On a Quick Expansion Method for Pythagorus Triples

by Frank C. Fung ( 1st published in December, 2021. )

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Section XXXX - Concluding Remarks

Concluding Remarks :

• The question then arises as to whether [ OMEGA ] / [ LAMBDA ] / [ THETA ] in the [ One-On-One Expansion Formula ] :
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    may have any physcial meaning attached to them .

  And the answer is YES .

• To explain , let us first mark off [ Point E ] as the mid-point of { Side B-C } of { Triangle A-B-C } .

  

  And we see here that [ 3*W ] is in fact the length of [ Median A-E ] :

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  Consequently , [ OMEGA ] is therefore an expression of an area , i.e. :

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• Let us now take quick look at [ LAMBDA ] :
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  Consequently , we have :

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  This then tell us that the { Square-root of [ LAMBDA ] } is the { length of Line Segment B-E } :

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    as marked-off via the cyan-color triangle in the diagrams above .

  And therefore :

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• Let us take a quick look at the angle [ Angle THETA ] .

  We then bring in again this relation for [ LAMBDA ] :

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  This then yields us this relation below :

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  We then set up :

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  • 

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  As such , [ Angle THETA ] is a derived angle arising from the { Triangle as marked in cyan-color } in the diagrams above .

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An Observation in passing :

• Let us bring back this expression for [ Z ] from above :
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  Since [ OMEGA ] and [ LAMBDA ] are both 2-dimensioned ,

  • the format for [ Z ] in the above expression might very well be 6-dimensioned .

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Original dated 2021-12-27