Author: Renzo Diomedi

3D COMPLEX NUMBER

PREFACE

The fundamental theorem of algebra describes the advantages of the utilzation of the complex numbers and their conjugates.

Let another kind of complex number be as:
, where , = horizontal axes ; = vertical axis, and where

is not a Quaternion or other Cayley-Dickson construction.

we note that

But, Let assume , a complex number that replaces in a 2dimensional complex plane.

Then we assume that the modulus can rotate in the closed set .
To use instead of allows us to use three-dimensional space while maintaining the features of complex numbers.
So, that intrinsically includes except the cases where the interaxles angle is ,
= integer, is a dependent variable and it is mobile in
all + plane ,
by assigning appropriate values at and . The position of can range in
by the infinite factors of

Then

Moreover if

, , ,

also by we have =

also by we have =

As hence we have

Holomorphy Conditions

and deriving partially by on , on ,
on , we obtain

But if

in the Holomorphy conditions are the same in 2d-complex number:

the Cauchy-Riemann equations in are ,

deriving partially the first equation by and the second equation by then
deriving partially the first equation by and the second equation by , we obtain

and about deriving partially the first equation by and the second equation by , we obtain ,

instead, deriving partially the first equation by and the second equation by , we obtain

So we obtain the Laplacian on 3D on ** v-vector ** but

CHAPTER 1

Trigonometric coordinates and Eulerian equations:

then

,

then

If , then = by Euler's identity

then also by De Moivre equations , with = integer

, (dot product)

, (dot product)

Since

, considering hence

And if where

How wrong are we in calculating the position of the point and its vector? we need to neutralize, as far as possible, the irrationality of to reduce the margin of error.

eg: the angle have value , then assuming is not a transcendental number and not even an irrational, then, the calculation of the value of the angle would be more exact. in this case, the argument to be used for the calculation of the angles should be instead of .

So in other case which could be non-irrational, eg , , the identification of the point position will be more precise.

CHAPTER 2

with

If

Let us examine the effects that it produces on

, then shifts its values and becomes

If are then as seen above, so =

Let be where and

The

,,, ,,,

a)

b)

All the proofs relating to Eisenstein Numbers are valid on , all points (or vectors) of the Lattices are algebraic entities.

Let us calculate with the 3 orthogonal coordinates

on

then , so this output has

These 6 points can coincide, id est they can occupy the same point in the 5D-space, eg all points can coincid if the variables on 2 axes have value = zero or

the first point is calculated by the formula a), the second point using the square root on the sum of the squares of

EG : , , , , , ,

; on the axis

; on the axis

; on the axis

; on the axis

The augmented Input creates 6 corner points of a Hexagone. Let consider there is actually only one point, defined by the coordinates

Below we perform operations on the Points (not vector operations).

(carnot)

the point has coordinates:

the point has coordinates:

the point has coordinates:

the point has coordinates:

the point has coordinates: , the point has coordinates:

eg:

eg:

..............................to be continued