On the 30-Rhombus Symmetric Solid Object

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

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Section XII - [ Axis of Bi-Symmetry ] within the [ 30-Rhombus S-S-O ]

Summary for the Section :

• In this Section , we shall identify the [ Axis of Bi-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 from the last Section :
  • the first set of 3 diagrams arising from the [ Third Attempt ] , and

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

  

  

  Question :

  • do we see an [ axis of bi-symmetry ] here .

• In fact , we do :
  • as per this set of six (6) diagrams below .

  

  

• Very simply :
  • any straight-line passing thru the centroids of a pair of [ opposite rhombuses ] on the [ 30-Rhombus S-S-O ]
    • is an [ axis of bi-symmetry ] .

  And in the above set of 6 diagrams ,

  • when we rotate the [ 30-Rhombus S-S-O ] thru an angle of [ 180-degrees ] ,
    • using the [ axis of bi-symmetry ] as marked-off in [ red-color ] as the [ axis-of-rotation ] ;

    the following happens :

    • the 2 rhombuses as marked-off in [ yellow-color ] do [ self-rotations about its centroid-point ] to retain the same posture ,

    • the 10 rhombuses as marked-off in [ cyan-color ] exhange positions ,

    • the 10 rhombuses as marked-off in [ purple-color ] exhange positions ,

    • the 4 rhombuses as marked-off in [ light-green-color ] exhange positions ,

    • the 4 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 [ 180-degrees rotation ] .

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

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

  Very simply :

  • There are exactly 30 Rhombuses in the [ 30-Rhombus S-S-O ] and :
    • these 30 rhombuses can be divided to form exactly 15 pairs of [ opposite rhombuses ] .

  As such ,

  • there are exactly 15 [ Axis of Bi-Symmetry ] within the [ 30-Rhombus S-S-O ] .

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One last look at the [ Axis of Bi-Symmetry ] :

• It is appropriate to bring-in , at this point , this set of 9 diagrams , with :
  • the 1st row of 3 diagrams being old diagrams from above ,

  • the 2nd row of 3 diagrams being new diagrams looking-in from directly in-front of the [ Axis of Bi-Symmetry ] ,

  • the 3rd row of 3 diagrams being old diagrams from above .

  
  
  

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go to the next section : Section XIII - Five (5) [ Hexa-Symmetric Structures ] within the [ 30-Rhombus S-S-O ]

go to the last section : Section XI - [ Axis of Penta-Symmetry ] within the [ 30-Rhombus Symmetric Solid Object ]

return to the 30-Rhombus Symmetric Solid Object HomePage

Original dated 2013-2-04