On the 30-Rhombus Symmetric Solid Object

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

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Section XIII - Five (5) Hexa-Symmetric Structures within the [ 30-Rhombus S-S-O ]

Summary for the Section :

• In this Section , we shall attempt to break-down the [ 30-Rhombus Symmetric Solid Object ]
  • into 5 identical [ Hexa-Symmetric Structures ] .

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

• Let us first bring-in this [ 30-Rhombus S-S-O ] for our use in this Section , and :
  • we again identify :
    • the point [ Point A ] as the point closest to the viewer's eye ,

      and

    • the point [ Point B ] as the point farthest-away .

  

  And we designate , for our discussion purposes below :

  • the 10 [ white-color pieces ] as the [ Top-5 ] and [ Bottom-5 ] pieces ,

  • the 10 [ brown-color pieces ] as the [ Upper-5 ] and [ Lower-5 ] pieces ,

  • the 10 [ yellow-color pieces ] as the [ Middle Ring-of-10 ] .

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Drawing in the 15 Line-Segments :

• In order to show the 5 [ Hexa-Symmetric Structures ] properly ,
  • we must first draw in the 15 line-segments connecting the centroidal-points of each pair of opposite rhombuses .

  And we shall execute this drawing exercise in three (3) stages .

  STAGE I :

  • We shall first draw in the line-segments for the { [ Top-5 ] - [ Bottom-5 ] pairing combinations } .

  

  STAGE II :

  • We shall next draw in the line-segments for the { [ Upper-5 ] - [ Lower-5 ] pairing combinations } .

  

  STAGE III :

  • We can then draw in the line-segments for the pairs within the [ Middle Ring-of-10 ] .

  

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Formation of the 5 identical Structures :

• Let us now consolidate the same-color pieces in the 15 diagrams above ,
  • to arrive at this set of 5 diagrams below .

  

  We see here , that :

  • because the relative positions of the [ 6 pieces-of-the-same-color ] within the [ 30-Rhombus S-S-O ] are always the same ,
    • each of the five (5) Structures above is indeed Hexa-Symmetric and identical to one-another .

• Let us also take note here that :
  • for each of the 5 [ Hexa-Symmetric Structures ] above :
    • the 3 Line-Segments joining the centroidal-points of each [ pair of opposite rhombuses ]

      do in fact intersect one-another at right-angles .

  And we now bring back this set of 6 diagrams from our [ First Attempt ] to construct the [ 30-Rhombus S-S-O ] :

  • for the reader's further review .

  

  

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The 5-Color [ 30-Rhombus S-S-O ] :

• Let us now rbing-in this 5-color [ 30-Rhombus S-S-O ] ,
  • identifying each set of [ 6 same-color-pieces ] forming a [ Hexa-Symmetric Structure ] .

  

• Real Neat , I would say .

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go to the next section : Section XIV - Six (6) sets of Parallel Edges within the [ 30-Rhombus S-S-O ]

go to the last section : Section XII - [ Axis of Bi-Symmetry ] within the [ 30-Rhombus S-S-O ]

return to the 30-Rhombus Symmetric Solid Object HomePage

Original dated 2013-2-04