Question about 3 || gybef

[ndl]

I put the following coordinates into stella for the 3 || gybef:

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1.00000000000000000  0.70710678118654752  0.00000000000000000  0.00000000000000000 
-1.00000000000000000  0.70710678118654752  0.00000000000000000  0.00000000000000000 
0.00000000000000000 -0.70710678118654752  1.00000000000000000  0.00000000000000000 
0.00000000000000000 -0.70710678118654752 -1.00000000000000000  0.00000000000000000 
1.00000000000000000  0.00000000000000000  1.00000000000000000  1.58113883008418967
1.00000000000000000  0.00000000000000000 -1.00000000000000000  1.58113883008418967
-1.00000000000000000  0.00000000000000000  1.00000000000000000  1.58113883008418967
-1.00000000000000000  0.00000000000000000 -1.00000000000000000  1.58113883008418967
0.00000000000000000  1.41421356237309505  0.00000000000000000  1.58113883008418967
1.00000000000000000  0.70710678118654752  0.00000000000000000  3.16227766016837933
-1.00000000000000000  0.70710678118654752  0.00000000000000000  3.16227766016837933


according to lace city on Klitzing's website, but stella's convex hull algorithm cut the tets running in between the 2nd 2 layers lengthwise to create 3+3 non-CRF tets with one line longer. Is there something I did wrong or is the algorithm no good or is this really not CRF?

Offending tets are in yellow and red:

3=gybef.JPG (32.57 KiB) Viewed 14901 times

[Klitzing]

oops, 3||gybef in fact once was derived as a stack of segmentochora, cf. the according lace city:

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x o   o x   
               traf
   x x   o o
               trippy
      o x   


And it happened that the dihedral angle between the upper lacing trip (visible in the upper part of the left slope) and the lower lacing trip (visible at the lower part of the left slope) - each wrt. to the medial squippy (the horizontally displayed connection cell) combine just to 180 degrees and therefore can be united into a gybef cell.

Sadly so far no-one has calculated the other dihedral angles across that connecting (thus pseudo) squippy. - Now I've just done so: In fact the dihedral angle between the upper oct (visible in the upper part as right nodes each) across the connecting triangle towards the lower squippy (visible in the lower part as right nodes each) happens to be arccos(-7/8) = 151.044976 degrees. But the dihedral angle between the upper tet (visible in the central triangle of the lace city as left nodes each) across the connecting triangle towards the lower tet (visible in the lower part as left nodes each) happens to be 360 degrees - arccos(-7/8) = 208.955024 degrees, i.e. 3||gybef comes out to be NOT convex after all.

Thanx for pointing out!

--- rk

[Klitzing]

Just considered next the related figure trip||gybef too, having lace city

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x o   x x   
               dygytpuf
   x x   o x
               tepe
      o x   


Here too the corresponding, above coming out to be offending dihedrals likewise never had been considered. Only the left slope, resulting in the gybef cell, had been considered. - I just did the corresponding calculations. And, in contrast to the above case, the dihedrals do not turn out to become concave. But sadly, this stack of segmentochora still fails to be a CRF!

This again can be seen from the left nodes each from the central triangle and the lower triangle, i.e. the dihedral between the squippy (above) and tet (below) across the connecting triangle. That angle turns out to be 180 degrees too. Thus those cells are co-realmic and unite into a figure featuring rhombs! And for the right nodes each in those triangles of the display represent 2 trips (above and below), connecting at a square. Those too become co-realmic and thus unite into a rhombic prism cell. - Right that occurrence of rhombic faces prevents from being CRF.

--- rk