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 not on u-vector neither w-vector . The laplacian equation is only on direction of axis that links the north pole and south pole of the sphere obtained by the Normed space consisting of vectors and is the harmonic function of the laplacian equation on the v-vector









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

PI/4 approximately is 3.14159265358979/4 = 0.785398163397448



, 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 k and n natural numbers, is a root of unity id est a complex number = 1 if raised to a positive integer exponent.
If n=3 we have a cubic root of unity that produces an Equilateral (it is immediate to verify that ) Triangular lattice, then a Hexagonal Lattice.


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 x, y, z are integers. Then the cartesian axes are not orthogonal.

The z axis shifts thus is inclined of 120 relating x axis , and the y axis shifts thus is inclined by 120 relating s axis



,,, ,,,





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 x, y, z integers , and their corresponding coordinates on equilateral triangular lattice. An input on three axes x, y, z only, produces a single point (or vector). An augmented input x,y,z, extended to the corresponding equilateral triangular coordinates, has 6 variables:

on 5 space-dimensions, id est the axes:

then , so this output has 6 points (or 6 vectors) of which 2 lie on the s-plane.

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 3 points coincide if the variables on an axis have values = zero

the first point is calculated by the formula a), the second point using the square root on the sum of the squares of x and z . Ulterior 2 points are calculated by the formula b) using the two vectors lying on s-planes already created as described above. We create ulterior 2 points using the square root on the sum of the squares of each of the points on the 2 axes s and y .
EG : , , , , , ,







; on the axis

; on the axis

; on the axis

; on the axis


3 space coordinates generate one point and one vector. Same coordinates extended on a 5-dimensional space (2 additional dimensions are generated by the cubic root of unity that replaces the imaginary unit on a 3d complex number) generate 6 points and 6 vectors.

The augmented Input creates 6 corner points of a Hexagone. Let consider there is actually only one point, defined by the coordinates x y z , which in a 5-dimensional space is in 6 different positions. So this point creates an object similar to an irregular 3Dimensional hexagone. But in nature we know the hexagon as a regular 2-manifold and its angles are 2/3 PI 2,094395. Then we calculate the constant values of the six angles of our irregular multidimensional hexagon and we can compare them with 2/3 PI, and verify the distortion.

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




















(carnot)










COORDINATES OF THE POINTS:





the point has coordinates:







the point has coordinates:







the point has coordinates:





the point has coordinates:





the point has coordinates: , the point has coordinates:












d =















c =















e =















f =









































































































































































eg: x=4 , y=3 , z=2 2,052051


eg: x=1,5 , y=2 , z=8 1,97901


2/3 PI 2,094395 - 2,052051 = 0,042344


2/3 PI 2,094395 - 1,97901 = 0,115385


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








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