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Geometrical interpretation and generation of Benford Sets

A new approach to Benford 's Law.

At last , there is a (published method of generation , for Benford Sets)

Theorem of Elmuhsi

Part 1

a-

b-

r , may be taken any value , in that case (A) will be altered to : A = [ r . tan(t) ]2 
and nothing will be changed as far as the theorem is concerned , since Benford's law is a SCALE INVARIANT law , and we have a new scale as (r2 ) to be multiplied by.

c-

Instead of using the square as the target area , some special geometrical shapes may be INSCRIBED into the square, such as a (circle) or (a triangle with the base and height equal to h ) . Then the (areas) of these special shapes will again obey to BENFORD'S LAW , because we have new scales coming from the appropriate shapes of the system , and Benford's law is scale invariant.

d-

The facts told in proposition (c) , is true for the (surface areas and volumes) of some special 3D shapes which are inscribed into the (cube volume h3 ) formed by the length (h) . (example ; a sphere)

How to check the theorem :

you may download two zip files ;

If you have celeron processor and 32 MB of ram then you can download the file;

Elmuhsi (10000 element sets)

If you have higher than celeron processor and 64 MB of ram then you can download the following files both.

Elmuhsi (10000 element sets)

Elmuhsi (65000 element sets)

after loading , unzip the files by  WinZip or another suitable unzip facility

there are three files in both packages;

 

1-analys1  (an exe program where you can execute from DOS prompt or WINDOWS

makes analysis for leading digit distribution )

 

2-analys2  (an exe program where you can execute from DOS prompt or WINDOWS

makes analysis for second digit distribution )

 

3-An Excel 2000 program written for 10000 or 65000 element Benford Sets

If you don't have "excel 2000" or have a lower version of microsoft Excel , then you 

can download excel viewer from the following site  (or from other similar sites);

http://support.microsoft.com/support/downloads/DP1846.ASP

 

But please remember that you cannot be able to (execute the macros or see the formulas in the cells)

by a viewer ,  instead you'll only see the sheets as if looking to a printout .

If you want to use all the facilities provided in these excel sheets , then you have to install "Excel 2000"

or bring the sheets to a machine where you have already "Excel 2000" installed.

 

Part 2

 

Generation of Benford sets by dividing a fixed line of length (d) into two ; 
( x , d-x ) and taking the ratio   r=(d/x)-1  as a generator of  Benford set.

 

 

 

How to check the theorem :

you may download a zip file ;

 Elmuhsi Prt 2 (10000 element sets)

after loading , unzip the file by  WinZip or another suitable unzip facility

there are three files in the package;

 

1- lratio1  (an exe program where you can execute from DOS prompt or WINDOWS

makes analysis for leading digit distribution )

 

2- lratio2  (an exe program where you can execute from DOS prompt or WINDOWS

makes analysis for second digit distribution )

 

3-An Excel 2000 program written for 10000  element Benford Sets

you can do the same things as in part1 to view these files.

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