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By Kardi Teknomo, PhD. 
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The power of digital roots has complexity pattern. On the left side, it is almost regular but getting to the right (higher number of exponent) the pattern is irregular. The boundary between irregular and regular pattern is a curve with two asymptotes in the bases and in the power. The part of irregular pattern is really complex that one cannot predict what will be the next pattern. It is quite amazing that such random pattern can be produced using simple deterministic formula of digital root 
Practically, the pattern of digital root power cannot be produce using simple formula  because of Mod function in most programming language usually cannot take very large number. One must use iteration method (macro) to produce such pattern [See the computer code].
Comments: The apparent complex pattern above may be generated due to inaccuracy of
round off error as suggested by the comment below.
Hi Kardi,
I have a general interest in complexity and was intrigued by the
pattern that appeared to be associated with the powers of digital
roots that you published. However, the breakdown of the pattern was
not like the normal onset of chaos.
It was noticeable in the pattern that the solid colour pattern
associated with multiples of nine indicating a digital root of nine
broke down more quickly the larger the multiple. I counted the pixels
and checked the first break down for several multiples of nine on my
calculator. Each time I got a standard representation instead of the
full integer representation. When I reduced the power by one I got
the full integer number and when I multiplied this by nine manually
and calculated the digital root the answer was nine - not the
deviation the pattern indicated.
The maximum number of digits before swapping to standard notation was
14. In .NET and most recent versions of visual basic a long integer
can have up to 19 digits. However, this would not give a sufficient
range of numbers for the pattern you were trying to create. I suspect
therefore that you used double precision numbers which given a much
greater range but unfortunately are only accurate to 14 significant
figures.
You could get around the problem by using the big number class in Java
or writing your own long hand multiplication routines, a language like
Perl or Lisp would probably be ideal although I am sure you could do
it easily enough in VB.
Nigel Phillips
Programme Director BIT
Information Systems & Information Technology
Faculty of Business, Computing and Information Management
London South Bank University
London SE1 0AA
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Download the MS Excel companion of this tutorial here
This tutorial is copyrighted.
Preferable reference for this tutorial is
Teknomo, Kardi. Digital Root. http:\\people.revoledu.com\kardi\
tutorial\DigitSum\
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