On 8-Colors Combinatorics and [ Quasi 24-Symmetry ]

by Frank C. Fung ( 1st published in March, 2011. )

Top Of Page

Section XVIII - The Reduced Set of 5,040 [ 8-colors configurations ]

Summary for the Section :

• In this Section , we shall also start with [ Configuration Annapurna ] , but :
  • will only permit rotations using the [ -X / -Y / -Z Axis ] as the axis-of-rotation .

go to Top Of Page

The reduced set of 5,040 of [ 8-colors configurations ] :

• Let us start again with [ Configuration Annapurna ] and we take special notice here that :
  • If we permit only :
    • [ 1 / 2 / 3 / 4-notch rotations ] using the [ -X / -Y / -Z Axis ] as the axis-of-rotation ;

    the [ red-color piece ] will always stay in the same position and does not move at-all .

• Let us now do a quick illustration here for a better understanding of the mechanics involved :

  For the [ -X Axis rotations ] , we have :

  

  • And the [ red-color piece ] did not move here .

  For the [ -Y Axis rotations ] , we have :

  

  • And the [ red-color piece ] did not move here either .

  For the [ -Z Axis rotations ] , we have :

  

  • And the [ red-color piece ] also did not move here .

• Since the [ Red-Color Piece ] remains in the same position and does not move :
  • when we allow only rotations using the [ -X / -Y / -Z Axis ] as the axis-of-rotation ;

  we would therefore be looking at a reduced set of 5,040 possible [ 8-colors configurations ] ,

  • under such circumstances and restrictions .

    ( 5,040 = 7 x 6 x 5 x 4 x 3 x 2 x 1 )

go to Top Of Page

The First Question on the 5,040 [ 8-colors configurations ] :

• Our First Question here on the 5,040 [ 8-colors configurations ] is simply this :
  • If we were to start with [ Configuration Annapurna ] and permit an unlimited number of :
    • [ 1 / 2 / 3 / 4-notch rotations ] using the [ -X / -Y / -Z Axis ] as the axis-of-rotation ,

    would be be able to reach each-and-every one of the reduced set of 5,040 [ 8-colors configurations ] ?

  And the answer here is YES .

  ( The analysis here is done thru a computer programming procedure . )

go to Top Of Page

The Second Question on the 5,040 [ 8-colors configurations ] :

• Our Second Question here is this :
  • If we were to start with [ Configuration Annapurna ] and permit an unlimited number of :
    • [ 2-notch rotations ] using the [ -X / -Y / -Z Axis ] as the axis-of-rotation ,

    would be be able to reach each-and-every one of the reduced set of 5,040 [ 8-colors configurations ] ?

  And the answer here is NO .

• In fact :
  • we would only reach a set of 24 [ 8-colors configurations ] :
    • [ Configuration Annapurna ] including ;

    if we were to permit only [ 2-notch rotations ] in the [ -X / -Y / -Z Axial-Directions ] .

  ( Again , the analysis here is done thru a computer programming procedure . )

go to Top Of Page

A First Look at the 24 [ 8-colors configurations ] :

• Let us now take a first look at these 24 [ 8-colors configurations ] ,
  • as per the diagram on-the-left below .

  ( [ Configuration A ] here is this diagram is the same as the configuration [ Configuration Annapurna ] . )

  

• And the diagram on-the-right above is then the Linkage Diagram :
  • showing the 36 links linking the 24 [ 8-colors configurations ] .

  And a few quick words here on the Linkage Diagram :

  • We have started with [ Configuration A / Configuration Annapurna ] as the core-configuration here ,
    • at [ Position A ] at the very-left of the said diagram .

  • There are three (3) types of links here :
    • each [ Brown-Color Line ] represents a [ 2-notch -X Axis rotation ] link ,

    • each [ Dark-Green-Color Line ] represents a [ 2-notch -Y Axis rotation ] link , and

    • each [ Deep-Blue-Color Line ] represents a [ 2-notch -Z Axis rotation ] link .

  • Each of the 36 links is [ reversible ] , in the sense that :
    • both the [ forward link ] and the [ backward link ] may be simply treated as [ 2-notch rotations ] on the same axis-of-rotation .

go to Top Of Page

Why are the 2-notch [ forward link ] and [ backward link ] reversible :

• A quick explanation here on why the 2-notch [ forward link ] / [ backward link ] are reversible .

  And the explanation is as follows :

  • IF :
    • the [ forward-link ] is a [ 180-degrees rotation in the counter-clockwise direction ] on any given axis-of-rotation ,

    THEN :

    • the [ reverse-link ] should be a [ 180-degrees rotation in the clockwise direction ] on the same axis-of-rotation .

    And vice versa .

  • BUT :
    • the result of a [ 180-degrees rotation in the counter-clockwise direction ] on a given axis-of-rotation

      and

    • the result of a [ 180-degrees rotation in the clockwise direction ] on the same axis-of-rotation

      are always the same ;

      • i.e :
        • you will always arrive at the exact-same new [ 8-colors configuration ] going either way ,

          counter-clockwise and clockwise .

  • As such , for our Linkage Diagram purposes here , the said 2 rotations :
    • the [ 2-notch counter-clockwise rotation ] vs. the [ 2-notch clockwise rotation ] ,

      are indeed freely interchangeable .

    Hence , we can now use a [ 2-notch rotation in the counter-clokcwise direction ] :

    • to cover both the [ forward link ] and the [ backward link ] at-the-same-time .

• But more on the Linkage Diagrams later-on in Section XXV .

go to Top Of Page

Naming the set of 24 [ 8-colors configurations] here :

• Let us now give the above-said set of 24 [ 8-Colors configurations ] a specific name ,
  • for identification purposes and easier referrals later-on .

  And we shall call this set of 24 [ 8-colors configurations ] :

  • the [ 24 QSM-Configurations --- Himalaya Series ] .

  

  ( QSM stands for Quasi-SymMetric and Annapurna is a mountain in the Himalaya Mountain Range . )

• And our analysis from-here-on-in shall be based primarily on this set 2 diagrams immediately above .

go to Top Of Page

go to the next section : Section XIX - Converting the [ 8-colors configurations ] into [ 8-numbers strings ]

go to the last section : Section XVII - The Global Set of 40,320 [ 8-colors configurations ]

return to the 8-Colors Combinatorics / Quasi 24-Symmetry HomePage

Original dated 2011-3-06