SPIRALOGARITHM HAND DRAWN By Panagiotis Stefanides
http://www.stefanides.gr/why_logarithm.htm


SPIRALOGORITHM [ name coinned by Panagiotis Stefanides].
Configuration giving simple relationships between spirals and logarithms.

Introduced here is the STEPHANOID CURVE[ authors proposal of nomination] , the mirror images of two Archimedes curves of a special form,
passing Zero x-y point and on which lie vector length logarithms, of vector line lengths on the spiral[ one on top of the other].

Let X=e^[ Θ/90], be a point on the curve of this Spiral of Logarithmic base e ,on the plane of x-y co-ordinate system of Cartesian axes.
Phasor OX is at angle Θ from the x-axis taking anticlockwise direction as positive(Θ on x-axis is taken zero )and clockwise negative.
This Spiral crosses the [-y ] axis at point [ 1/e ],then the x axis at point 1 [moving anticlockwise] ,following crosses the y axis at point with value e ,then the [-x] axis at e^2... etc...
This gives ln(X)=[ Θ/90], OR in RADIANS ln(X)= Θ/[Pi/2].

The 0 of the x y cross section is the assymptotic point of the Spiral.

This theory is illustrated under:
http://www.stefanides.gr/why_logarithm.htm


TRANSFORMING [e^x] EXPANSION INVOLVING OTHER THAN [e ] BASES :

e^x = 1+x+(x^2/2!)+(x^3/3!)+(x^4/4!)+(x^5/5!)
may be transformed to other than e bases.
For instance,
Lets for reasons that serve my web work(references below) , instead of x use Z
where Z is a ral positive or negative number.

So e^Z= 1+ Z+ [Z^2]/2!+…+.....

Let X=e^Z , AND Log_e[X]=Z

So X= 1 + Log_e[X] + {[Log_e[X] ]^2}/2!+....

Raise e^Z to the power of N (POSITIVE REAL NUMBER),we get

e^[NZ]=1+NZ+{[NZ]^2}/2!+...

Let e^N = T SO THAT Log_e[T]=N

WHERE T is a POSITIVE REAL NUMBER
[BASE OF ANOTHER THAN e LOGARITHM ] , then ,

e^[NZ]= T^Z=1+Z[Log_e[T]] + [{Z[Log_e[T]]}^2]/2!+.....

Let this T^Z = Y, or Log_T[Y]=Z

AND ( from above ) Log_e[X]=Z = Log_T[Y] ,so

X=e^{Log_T[Y]} =1+{Log_T[Y]} + =…. AND

Y=T^{ Log_e[X] }=1+ [{ Log_e[X]}*{Log_e[T]} ]+…+….or

Y= 1+[{ Log_T[Y]}*{Log_e[T]} ] +…+…


According to this involved theory :
Log_e[X] = Log_T[Y] ==Z= Θ( deg)/90=Θ(rad)/[Pi/2]

For BASE T=1 then Y= T^Z=1 [ MODULUS (1) ,ARGUMENT (Θ) ] or
1^Z=1 [ MODULUS{sqrt[cos^2[theta] +sin^2[theta] }=1 ] ,
ARGUMENT (Θ)

or 1^{ Θ/[Pi/2] }=1[ MODULUS ( 1) ,ARGUMENT (Θ) ]
i.e. the ROOTS and POWERS of UNITY ,
And Log_1[1] = Z = Θ/90=Θ/[Pi/2].

This is the case of a CIRCLE, a DEGENERATED SPIRAL OF BASE 1
This CIRCLE passes points on x and y axes :
x=1 , y=1

REF:
http://www.stefanides.gr/why_logarithm.htm

http://mathforum.org/kb/thread.jspa?threadID=1191778&messageID=3882495#3882495
http://www.stefanides.gr/Html/logarithm.htm
http://www.stefanides.gr/Html/Nautilus.htm


© Copyright Panagiotis Stefanides

Panagiotis Stefanides
http://www.stefganides.gr