Octo-Symmetry for the { 4-Dimension Quasi-Octahedron } ,

and 2 Identical { 8-D Structures } oriented Symmetrically

- This paper started out as an attempt for a better understanding of :
- the { 28 Symmetric Lines in 7-Dimension } intersecting one-another at the angle of [ 70.53 degrees ] ,
- i.e. :
- arccos [ 1 / 3 ] = [ 70.53 degrees ] .

- i.e. :

We came around full-circles to establish that there are :

- { 16 Symmetric Lines in 6-Dimension } intersecting one-another at the angle of [ 70.53 degrees ] ,
- { 10 Symmetric Lines in 5-Dimension } intersecting one-another at the angle of [ 70.53 degrees ] ,
- { 6 Symmetric Lines in 4-Dimension } intersecting one-another at the angle of [ 70.53 degrees ] ,
- { 4 Symmetric Lines in 3-Dimension } intersecting one-another at the angle of [ 70.53 degrees ] ,
with :

- each { set of symmetric lines in a lower-Dimension } being a sub-set of the { set of symmetric lines in the next-higher-Dimension } .

- the { 28 Symmetric Lines in 7-Dimension } intersecting one-another at the angle of [ 70.53 degrees ] ,

- We were able to split up the 32 Cross-Diagonal Lines of the { 6-Dimension Quasi-Cube } into 2 identical structures :
- { Structure A } with 16 Lines intersecting one-another at the angle of [ 70.53 degrees ] , and
- { Structure B } with 16 Lines intersecting one-another at the angle of [ 70.53 degrees ] .

Interesting enough , we were able to identify :

- a specifing set of 4 Lines in { Structure A } in the formation of { the 4 Cross-Diagonal Lines of a 3-D Cube } ,
and

- a matching set of 4 Lines in { Structure B } in the formation of { the 4 Cross-Diagonal Lines of a 3-D Cube } ;
so that :

- the 1st [ set of 4 Lines in Structure A ] and the 2nd [ set of 4 Lines in Structure B ] are orthogonal .

60 pairs of such matching { sets of 4-Lines } were found .

- We were able to split up the 128 Cross-Diagonal Lines of the { 8-Dimension Quasi-Cube } into 2 identical Structures ,
- and the 2 { 64-Lines Strutctures } are oriented symmetrically .

- We were able to verify that the 96-triangles { 4-D Quasi-Octahedron } is also Octo-Symmetric in nature .

- For each-and-every { 16 Symmetric-Lines Structure in 6-D } :
- we can always construct another { 16 Symmetric-Lines Structure in 6-D } based there-upon ,
so that :

- the 2 { 16-Lines Structures } will share exactly one Line in common .

- we can always construct another { 16 Symmetric-Lines Structure in 6-D } based there-upon ,