The relation to ex runs as follows: ex in axial tetrahedral subsymmetry can be given as xoofoxFfoofxofo 3 oofoxfooofxofoo 3 ofoxfoofFxofoox &#xt (F = ff = f+x). There the fourth layer will be transformed according to: f3o3x → f3x3(-x) → F3(-x)3o, the fifth layer will be transformed according to: o3x3f → x3(-x)3F, the eleventh layer according to: f3x3o → F3(-x)3x, and the twelfth layer according to: x3o3f → (-x)3x3f → o3(-x)3F. This then results in the kaleido-faceting xooFxxFfooFoofo 3 oof(-x)(-x)fooof(-x)(-x)foo 3 ofooFoofFxxFoox &#xt. Then a (uniformly to all layers) applied Stott expansion wrt. the second node produces this polychoron.

There are other axial EKF which re-use one half of this polychoron, but then continue differently, still having the same vertex layers throughout: xooFxxFfooFxofo3xxFooFxxxFoxFxx3ofooFoofFxxfoox&#xt and ooofxxFfooFoofo3oxFxoFxxxFooFxx3xfoxFoofFxxFoox&#xt.

Incidence matrix according to Dynkin symbol

xooFxxFfooFoofo3xxFooFxxxFooFxx3ofooFoofFxxFoox&#xt   → height(1,2) = height(3,4) = height(5,6) = height(7,8) = height(8,9) = height(10,11) = height(12,13) = height(14,15) = sqrt[3-sqrt(5)]/4 = 0.218508
                                                        height(2,3) = height(6,7) = height(9,10) = height(13,14) = sqrt[7-3 sqrt(5)]/4 = 0.135045
                                                        height(4,5) = height(11,12) = 1/sqrt(8) = 0.353553
(tut || pseudo inv (f,x)-tut || pseudo F-oct || pseudo F-tet || pseudo (x,F)-co || pseudo (x,F)-tut || pseudo (F,x)-tut || pseudo (f,x,f)-toe || pseudo inv (F,x)-tut || pseudo inv (x,F)-tut || pseudo (F,x)-co || pseudo dual F-tet || pseudo F-oct || pseudo (f,x)-tut || inv tut)

o..............3o..............3o..............     & | 24  *  * *  *  *  *  * |  1  2  2  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 2 1  2  2  1  2  2  1 0  0  0  0  0  0 0  0  0  0  0  0  0  0 0  0 | 1 2 1  2  2 0  0 0  0 0 0
.o.............3.o.............3.o.............     & |  * 24  * *  *  *  *  * |  0  0  2  0  0  2  1  2  0  0  0  0  0  0  0  0  0  0  0  0 | 0 0  1  2  0  0  2  0 1  1  2  2  0  0 0  0  0  0  0  0  0  0 0  0 | 0 1 0  1  2 1  1 0  0 0 0
..o............3..o............3..o............     & |  *  * 12 *  *  *  *  * |  0  0  0  2  0  0  2  0  2  0  0  0  0  0  0  0  0  0  0  0 | 0 0  0  0  1  0  4  2 0  0  0  4  1  0 0  0  0  0  0  0  0  0 0  0 | 0 0 0  2  4 0  2 0  0 0 0
...o...........3...o...........3...o...........     & |  *  *  * 8  *  *  *  * ♦  0  0  0  0  3  0  0  0  0  3  0  0  0  0  0  0  0  0  0  0 | 0 0  0  0  0  3  0  3 0  0  0  0  0  3 0  0  0  0  0  0  0  0 0  0 | 0 0 1  0  3 0  0 1  0 0 0
....o..........3....o..........3....o..........     & |  *  *  * * 24  *  *  * |  0  0  0  0  0  0  0  2  0  0  2  2  2  0  0  0  0  0  0  0 | 0 0  0  0  0  0  0  0 0  2  1  2  0  0 1  1  2  1  2  0  0  0 0  0 | 0 0 0  0  1 1  2 0  1 1 0
.....o.........3.....o.........3.....o.........     & |  *  *  * *  * 24  *  * |  0  0  0  0  0  0  0  0  1  0  0  0  0  1  1  2  2  0  0  0 | 0 0  0  0  0  0  0  1 0  0  0  2  1  0 0  0  0  0  0  3  2  2 0  0 | 0 0 0  0  2 0  3 0  0 0 1
......o........3......o........3......o........     & |  *  *  * *  *  * 24  * |  0  0  0  0  0  0  0  0  0  1  0  0  2  0  1  0  0  2  2  0 | 0 0  0  0  0  0  0  1 0  0  0  0  0  2 0  0  1  2  2  0  2  0 1  2 | 0 0 0  0  2 0  1 1  2 1 0
.......o.......3.......o.......3.......o.......       |  *  *  * *  *  *  * 24 |  0  0  0  0  0  0  0  0  0  0  0  2  0  0  0  2  0  0  2  1 | 0 0  0  0  0  0  0  0 0  0  0  2  0  0 0  2  0  0  2  0  2  1 0  2 | 0 0 0  0  2 0  2 0  2 0 0
------------------------------------------------------+------------------------+-------------------------------------------------------------+--------------------------------------------------------------------+--------------------------
x.............. ............... ...............     & |  2  0  0 0  0  0  0  0 | 12  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 2 0  2  0  1  0  0  0 0  0  0  0  0  0 0  0  0  0  0  0  0  0 0  0 | 1 2 0  2  0 0  0 0  0 0 0
............... x.............. ...............     & |  2  0  0 0  0  0  0  0 |  * 24  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 1 1  0  1  0  1  0  0 0  0  0  0  0  0 0  0  0  0  0  0  0  0 0  0 | 1 1 1  0  1 0  0 0  0 0 0
oo.............3oo.............3oo.............&#x  & |  1  1  0 0  0  0  0  0 |  *  * 48  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 0 0  1  1  0  0  1  0 0  0  0  0  0  0 0  0  0  0  0  0  0  0 0  0 | 0 1 0  1  1 0  0 0  0 0 0
o.o............3o.o............3o.o............&#x  & |  1  0  1 0  0  0  0  0 |  *  *  * 24  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 0 0  0  0  1  0  2  1 0  0  0  0  0  0 0  0  0  0  0  0  0  0 0  0 | 0 0 0  2  2 0  0 0  0 0 0
o..o...........3o..o...........3o..o...........&#x  & |  1  0  0 1  0  0  0  0 |  *  *  *  * 24  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 0 0  0  0  0  2  0  1 0  0  0  0  0  0 0  0  0  0  0  0  0  0 0  0 | 0 0 1  0  2 0  0 0  0 0 0
............... .x............. ...............     & |  0  2  0 0  0  0  0  0 |  *  *  *  *  * 24  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 0 0  0  1  0  0  0  0 1  0  1  0  0  0 0  0  0  0  0  0  0  0 0  0 | 0 1 0  0  1 1  0 0  0 0 0
.oo............3.oo............3.oo............&#x  & |  0  1  1 0  0  0  0  0 |  *  *  *  *  *  * 24  *  *  *  *  *  *  *  *  *  *  *  *  * | 0 0  0  0  0  0  2  0 0  0  0  2  0  0 0  0  0  0  0  0  0  0 0  0 | 0 0 0  1  2 0  1 0  0 0 0
.o..o..........3.o..o..........3.o..o..........&#x  & |  0  1  0 0  1  0  0  0 |  *  *  *  *  *  *  * 48  *  *  *  *  *  *  *  *  *  *  *  * | 0 0  0  0  0  0  0  0 0  1  1  1  0  0 0  0  0  0  0  0  0  0 0  0 | 0 0 0  0  1 1  1 0  0 0 0
..o..o.........3..o..o.........3..o..o.........&#x  & |  0  0  1 0  0  1  0  0 |  *  *  *  *  *  *  *  * 24  *  *  *  *  *  *  *  *  *  *  * | 0 0  0  0  0  0  0  1 0  0  0  2  1  0 0  0  0  0  0  0  0  0 0  0 | 0 0 0  0  2 0  2 0  0 0 0
...o..o........3...o..o........3...o..o........&#x  & |  0  0  0 1  0  0  1  0 |  *  *  *  *  *  *  *  *  * 24  *  *  *  *  *  *  *  *  *  * | 0 0  0  0  0  0  0  1 0  0  0  0  0  2 0  0  0  0  0  0  0  0 0  0 | 0 0 0  0  2 0  0 1  0 0 0
....x.......... ............... ...............     & |  0  0  0 0  2  0  0  0 |  *  *  *  *  *  *  *  *  *  * 24  *  *  *  *  *  *  *  *  * | 0 0  0  0  0  0  0  0 0  1  0  0  0  0 1  0  1  0  0  0  0  0 0  0 | 0 0 0  0  0 1  1 0  0 1 0
....o..o.......3....o..o.......3....o..o.......&#x  & |  0  0  0 0  1  0  0  1 |  *  *  *  *  *  *  *  *  *  *  * 48  *  *  *  *  *  *  *  * | 0 0  0  0  0  0  0  0 0  0  0  1  0  0 0  1  0  0  1  0  0  0 0  0 | 0 0 0  0  1 0  1 0  1 0 0
....o...o......3....o...o......3....o...o......&#x  & |  0  0  0 0  1  0  1  0 |  *  *  *  *  *  *  *  *  *  *  *  * 48  *  *  *  *  *  *  * | 0 0  0  0  0  0  0  0 0  0  0  0  0  0 0  0  1  1  1  0  0  0 0  0 | 0 0 0  0  0 0  1 0  1 1 0
.....x......... ............... ...............     & |  0  0  0 0  0  2  0  0 |  *  *  *  *  *  *  *  *  *  *  *  *  * 12  *  *  *  *  *  * | 0 0  0  0  0  0  0  0 0  0  0  0  1  0 0  0  0  0  0  2  0  0 0  0 | 0 0 0  0  0 0  2 0  0 0 1
.....oo........3.....oo........3.....oo........&#x  & |  0  0  0 0  0  1  1  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  * 24  *  *  *  *  * | 0 0  0  0  0  0  0  1 0  0  0  0  0  1 0  0  0  0  0  0  2  0 0  0 | 0 0 0  0  2 0  1 1  0 0 0
.....o.o.......3.....o.o.......3.....o.o.......&#x  & |  0  0  0 0  0  1  0  1 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 48  *  *  *  * | 0 0  0  0  0  0  0  0 0  0  0  1  0  0 0  0  0  0  0  0  1  1 0  0 | 0 0 0  0  1 0  2 0  0 0 0
.....o...o.....3.....o...o.....3.....o...o.....&#x    |  0  0  0 0  0  2  0  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 24  *  *  * | 0 0  0  0  0  0  0  0 0  0  0  0  0  0 0  0  0  0  0  2  0  1 0  0 | 0 0 0  0  0 0  2 0  0 0 1
............... ......x........ ...............     & |  0  0  0 0  0  0  2  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 24  *  * | 0 0  0  0  0  0  0  0 0  0  0  0  0  0 0  0  0  1  0  0  0  0 1  1 | 0 0 0  0  1 0  0 0  1 1 0
......oo.......3......oo.......3......oo.......&#x  & |  0  0  0 0  0  0  1  1 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 48  * | 0 0  0  0  0  0  0  0 0  0  0  0  0  0 0  0  0  0  1  0  1  0 0  1 | 0 0 0  0  1 0  1 0  1 0 0
............... .......x....... ...............       |  0  0  0 0  0  0  0  2 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 12 | 0 0  0  0  0  0  0  0 0  0  0  0  0  0 0  2  0  0  0  0  0  0 0  2 | 0 0 0  0  2 0  0 0  2 0 0
------------------------------------------------------+------------------------+-------------------------------------------------------------+--------------------------------------------------------------------+--------------------------
x..............3x.............. ...............     & |  6  0  0 0  0  0  0  0 |  3  3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 8 *  *  *  *  *  *  * *  *  *  *  *  * *  *  *  *  *  *  *  * *  * | 1 1 0  0  0 0  0 0  0 0 0
............... x..............3o..............     & |  3  0  0 0  0  0  0  0 |  0  3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | * 8  *  *  *  *  *  * *  *  *  *  *  * *  *  *  *  *  *  *  * *  * | 1 0 1  0  0 0  0 0  0 0 0
xo............. ............... ...............&#x  & |  2  1  0 0  0  0  0  0 |  1  0  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | * * 24  *  *  *  *  * *  *  *  *  *  * *  *  *  *  *  *  *  * *  * | 0 1 0  1  0 0  0 0  0 0 0
............... xx............. ...............&#x  & |  2  2  0 0  0  0  0  0 |  0  1  2  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | * *  * 24  *  *  *  * *  *  *  *  *  * *  *  *  *  *  *  *  * *  * | 0 1 0  0  1 0  0 0  0 0 0
x.o............ ............... ...............&#x  & |  2  0  1 0  0  0  0  0 |  1  0  0  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | * *  *  * 12  *  *  * *  *  *  *  *  * *  *  *  *  *  *  *  * *  * | 0 0 0  2  0 0  0 0  0 0 0
............... x..o........... ...............&#x  & |  2  0  0 1  0  0  0  0 |  0  1  0  0  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | * *  *  *  * 24  *  * *  *  *  *  *  * *  *  *  *  *  *  *  * *  * | 0 0 1  0  1 0  0 0  0 0 0
ooo............3ooo............3ooo............&#x  & |  1  1  1 0  0  0  0  0 |  0  0  1  1  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0 | * *  *  *  *  * 48  * *  *  *  *  *  * *  *  *  *  *  *  *  * *  * | 0 0 0  1  1 0  0 0  0 0 0
o.oo.oo........3o.oo.oo........3o.oo.oo........&#xr & |  1  0  1 1  0  1  1  0 |  0  0  0  1  1  0  0  0  1  1  0  0  0  0  1  0  0  0  0  0 | * *  *  *  *  *  * 24 *  *  *  *  *  * *  *  *  *  *  *  *  * *  * | 0 0 0  0  2 0  0 0  0 0 0 cycle: (ACFGD)
.o.............3.x............. ...............     & |  0  3  0 0  0  0  0  0 |  0  0  0  0  0  3  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | * *  *  *  *  *  *  * 8  *  *  *  *  * *  *  *  *  *  *  *  * *  * | 0 1 0  0  0 1  0 0  0 0 0
.o..x.......... ............... ...............&#x  & |  0  1  0 0  2  0  0  0 |  0  0  0  0  0  0  0  2  0  0  1  0  0  0  0  0  0  0  0  0 | * *  *  *  *  *  *  * * 24  *  *  *  * *  *  *  *  *  *  *  * *  * | 0 0 0  0  0 1  1 0  0 0 0
............... .x..o.......... ...............&#x  & |  0  2  0 0  1  0  0  0 |  0  0  0  0  0  1  0  2  0  0  0  0  0  0  0  0  0  0  0  0 | * *  *  *  *  *  *  * *  * 24  *  *  * *  *  *  *  *  *  *  * *  * | 0 0 0  0  1 1  0 0  0 0 0
.oo.oo.o.......3.oo.oo.o.......3.oo.oo.o.......&#xr & |  0  1  1 0  1  1  0  1 |  0  0  0  0  0  0  1  1  1  0  0  1  0  0  0  1  0  0  0  0 | * *  *  *  *  *  *  * *  *  * 48  *  * *  *  *  *  *  *  *  * *  * | 0 0 0  0  1 0  1 0  0 0 0 cycle: (BCFHE)
..o..x......... ............... ...............&#x  & |  0  0  1 0  0  2  0  0 |  0  0  0  0  0  0  0  0  2  0  0  0  0  1  0  0  0  0  0  0 | * *  *  *  *  *  *  * *  *  *  * 12  * *  *  *  *  *  *  *  * *  * | 0 0 0  0  0 0  2 0  0 0 0
............... ...o..x........ ...............&#x  & |  0  0  0 1  0  0  2  0 |  0  0  0  0  0  0  0  0  0  2  0  0  0  0  1  0  0  0  0  0 | * *  *  *  *  *  *  * *  *  *  *  * 24 *  *  *  *  *  *  *  * *  * | 0 0 0  0  1 0  0 1  0 0 0
....x..........3....o.......... ...............     & |  0  0  0 0  3  0  0  0 |  0  0  0  0  0  0  0  0  0  0  3  0  0  0  0  0  0  0  0  0 | * *  *  *  *  *  *  * *  *  *  *  *  * 8  *  *  *  *  *  *  * *  * | 0 0 0  0  0 1  0 0  0 1 0
............... ....o..x....... ...............&#x  & |  0  0  0 0  1  0  0  2 |  0  0  0  0  0  0  0  0  0  0  0  2  0  0  0  0  0  0  0  1 | * *  *  *  *  *  *  * *  *  *  *  *  * * 24  *  *  *  *  *  * *  * | 0 0 0  0  1 0  0 0  1 0 0
....x...o...... ............... ...............&#x  & |  0  0  0 0  2  0  1  0 |  0  0  0  0  0  0  0  0  0  0  1  0  2  0  0  0  0  0  0  0 | * *  *  *  *  *  *  * *  *  *  *  *  * *  * 24  *  *  *  *  * *  * | 0 0 0  0  0 0  1 0  0 1 0
............... ....o...x...... ...............&#x  & |  0  0  0 0  1  0  2  0 |  0  0  0  0  0  0  0  0  0  0  0  0  2  0  0  0  0  1  0  0 | * *  *  *  *  *  *  * *  *  *  *  *  * *  *  * 24  *  *  *  * *  * | 0 0 0  0  0 0  0 0  1 1 0
....o..oo......3....o..oo......3....o..oo......&#x  & |  0  0  0 0  1  0  1  1 |  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  1  0 | * *  *  *  *  *  *  * *  *  *  *  *  * *  *  *  * 48  *  *  * *  * | 0 0 0  0  0 0  1 0  1 0 0
.....x...o..... ............... ...............&#x  & |  0  0  0 0  0  3  0  0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  2  0  0  0 | * *  *  *  *  *  *  * *  *  *  *  *  * *  *  *  *  * 24  *  * *  * | 0 0 0  0  0 0  1 0  0 0 1
.....ooo.......3.....ooo.......3.....ooo.......&#x  & |  0  0  0 0  0  1  1  1 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0  1  0 | * *  *  *  *  *  *  * *  *  *  *  *  * *  *  *  *  *  * 48  * *  * | 0 0 0  0  1 0  1 0  0 0 0
.....o.o.o.....3.....o.o.o.....3.....o.o.o.....&#x    |  0  0  0 0  0  2  0  1 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  2  1  0  0  0 | * *  *  *  *  *  *  * *  *  *  *  *  * *  *  *  *  *  *  * 24 *  * | 0 0 0  0  0 0  2 0  0 0 0
............... ......x........3......o........     & |  0  0  0 0  0  0  3  0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  3  0  0 | * *  *  *  *  *  *  * *  *  *  *  *  * *  *  *  *  *  *  *  * 8  * | 0 0 0  0  0 0  0 1  0 1 0
............... ......xx....... ...............&#x  & |  0  0  0 0  0  0  2  2 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  2  1 | * *  *  *  *  *  *  * *  *  *  *  *  * *  *  *  *  *  *  *  * * 24 | 0 0 0  0  1 0  0 0  1 0 0
------------------------------------------------------+------------------------+-------------------------------------------------------------+--------------------------------------------------------------------+--------------------------
x..............3x..............3o..............     & ♦ 12  0  0 0  0  0  0  0 |  6 12  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 4 4  0  0  0  0  0  0 0  0  0  0  0  0 0  0  0  0  0  0  0  0 0  0 | 2 * *  *  * *  * *  * * *
xo.............3xx............. ...............&#x  & ♦  6  3  0 0  0  0  0  0 |  3  3  6  0  0  3  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 1 0  3  3  0  0  0  0 1  0  0  0  0  0 0  0  0  0  0  0  0  0 0  0 | * 8 *  *  * *  * *  * * *
............... x..o...........3o..o...........&#x  & ♦  3  0  0 1  0  0  0  0 |  0  3  0  0  3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 0 1  0  0  0  3  0  0 0  0  0  0  0  0 0  0  0  0  0  0  0  0 0  0 | * * 8  *  * *  * *  * * *
xoo............ ............... ...............&#x  & ♦  2  1  1 0  0  0  0  0 |  1  0  2  2  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0 | 0 0  1  0  1  0  2  0 0  0  0  0  0  0 0  0  0  0  0  0  0  0 0  0 | * * * 24  * *  * *  * * *
............... xxFooFxx....... ...............&#xt & ♦  2  2  2 1  1  2  2  2 |  0  1  2  2  2  1  2  2  2  2  0  2  0  0  2  2  0  1  2  1 | 0 0  0  1  0  1  2  2 0  0  1  2  0  1 0  1  0  0  0  0  2  0 0  1 | * * *  * 24 *  * *  * * *
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............... ....o..xx...... ...............&#x  & ♦  0  0  0 0  1  0  2  2 |  0  0  0  0  0  0  0  0  0  0  0  2  2  0  0  0  0  1  2  1 | 0 0  0  0  0  0  0  0 0  0  0  0  0  0 0  1  0  1  2  0  0  0 0  1 | * * *  *  * *  * * 24 * *
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