On the 30-Rhombus Symmetric Solid Object

by Frank C. Fung ( 1st published in February, 2013. )

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Section XVII - Reason for calculating the [ Tangent of 9-degrees ]

Summary for the Section :

• In this Section , we shall give an assessment on the Shape of the Rhombus ,
  • and also explain why we might want to find an expression for the [ tangent of 9-degrees ] based on [ radicals ] .

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Assessing the Shape of the Rhombus :

• Let us recall that the shape of the rhombus may be defined by :
  • the ratio of the lengths of its Major-Axis and Minor-Axis , i.e. :
    • 

  

• Let us also recall that we used our Standard Models for the Dodecahedron and Icosahedron
  • in our [ First Attempt ] to construct the [ 30-Rhombus Symmetric Solid Object ] .

  And :

  • For our Standard Model of the Dodecahedron :
    • 

    • 

  • For our Standard Model of the Icosahedron :
    • 

    • 

  Therefore , for the 30 rhombuses in the [ 30-RHombus S-S-O ] :

  • 

    yielding :

  • 

• And we shall now take a closer look at this [ ratio ] , next .

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On the [ Tangent of PSI ] :

• Let us now bring-in the latest equation on the [ Tangent of PSI ] from Section IV :
  • 

  And we take note here that :

  • this equation is based on the [ Sine of 18-degrees ] .

• Let us now express the [ Sine of 18-degrees ] via the following :

  First :

  • 

  Second :

  • 

  • 

  Consequently , combining the 3 equations above then yields :

  • 

  Thus , once we have found an expression for the [ Tangent of 9-degrees ] based on [ radicals ] ,

  • the [ Tangent of PSI ] would then also be expressible via [ radicals ] .

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On the [ Tangent of PHI ] :

• Let us now bring-in the latest equation on the [ Tangent of PHI ] from Section VI :
  • 

    where :

    • 

  Let us now elaborate on this equation a bit more , and we have :

  • 

    yielding :

  • 

  Consequently :

  • 

  Let us now also do the following manupulations :

  • 

• And the Final Equation here is then :
  • 

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Summarizing :

• In the Structural Dimensioning of the [ 30-Rhombus S-S-O ] , the following 3 items :
  • the [ Tangent of PSI ] ,

  • the [ Tangent of PHI ] , and

  • the [ Ratio defining the shape of the Rhombus ] ;

    are always of utmost importance .

  Since :

  • the [ Tangent of PHI ] is already expressible via [ radicals ] ,

  the only thing left to do :

  • is to find an expression for the [ Tangent of PSI ] based on [ radicals ] .

• In the next 3 Sections to come ,
  • we shall be attempting to find an expression for the [ Tangent of 9-degrees ] based on [ radicals ] .

  And once that is done ,

  • the Structural Dimensioning process for the [ 30-Rhombus S-S-O ] can be deemed completed .

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go to the next section : Section XVIII - Bringing-in the Tangent Addition Formula

go to the last section : Section XVI - Co-Planar Axis within the [ 30-Rhombus S-S-O ]

return to the 30-Rhombus Symmetric Solid Object HomePage

Original dated 2013-2-04