On the 30-Rhombus Symmetric Solid Object

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

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Section XV - [ Axis of Tri-Symmetry ] within the [ 30-Rhombus S-S-O ]

Summary for the Section :

• In this Section , we shall identify the [ Axis of Tri-Symmetry ] within the [ 30-Rhombus S-S-O ] .

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Bringing-in the [ 30-Rhombus S-S-O ] :

• Let us now bring-in these 6 diagrams based on diagrams we have seen before :
  • the first set of 3 diagrams arising from the [ Third Attempt ] , and

  • the second set of 3 diagrams arising from the [ First Attempt ] .

  

  

• Very simply , an [ Axis of Tri-Symmetry ] may be constructed as follows :
  • First ,
    • we identify any nodal-point that is common to exactly three (3) rhombuses as the point [ Point A ] .

  • Next ,
    • we identify the nodal-point opposite to [ Point A ] as the point [ Point B ] .

  • Then ,
    • the straight-line passing thru [ Point A ] and [ Point B ] is an [ Axis of Tri-Symmetry ] .

  And in the above set of 6 diagrams ,

  • when we rotate the [ 30-Rhombus S-S-O ] thru an angle of [ 120 / 240-degrees ] ,
    • using the [ axis of tri-symmetry ] as marked-off in [ red-color ] as the [ axis-of-rotation ] ;

    the following happens :

    • the 6 rhombuses as marked-off in [ blue-color ] exchange positions ,

    • the 6 rhombuses as marked-off in [ cyan-color ] exhange positions ,

    • the 3 rhombuses as marked-off in [ magenta-color ] exhange positions ,

    • the 3 rhombuses as marked-off in [ dark-yellow-color ] exhange positions ,

    • the 3 rhombuses as marked-off in [ purple-color ] exhange positions ,

    • the 3 rhombuses as marked-off in [ yellow-color ] exhange positions ,

    • the 3 rhombuses as marked-off in [ light-green-color ] exhange positions , and

    • the 3 rhombuses as marked-off in [ dark-green-color ] exhange positions .

  And the [ 30-Rhombus S-S-O ] will look exactly the same before-and-after the said [ 120 / 240-degrees rotations ] .

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How many [ Axis of Tri-Symmetry ] are there :

• The question then arises as to how many [ Axis of Tri-Symmetry ] are there ?

  And we see from this set of 3 diagrams below that :

  • the [ nodal-points in-question ] here are originally the 20 nodal-points of the Dodecahedron .

  

  Since :

  • there are exactly 20 nodal-points associated with the Dodecahedron ,

  it then follows that :

  • these 20 nodal-points can be divided to form exactly 10 pairs of [ opposite nodal-points ] .

  As such ,

  • there are exactly 10 [ Axis of Tri-Symmetry ] within the [ 30-Rhombus S-S-O ] .

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

go to the last section : Section XIV - Six (6) sets of Parallel Edges within the [ 30-Rhombus Symmetric Solid Object ]

return to the 30-Rhombus Symmetric Solid Object HomePage

Original dated 2013-2-04