On the 30-Rhombus Symmetric Solid Object

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

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Section IX - Second Attempt at the [ 30-Rhombus Symmetric Solid Object ]

Summary for the Section :

• In this Section , we shall make our 2nd attempt to construct the [ 30-Rhombus Symmetric Solid Object ] .

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Bringing-in the Dodecahedron and the Icosahedron :

• Let us now bring-in the Dodecahedron as shown below in this special orientation .

  

  And we also bring-in the Icosahedron as shown below in a corresponding / matching orientation .

  

• We then combine the 20 Nodal-Points of the Dodecahedron and the 12 Nodal-Points of the Icosahedron :
  • to arrive at this 32 Nodal-Points Object as shown in the diagram in-the-middle below .

  

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The Resultant [ 30-Rhombus Symmetric Solid Object ] :

• We then present this Second Version of the resultant [ 30-Rhombus Symmetric Solid Object ] :

  

  Again , we have a problem here :

  • There are 30 Rhombuses here but :
    • only 10 Rhombuses showed up in the [ Front View ] ,

      and

    • another 10 Rhombuses showed up in the [ Transparent / See-Thru Back View ] .

    Ten (10) Rhombuses are in fact hidden-from-view :

  

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A first look at the [ Axis of Penta-Symmetry ] :

• However , our 2nd attempt here also serves a specific purpose .

  Let us now identify :

  • the point closest to the viewer's eye as the point [ Point A ] ,

    and

  • the point farthest-away from the viewer's eye as the point [ Point B ] ;
    • as shown below .

  

• Let us just state , at this point , that :
  • the straight-line passing thru [ Point A ] and [ Point B ] is in fact an [ Axis of Penta-Symmetry ] .

  And we shall now do the following :

  • color the 5 Rhombuses sharing [ Point A ] as a common Nodal-Point in [ cyan-color ] ,

  • color the 5 Rhombuses sharing [ Point B ] as a common Nodal-Point in [ blue-color ] ,

  • color the remaining 5 rhombuses in the Front View in [ purple-color ] ,

  • color the remaining 5 rhombuses in the Transparent/See-Thru View in [ magenta-color ] ,

  • color the 10 rhombuses hidden-from-view alternately in [ light-green / dark-green color ] ;
    • as shown below .

  

  Very simply :

  • when we rotate the [ 30-Rhombus Symmetric Solid Object ] thru the angles of [ 72 / 144 / 216 / 288 / 360 degrees ] ,
    • using an [ axis of penta-symmetry ] as the axis-of-rotation ,

    the resultant [ 30-Rhombus Symmetric Solid Object ] will look exactly the same as before the said rotation .

• But more on this later in Section XI .

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go to the next section : Section X - Third Attempt at the [ 30-Rhombus Symmetric Solid Object ]

go to the last section : Section VIII - First Attempt at the [ 30-Rhombus Symmetric Solid Object ]

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Original dated 2013-2-04