quaternions.

Ivars