On Symmetric Structures within Symmetric Solid Objects

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

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Section XXXII - Ratios-Of-Areas and the Egyptian Fractions

Summary for the Section :

• In this Section , we shall take a quick look at the Ratio-Of-Areas for the Spherical Solid Objects ,
  • leading onto the [ Egyptian Fractions ] .

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Setting up the [ Base Area ] for comparison purposes :

• We take note here that :
  • with only three (3) [ Nodal Points ] , the best we can do is to construct a [ Triangle ] ,
    • which is a [ planar structure ] .

  Therefore , we need a minimum of at least four (4) [ Nodal Points ] to construct a [ 3-Dimensional Object ] , so that :

  • the Tetrahedron with 4 [ nodal points ] is indeed basic to all [ 3-Dimesional Solid Objects ] .

• Thus , we shall choose the Tetrahedron and the associated [ Tetrahedral Sphere ] as our [ base structure ] for comparison purposes .

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Ratios-Of-Area Comparisons :

• For our comparison purposes here , we shall now use :
  • the area of an [ individual piece ] for the 4-Split [ Tetrahedral Sphere ] as the [ Base Area ] for comparison .

  And the [ Base Area ] is then given by :

  • [ pi ] * [ R-Square ] ,

    which has same format as the formula we use for the [ area-of-a-circle ] .

• We then have this table below for the Comparison :

  Name of Spherical Solid Object	Area of an Indivdual Piece	Ratio-Of-Areas Individual Piece vs. Base Area
  Tetrahedral Sphere ( 4-Split )	[ pi ] * [ R-Square ] * [ 1 / 1 ]	[ 1 / 1 ] : [ 1 ]
  Quasi-Cubic Sphere ( 6-Split )	[ pi ] * [ R-Square ] * [ 2 / 3 ]	[ 2 / 3 ] : [ 1 ]
  Octahedral Sphere ( 8-Split )	[ pi ] * [ R-Square ] * [ 1 / 2 ]	[ 1 / 2 ] : [ 1 ]
  Dodecahedral Sphere ( 12-Split )	[ pi ] * [ R-Square ] * [ 1 / 3 ]	[ 1 / 3 ] : [ 1 ]
  Icosahedral Sphere ( 20-Split )	[ pi ] * [ R-Square ] * [ 1 / 5 ]	[ 1 / 5 ] : [ 1 ]

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The Egyptian Fractions :

• We take note that the Egyptian Fraction is comprised mainly of :
  • the number [ 1 ] , being equal to [ 1 / 1 ] ;
  • the special fraction [ 2 / 3 ] ; and :
  • all [ recipricals ] .

  And all other { [ fractions ] less than [ 1 ] } may be expressed as :

  • the sum of Egyptian Fractions .

• And in the table above , we have identified the most important 5 of the [ Eqyptian Fractions ] , namely :
  • [ 1 / 1 ] , [ 2 / 3 ] , [ 1 / 2 ] , [ 1 / 3 ] , and [ 1 / 5 ] .

  Were we able to put a [ geometric meaning ] to these five (5) [ fractions ] ?

  • Yes ! The Author think so .

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go to the next section : Section XXXIII - Duality : Tetrahedron vs. Tetrahedron

go to the last section : Section XXXI - The Icosahedral Sphere ( 20-Split )

return to the Symmetric Solid Objects HomePage

Original dated 2010-3-31