Tetrahedron and Full-Body-Rolls

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

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

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Section XXXIII - The Tetrahedron and the Full-Body-Rolls

Summary for the Section :

In this Section , we shall be looking into Full-Body-Rolls ,
- and the demonstration shall be via the Tetrahedron and the Cube .

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Setting up the Tetrahedron :

Let us now set up the 3 orthogonal axis , the Cube and the Tetrahedron for our use in this Section ,
- as per this set of 4 diagrams below .
Let us now also identify the 4 [ Axis of Tri-Symmetry ] for the Tetrahedron , as follows :

And we simply note here that the [ Axis TR-1 / TR-2 / TR-3 / TR-4 ] ,
- are in fact the 4 [ cross-diagonals ] of the Cube .
Let us also identify , at this point , the 3 [ Axis of Bi-Symmetry ] for the Tetrahedron , as follows :
- And these 3 axis , [ Axis BI-1 / BI-2 / BI-3 ] are in fact the [ X / Y / Z Axis ] .
  - as per the diagram on-the-very-right above .

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Rotations of 120 / 240 / 360 degrees on [ Axis TR-1 ] :

Let us now set up the standard for comparison for the remainder of this Section :

First ,
- we identify the Tetrahedron in [ Figure 33-0 ] on-the-left , below ,
  - as the [ Starting Position ] before any rotations of any sort .
Secondly ,
- we rotate the Tetrahedron thru the angles of [ 120 / 240 / 360 degrees ] ,
  - using [ Axis TR-1 ] as the axis-of-rotation ,
    
    to arrive at [ Figure 33-1 ] , [ Figure 33-2 ] , and [ Figure 33-3 ] as shown above .

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The First Conjecture on Full-Body-Rolls :

Let us now bring-in the [ Starting Position ] for the Tetrahedron from above and do the following :

Step One :
- We do a [ 90 degrees counter-clockwise rotation ] on the Tetrahedron ,
  - using [ Axis BI-1 ] as the axis-of-rotation , ( i.e. : the [ +X Axis ] )
    - to arrive at [ Figure 33-4F ] and [ Figure 33-4-T ] below .
Step Two :
- We do a [ 90 degrees counter-clockwise rotation ] on the resultant Tetrahedron ,
  - using [ Axis BI-3 ] as the axis-of-rotation , ( i.e. the [ +Z Axis ] )
    - to arrive at [ Figure 33-5F ] and [ Figure 33-5T ] below .
And we simply note here that [ Figure 33-5F ] is the same as [ Figure 33-1 ] above when we did the [ 120 degrees rotation on Axis-TR-1 ] .
Here come the conjecture :

First Conjecture on Full-Body-Rolls :
- Any 2 consecutive [ 90-degrees rotations ] in any 2 orthogonal-axis in the [ Axis BI-1 / BI-2 / BI-3 ] 3-axis-set ,
  
  is equivalent to :
  - a single [ 120 degrees rotation ] in a selected axis from the [ Axis TR-1 / TR-2 / TR-3 / TR-4 ] 4-axis-set .
And here is where the fun begins !

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The [ 6-elements loop ] for [ Full-Body-Rolls ] :

Let us now do this [ 6-rolls ] rotation-sequence for the Full-Body-Rolls :
- a [ 90-degrees counter-clockwise rotation ] using [ Axis BI-1 ] as the axis-of-rotation ;
- a [ 90-degrees counter-clockwise rotation ] using [ Axis BI-3 ] as the axis-of-rotation ;
- a [ 90-degrees counter-clockwise rotation ] using [ Axis BI-2 ] as the axis-of-rotation ;
- a [ 90-degrees counter-clockwise rotation ] using [ Axis BI-1 ] as the axis-of-rotation ;
- a [ 90-degrees counter-clockwise rotation ] using [ Axis BI-3 ] as the axis-of-rotation ;
- a [ 90-degrees counter-clockwise rotation ] using [ Axis BI-2 ] as the axis-of-rotation .
We then bring-in this set of diagrams for the 6 rotations :

And we see here that the final [ Figure 33-11 ] here is the same as the starting-position [ Figure 33-0 ] ,
- which actually confirms that the above 6-Full-Body-Rolls rotation-sequence is indeed a [ 6-elements loop ] .
And here comes the very special catch ; so watch-out !

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Bringing-in the Strobe-Light :

Let us suppose for the moment that we were able to do the above 6 rotations in a time-span of [ 6 seconds ] :
- i.e. : 1 rotation per second .
Let us also suppose that we did the 6 rotations in a very-dark room and used a Strobe-Light flashing at [ 1 second intervals ] :
- to take the these 7 pictures :
Just suppose for the moment that the Strobe-Light malfunctioned and flashed at [ 2 seconds intervals ] ,
- so that we only get this set of 4 pictures below .
Question :
- Is there any possibility of us mistaking this set of 4 pictures above ,
  - for the series we have previously for the [ 120 / 240 / 360 degrees rotations on Axis-TR-1 ] ?
Answer ???
- No way Jose ! Just fix the Strobe-Light , huh ?

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Second Conjecture on the [ Full-Body-Rolls ] :

And for the finale of this Section , we bring-in another conjecture for the Full-Body-Rolls .

Second Conjecture on Full-Body-Rolls :
- Step One :
  - Take any Solid Object of any size , shape and colors ,
    - and identify any point within the body of the Solid Object as the [ stationary / non-moving point ] ,
      - and label this as the [ Point-Of-Origin ] .
- Step Two :
  - Set up three (3) orthogonal axis passing thru the [ Point-Of-Origin ] ,
    - and label these as the [ 1st-Axis ] , the [ 2nd-Axis ] and the [ 3rd-Axis ] , in any order you wish .
- Step Three :
  - Do 3 consecutive [ 180 degrees rotations ] on the Solid Object :
    - the 1st rotation in the [ 1st-Axis ] ,
    - the 2nd rotation in the [ 2nd-Axis ] , and
    - the 3rd rotation in the [ 3rd Axis ] .
- And you will always end-up at exactly where you started-from originally-in-the-first-place .
See , [ Change-But-No-Change ] :
- that was what we said back in 2004 in An Approach to the I-CHING .

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go to the next section : Section XXXIV - Setting up the 210 [ sets-of-24 ] with Physical Symmetries

go to the last section : Section XXXII - Revisting Mirror Images

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

Original dated 2011-3-06