by **Marek14** » Sat Jan 07, 2012 11:16 am

Hmm, if I remember correctly, the vertices of a 24-cell can be expressed as (2,0,0,0) and all permutations and sign changes of it (8 vertices) plus (1,1,1,1) and all sign changes (16 vertices) with edge length 2, edges being of type (2,0,0,0)-(1,1,1,1) or (1,1,1,1)-(-1,1,1,1).

Since 24-cell is self-dual, we can use the same structure for cells/faces.

We should also notice that there are three tesseractic-symmetry groups of cells embedded in the structure. One of them are orthogonal faces ((2,0,0,0) and its ilk), second is (+-1,+-1,+-1,+-1) with even sign parity, third is the same, but with odd sign parity.

Let's start by augmenting the "top" face, (2,0,0,0). This eliminates all faces (+1,+-1,+-1,+-1), so next face to augment must be (0,<+-2,0,0>) [<+-2,0,0> indicates permutation], (-1,+-1,+-1,+-1) or (0,0,0,-2).

Let's make a handy table:

Fixed: (2,0,0,0)

Possible 1: (0,2,0,0),(0,-2,0,0),(0,0,2,0),(0,0,-2,0),(0,0,0,2),(0,0,0,-2) (first "ring", the same group as first cell)

Possible 2: (-1,1,1,1),(-1,-1,1,1),(-1,1,-1,1),(-1,-1,-1,1),(-1,1,1,-1),(-1,-1,1,-1),(-1,1,-1,-1),(-1,-1,-1,-1) (second "ring", different groups than first cell)

Possible 3: (-2,0,0,0) (antipode of first cell)

This leads to three biaugmented 24-cells:

1: orthobiaugmented 24-cell

Fixed: (2,0,0,0),(0,2,0,0)

Possible 1: (0,0,2,0),(0,0,-2,0),(0,0,0,2),(0,0,0,-2) (four cells in "ortho" position to both fixed cells)

Possible 2: (-1,-1,1,1),(-1,-1,-1,1),(-1,-1,1,-1),(-1,-1,-1,-1) (all remaining cells in other groups, in "meta" position to both fixed cells)

Possible 3: (-2,0,0,0),(0,-2,0,0) (antipodes to two fixed cells, "ortho" position to one and "para" to other)

2: metabiaugmented 24-cell:

Fixed: (2,0,0,0),(-1,1,1,1)

Possible 1: (0,-2,0,0),(0,0,-2,0),(0,0,0,-2),(-1,1,-1,-1),(-1,-1,1,-1),(-1,-1,-1,1) (cells in "ortho" position to one fixed cell and "meta" to the other)

Possible 2: (-1,-1,-1,-1) (sort of "antipode" to the particular "meta" combination, it has "meta" position to both fixed cell)

3: parabiaugmented 24-cell:

Fixed: (2,0,0,0),(-2,0,0,0)

Possible: (0,2,0,0),(0,-2,0,0),(0,0,2,0),(0,0,-2,0),(0,0,0,2),(0,0,0,-2) (the remaining group of the fixed cells, all in "ortho" position to both)

This leads to 4 possible triaugments:

1: tesedge-triaugmented 24-cell (ooo)

Name explanation: it augments three cells in one tesseractic group around the "edge" of this fictious tesseract.

Fixed: (2,0,0,0),(0,2,0,0),(0,0,2,0)

Possible 1:(0,0,0,2),(0,0,0,-2) (four cells in "ortho" position to all three fixed cells)

Possible 2: (-1,-1,-1,1),(-1,-1,-1,-1) (all remaining cells in other groups, in "meta" position to all three fixed cells)

Possible 3: (-2,0,0,0),(0,-2,0,0),(0,0,-2,0) (antipodes to the three fixed cells, "ortho" position to two of them and "para" to the third)

2: assymetrical triaugmented 24-cell (omm)

Name explanation: it's the least symmetrical of the four.

Fixed: (2,0,0,0),(0,2,0,0),(-1,-1,1,1)

Possible 1: (0,0,-2,0),(0,0,0,-2) ("ortho" position to two fixed cells, "meta" to one)

Possible 2: (-1,-1,-1,-1) ("ortho" position to one cell, "meta" to two)

3: arc-triaugmented 24-cell (oop)

Name explanation: The three augmented cells lie on the same tesseract and form an arc of two antipodes and one lateral cell

Fixed: (2,0,0,0),(0,2,0,0),(-2,0,0,0)

Possible 1: (0,0,2,0),(0,0,-2,0),(0,0,0,2),(0,0,0,-2) (four cells in "ortho" position to all three fixed cells)

Possible 3: (0,-2,0,0) (antipode to the unpaired cell, "ortho" to two fixed cell and "para" to the third)

4: symmetrical triaugmented 24-cell: (mmm)

Name explanation: the most symmetrical triaugmentation using one augment from each tesseractic group

Fixed: (2,0,0,0),(-1,1,1,1),(-1,-1,-1,-1)

This augment cannot be augmented any further.

Now for tetraaugments. There is 5 of those:

1: tesvertex-tetraaugmented 24-cell

Name explanation: it augments four cells in one tesseractic group around the "vertex" of this fictious tesseract.

Fixed: (2,0,0,0),(0,2,0,0),(0,0,2,0),(0,0,0,2)

Possible 1:(-2,0,0,0),(0,-2,0,0),(0,0,-2,0),(0,0,0,-2) (antipodes to the four fixed cells, "ortho" position to three of them and "para" to the fourth)

Possible 2:(-1,-1,-1,-1) (the only cell in other group, in "meta" position to all four fixed cells)

2: edge+1-tetraaugmented 24-cell

Name explanation: it augments three cells in one tesseractic group around the "edge" of this fictious tesseract plus one cell from another group.

Fixed: (2,0,0,0),(0,2,0,0),(0,0,2,0),(-1,-1,-1,1)

Possible:(0,0,0,-2) (the only possible augmentation cell, "ortho" position to three fixed cells and "meta" to the fourth)

3: tesassymetrical-tetraaugmented 24-cell

Name explanation: it augments four cells in one tesseractic group in "assymetrical" configuration as opposed to "vertex" or "ring" configuration

Fixed: (2,0,0,0),(0,2,0,0),(0,0,2,0),(-2,0,0,0)

Possible 1:(0,0,0,2),(0,0,0,-2) (two cells in "ortho" position to all four fixed cells)

Possible 2: (0,-2,0,0),(0,0,-2,0) (antipodes to two of the fixed cells, "ortho" position to three fixed cells and "para" to the fourth)

4: skewed tetraaugmented 24-cell

Name explanation: augments two and two cells in two different tesseractic groups

Fixed: (2,0,0,0),(0,2,0,0),(-1,-1,1,1),(-1,-1,-1,-1)

Cannot be augmented any further.

5: ring-triaugmented 24-cell

Name explanation: The four augmented cells lie on the same tesseract and form a ring aroung it.

Fixed: (2,0,0,0),(0,2,0,0),(-2,0,0,0),(0,-2,0,0)

Possible: (0,0,2,0),(0,0,-2,0),(0,0,0,2),(0,0,0,-2) (four cells in "ortho" position to all four fixed cells)

Now for pentaaugmentations. There are 3 of them.

1: tesantiedge-pentaaugmented 24-cell

Name explanation: it augments five cells in one tesseractic group such that three cells around one "edge" remains unaugmented.

Fixed: (2,0,0,0),(0,2,0,0),(0,0,2,0),(0,0,0,2),(-2,0,0,0)

Possible:(0,-2,0,0),(0,0,-2,0),(0,0,0,-2) (antipodes to three unpaired fixed cells, "ortho" position to four fixed cells and "para" to the fifth)

2: vertex+1-pentaaugmented 24-cell

Name explanation: it augments four cells in one tesseractic group around the "vertex" of this fictious tesseract plus one cell in another group.

Fixed: (2,0,0,0),(0,2,0,0),(0,0,2,0),(0,0,0,2),(-1,-1,-1,-1)

Cannot be augmented any further.

3: tesantiarc-pentaaugmented 24-cell

Name explanation: it augments five cells in one tesseractic group such that the unaugmented cells form an "arc".

Fixed: (2,0,0,0),(0,2,0,0),(0,0,2,0),(-2,0,0,0),(0,-2,0,0)

Possible 1:(0,0,0,2),(0,0,0,-2) (two cells in "ortho" position to all five fixed cells)

Possible 2: (0,0,-2,0) (antipode to the only unpaired cell, "ortho" position to four fixed cells and "para" to the fifth)

Now for hexaaaugmentations. There are 2 of them.

1: tesantiface-hexaaugmented 24-cell

Name explanation: it augments six cells in one tesseractic group such that two cells around one "face" remains unaugmented.

Fixed: (2,0,0,0),(0,2,0,0),(0,0,2,0),(0,0,0,2),(-2,0,0,0),(0,-2,0,0)

Possible: (0,0,-2,0),(0,0,0,-2) (antipodes to two unpaired fixed cells, "ortho" position to five fixed cells and "para" to the sixth)

2: tesantipode-hexaaugmented 24-cell

Name explanation: it augments six cells in one tesseractic group such that the unaugmented cells are antipodal.

Fixed: (2,0,0,0),(0,2,0,0),(0,0,2,0),(-2,0,0,0),(0,-2,0,0),(0,0,-2,0)

Possible 1:(0,0,0,2),(0,0,0,-2) (two cells in "ortho" position to all six fixed cells)

Now for the only possible heptaaugmentation:

1: heptaaugmented 24-cell

Fixed: (2,0,0,0),(0,2,0,0),(0,0,2,0),(0,0,0,2),(-2,0,0,0),(0,-2,0,0),(0,0,-2,0)

Possible: (0,0,0,-2) (antipodes to the only unpaired fixed cell, "ortho" position to six fixed cells and "para" to the seventh)

And the only octaaugmentation:

1: octaaugmented 24-cell

Fixed: (2,0,0,0),(0,2,0,0),(0,0,2,0),(0,0,0,2),(-2,0,0,0),(0,-2,0,0),(0,0,-2,0),(0,0,0,-2)

Cannot be augmented any further.

So, for number of augments:

1 - 1

2 - 3

3 - 4

4 - 5

5 - 3

6 - 2

7 - 1

8 - 1

total: 20