On Symmetric Structures within Symmetric Solid Objects

by Frank C. Fung ( 1st published in March, 2010. )

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Section II - A Special Note on the Notations

A Special Note on the words [ Triangle ] , [ Tetrahedron ] , and [ Pentagon ] :

• In the Author's previous papers , we have worked with several different types of tetrahedra :
  • the Equilateral-Triangle Tetrahedron , ( the ET-Tetrahedron ) ,
  • the Congruent-Triangle Tetrahedron , ( the CT-Tetrahedron ) ,
  • the general Tetrahedron ; and
  • the Flattened Tetrahedron .

  In this paper , we shall now use these three (3) words below with special and specific meanings :

  • the word [ Triangle ] will carry meaning of an Equilateral Triangle ,
  • the word [ Tetrahedron ] will carry the meaning of an Equilateral Tetrahedron , and
  • the word [ Pentagon ] will carry the meaning of an Equilateral Pentagon .

    unless specifically stated to the contrarary .

• And this is done as a matter of convenience :
  • since our primary concern in this paper here is with [ symmetric shapes ] only .

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A Special Note on the initial presentation of the Symmetric Solid Objects :

• For each of the five (5) Symmetric Solid Objects , we shall always initially provide a set of 3 views :
  • the [ Isometric View ] ,
  • the [ Tilted Z-Axis View ] , and
  • the [ Final View ] .

  

  

• From a technical stand-point , the procedure to arrive at the 3 view is as follows :
  • We always start with the [ Isometric View ] , as shown on-the-left above ,

  • We then tilt the Z-Axis backwards-into-the-paper , i.e. :
    • a clockwise-rotation of the Symmetric Solid Object thru an angle of [ 11.75 degrees ] ,
      • using the axis as marked in [ red-color ] above as the axis-of-rotation ;

      to arrive at the orientation for the [ Tilted Z-Axis View ] .

  • We then rotate the Symmetric Solid Object about the now-localized [ Z-axis ] , i.e. :
    • a counter-clockwise rotation of the Symmetric Solid Object thru an angle of [ 28.50 degrees ] ,
      • using the axis as marked in [ blue-color ] above as the axis-of-rotation ;

      to arrive at the orientation for the [ Final View ] .

• We do ask the reader to come-back to this Section to review the orientation of the [ X-Y-Z axis ] for the [ Final View ] ,
  • whenever problems in visualizing the Symmetric Solid Object may have arisen .

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go to the next section : Section III - Basic Properties of the Tetrahedron

go to the last section : Section I - Introduction

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31