Share this on:
 E-mail
13
VIEWS
0
COMMENTS
 
SHARES
About this iReport
  • Not verified by CNN

  • Click to view PCSTEFANIDES's profile
    Posted December 13, 2012 by
    PCSTEFANIDES
    Location
    KIFISSIA, ATHENS, Greece
    Assignment
    Assignment
    This iReport is part of an assignment:
    Tech talk

    More from PCSTEFANIDES

    SPIRALOGARITHM HAND DRAWN By Panagiotis Stefanides

     

    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

    What do you think of this story?

    Select one of the options below. Your feedback will help tell CNN producers what to do with this iReport. If you'd like, you can explain your choice in the comments below.
    Be and editor! Choose an option below:
      Awesome! Put this on TV! Almost! Needs work. This submission violates iReport's community guidelines.

    Comments

    Log in to comment

    iReport welcomes a lively discussion, so comments on iReports are not pre-screened before they post. See the iReport community guidelines for details about content that is not welcome on iReport.

    Add your Story Add your Story