Klitzing wrote:Just a minor question, quickfur:

what do you refer by:

quickfur wrote:...

- 1,1,(5,8)-tetradiminished cantellated tesseract (basically corresponding to a maximal tetradiminishing of the 24-cell)

- 1,(1,2,3),6-pentadiminished cantellated tesseract (corresponding to a maximal pentadiminishing of the 24-cell)

...

- 1,1,(5,8)-tetradiminished 24-cell

- 1,(1,2,3),6-pentadiminished 24-cell

...

--- rk

I knew that question would come up.

Basically, these numbers correspond with vertices on a 24-cell, and are assigned as follows. Take a 24-cell, and stratify it as point || cube || q-octahedron || cube || point. For the cube layers, we number the vertices thus:

- Code: Select all
` 1-----2`

/| /|

3-----4 |

| 5---|-6

|/ |/

7-----8

For the q-octahedron layer, we use this labelling:

- Code: Select all
` 1`

| 3'

|/

2---+---2'

/|

3 |

1'

Now, since a diminishing of the 24-cell always has at least one deleted vertex, we may simply assign that vertex to the first point. Thus the designation of the diminishing always begins with "1,...". Furthermore, only non-adjacent diminishings are admissible, so the second stratum (the first cube) is never diminished -- it is adjacent to the first point. So we may skip over the second stratum altogether. Thus, the second item in the designation refers to one of the q-octahedron vertices (which are all non-adjacent to each other, since they do not lie along a single cell, but are separated by the cube vertices on either side, so adjacent indices on the q-octahedron can be diminished). The third and fourth items, correspondingly, refer to the third cube, and the last point, if any. We also permute the vertices under 24-cell symmetry so that it gives us the smallest indices in the final designation.

Therefore, 1,1,(5,8) is to be understood as:

- 1,... : the first point is diminished

- ...,1,...: the top vertex of the q-octahedron is diminished

- ...,...,(5,8): the vertices marked 5 and 8 on the third cube are diminished.

Of course, when it comes to the cantellated tesseract x4o3x3o, the diminishings are not merely of single points, but of segmentochora 8-prism||4-gon. These segmentochora correspond with the square faces of the tesseract, and therefore corresponds with the cells of a 24-cell produced by augmenting the tesseract with cube pyramids, which in turn corresponds with the vertices of the dual 24-cell. Hence, 1,1,(5,8) here refers to 4 of the segmentochora of the corresponding positions to be deleted from x4o3x3o.

Similarly, 1,(1,2,3),6 is to be understood as:

- 1,...: the first point is diminished (this is always the case, since any diminishing can always be rotated under 24-cell symmetry to fall on the first point)

- ...,(1,2,3),...: the points marked 1, 2, and 3 on the q-octahedron are diminished;

- ...,...,6: the vertex marked 6 on the third cube is diminished.

The 1,1,(5,8)-tetradiminishing and the 1,(1,2,3),6-pentadiminishing are both maximal diminishings: the diminished vertices are such, that all of the remaining vertices are adjacent to one of them, so no more vertices can be deleted because that would introduce an adjacent diminishing which makes it non-CRF (we are not accounting for the 24-cell lunae here, since there is no corresponding construct for the x4o3x3o). In fact, these two diminishings are the only other maximal diminishings of the 24-cell besides the octadiminishing, which produces a tesseract (or resp., in the case of x4o3x3o, produces the 8,8-duoprism).