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The 3 medial vertex layers F3o3x, x3x3f, and o3F3o, together with connecting edges and faces, form a skew dissectioning surface. (That one is shown in the above pictures at outline of the near resp. far half.) It is interesting, that still there beyond 2 completely different continuations are known as well, one which uses in that half thawro cells too, but then in opposite orientation: oxFxoo3xooxFx3ofxfox&#xt, and one that gets rid of those fully: oxFxofo3xooxFxx3ofxfoox&#xt.

Also the nearer half can be replaced, then reversing the orientation of the teddies. Together with the second half from here becoming then: oxFxofx3xfoxFxo3ooxfooo&#xt.

Incidence matrix according to Dynkin symbol

oxFxofx3xooxFxo3ofxfooo&#xt   →  height(1,2) = height(4,5) = height(6,7) = sqrt[3-sqrt(5)]/4 = 0.218508
(F=ff=x+f)                       height(2,3) = height(3,4) = 1/sqrt(8) = 0.353553
                                 height(5,6) = sqrt[7-3 sqrt(5)]/4 = 0.135045
(oct || pseudo (x,f)-co || pseudo (F,x)-co || pseudo (x,x,f)-toe || pseudo F-oct || (f,x)-tut || tet)

o......3o......3o......     | 6  *  *  * *  * * |  4  4  0  0  0  0  0  0  0  0  0  0  0  0  0 0 | 2 2  2  4  2 0  0  0  0 0  0  0  0  0  0 0  0  0  0 0 0  0 0 | 1 2 1 2 0  0 0  0  0 0 0 0
.o.....3.o.....3.o.....     | * 12  *  * *  * * |  0  2  2  1  2  0  0  0  0  0  0  0  0  0  0 0 | 0 0  2  1  2 1  2  1  2 0  0  0  0  0  0 0  0  0  0 0 0  0 0 | 0 1 2 1 1  1 0  0  0 0 0 0
..o....3..o....3..o....     | *  * 12  * *  * * |  0  0  0  1  0  2  2  2  0  0  0  0  0  0  0 0 | 0 0  0  0  2 0  0  0  2 1  1  2  1  2  2 0  0  0  0 0 0  0 0 | 0 0 2 1 0  1 1  1  2 0 0 0
...o...3...o...3...o...     | *  *  * 24 *  * * |  0  0  0  0  1  0  1  0  1  1  1  1  0  0  0 0 | 0 0  0  0  0 0  1  1  1 0  1  1  0  0  1 1  1  1  1 0 0  0 0 | 0 0 1 0 1  1 0  1  1 1 0 0
....o..3....o..3....o..     | *  *  *  * 6  * * |  0  0  0  0  0  0  0  0  0  0  4  0  2  0  0 0 | 0 0  0  0  0 0  0  0  0 0  0  2  0  0  0 0  2  0  4 1 0  0 0 | 0 0 1 0 0  0 0  0  2 2 0 0
.....o.3.....o.3.....o.     | *  *  *  * * 12 * |  0  0  0  0  0  0  0  2  0  0  0  2  1  2  1 0 | 0 0  0  0  0 0  0  0  0 0  0  0  2  1  2 0  0  2  2 1 1  2 0 | 0 0 0 0 0  0 1  2  1 2 1 0
......o3......o3......o     | *  *  *  * *  * 4 ♦  0  0  0  0  0  0  0  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  0 3 0  3 3 | 0 0 0 0 0  0 0  0  0 3 1 1
----------------------------+-------------------+------------------------------------------------+--------------------------------------------------------------+---------------------------
....... x...... .......     | 2  0  0  0 0  0 0 | 12  *  *  *  *  *  *  *  *  *  *  *  *  *  * * | 1 1  0  1  0 0  0  0  0 0  0  0  0  0  0 0  0  0  0 0 0  0 0 | 1 1 0 1 0  0 0  0  0 0 0 0
oo.....3oo.....3oo.....&#x  | 1  1  0  0 0  0 0 |  * 24  *  *  *  *  *  *  *  *  *  *  *  *  * * | 0 0  1  1  1 0  0  0  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
.x..... ....... .......     | 0  2  0  0 0  0 0 |  *  * 12  *  *  *  *  *  *  *  *  *  *  *  * * | 0 0  1  0  0 1  1  0  0 0  0  0  0  0  0 0  0  0  0 0 0  0 0 | 0 1 1 0 1  0 0  0  0 0 0 0
.oo....3.oo....3.oo....&#x  | 0  1  1  0 0  0 0 |  *  *  * 12  *  *  *  *  *  *  *  *  *  *  * * | 0 0  0  0  2 0  0  0  2 0  0  0  0  0  0 0  0  0  0 0 0  0 0 | 0 0 2 1 0  1 0  0  0 0 0 0
.o.o...3.o.o...3.o.o...&#x  | 0  1  0  1 0  0 0 |  *  *  *  * 24  *  *  *  *  *  *  *  *  *  * * | 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 0 1  1 0  0  0 0 0 0
....... ....... ..x....     | 0  0  2  0 0  0 0 |  *  *  *  *  * 12  *  *  *  *  *  *  *  *  * * | 0 0  0  0  1 0  0  0  0 1  0  1  0  1  0 0  0  0  0 0 0  0 0 | 0 0 1 1 0  0 1  0  1 0 0 0
..oo...3..oo...3..oo...&#x  | 0  0  1  1 0  0 0 |  *  *  *  *  *  * 24  *  *  *  *  *  *  *  * * | 0 0  0  0  0 0  0  0  1 0  1  1  0  0  1 0  0  0  0 0 0  0 0 | 0 0 1 0 0  1 0  1  1 0 0 0
..o..o.3..o..o.3..o..o.&#x  | 0  0  1  0 0  1 0 |  *  *  *  *  *  *  * 24  *  *  *  *  *  *  * * | 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 0 0  0 1  1  1 0 0 0
...x... ....... .......     | 0  0  0  2 0  0 0 |  *  *  *  *  *  *  *  * 12  *  *  *  *  *  * * | 0 0  0  0  0 0  1  0  0 0  0  0  0  0  0 1  1  0  0 0 0  0 0 | 0 0 1 0 1  0 0  0  0 1 0 0
....... ...x... .......     | 0  0  0  2 0  0 0 |  *  *  *  *  *  *  *  *  * 12  *  *  *  *  * * | 0 0  0  0  0 0  0  1  0 0  1  0  0  0  0 1  0  1  0 0 0  0 0 | 0 0 0 0 1  1 0  1  0 1 0 0
...oo..3...oo..3...oo..&#x  | 0  0  0  1 1  0 0 |  *  *  *  *  *  *  *  *  *  * 24  *  *  *  * * | 0 0  0  0  0 0  0  0  0 0  0  1  0  0  0 0  1  0  1 0 0  0 0 | 0 0 1 0 0  0 0  0  1 1 0 0
...o.o.3...o.o.3...o.o.&#x  | 0  0  0  1 0  1 0 |  *  *  *  *  *  *  *  *  *  *  * 24  *  *  * * | 0 0  0  0  0 0  0  0  0 0  0  0  0  0  1 0  0  1  1 0 0  0 0 | 0 0 0 0 0  0 0  1  1 1 0 0
....oo.3....oo.3....oo.&#x  | 0  0  0  0 1  1 0 |  *  *  *  *  *  *  *  *  *  *  *  * 12  *  * * | 0 0  0  0  0 0  0  0  0 0  0  0  0  0  0 0  0  0  2 1 0  0 0 | 0 0 0 0 0  0 0  0  1 2 0 0
....... .....x. .......     | 0  0  0  0 0  2 0 |  *  *  *  *  *  *  *  *  *  *  *  *  * 12  * * | 0 0  0  0  0 0  0  0  0 0  0  0  1  0  0 0  0  1  0 0 1  1 0 | 0 0 0 0 0  0 1  1  0 1 1 0
.....oo3.....oo3.....oo&#x  | 0  0  0  0 0  1 1 |  *  *  *  *  *  *  *  *  *  *  *  *  *  * 12 * | 0 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  0 0  0  0 2 1 0
......x ....... .......     | 0  0  0  0 0  0 2 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 6 | 0 0  0  0  0 0  0  0  0 0  0  0  0  0  0 0  0  0  0 1 0  0 2 | 0 0 0 0 0  0 0  0  0 2 0 1
----------------------------+-------------------+------------------------------------------------+--------------------------------------------------------------+---------------------------
o......3x...... .......     | 3  0  0  0 0  0 0 |  3  0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 | 4 *  *  *  * *  *  *  * *  *  *  *  *  * *  *  *  * * *  * * | 1 1 0 0 0  0 0  0  0 0 0 0
....... x......3o......     | 3  0  0  0 0  0 0 |  3  0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 | * 4  *  *  * *  *  *  * *  *  *  *  *  * *  *  *  * * *  * * | 1 0 0 1 0  0 0  0  0 0 0 0
ox..... ....... .......&#x  | 1  2  0  0 0  0 0 |  0  2  1  0  0  0  0  0  0  0  0  0  0  0  0 0 | * * 12  *  * *  *  *  * *  *  *  *  *  * *  *  *  * * *  * * | 0 1 1 0 0  0 0  0  0 0 0 0
....... xo..... .......&#x  | 2  1  0  0 0  0 0 |  1  2  0  0  0  0  0  0  0  0  0  0  0  0  0 0 | * *  * 12  * *  *  *  * *  *  *  *  *  * *  *  *  * * *  * * | 0 1 0 1 0  0 0  0  0 0 0 0
....... ....... ofx....&#xt | 1  2  2  0 0  0 0 |  0  2  0  2  0  1  0  0  0  0  0  0  0  0  0 0 | * *  *  * 12 *  *  *  * *  *  *  *  *  * *  *  *  * * *  * * | 0 0 1 1 0  0 0  0  0 0 0 0
.x.....3.o..... .......     | 0  3  0  0 0  0 0 |  0  0  3  0  0  0  0  0  0  0  0  0  0  0  0 0 | * *  *  *  * 4  *  *  * *  *  *  *  *  * *  *  *  * * *  * * | 0 1 0 0 1  0 0  0  0 0 0 0
.x.x... ....... .......&#x  | 0  2  0  2 0  0 0 |  0  0  1  0  2  0  0  0  1  0  0  0  0  0  0 0 | * *  *  *  * * 12  *  * *  *  *  *  *  * *  *  *  * * *  * * | 0 0 1 0 1  0 0  0  0 0 0 0
....... .o.x... .......&#x  | 0  1  0  2 0  0 0 |  0  0  0  0  2  0  0  0  0  1  0  0  0  0  0 0 | * *  *  *  * *  * 12  * *  *  *  *  *  * *  *  *  * * *  * * | 0 0 0 0 1  1 0  0  0 0 0 0
.ooo...3.ooo...3.ooo...&#x  | 0  1  1  1 0  0 0 |  0  0  0  1  1  0  1  0  0  0  0  0  0  0  0 0 | * *  *  *  * *  *  * 24 *  *  *  *  *  * *  *  *  * * *  * * | 0 0 1 0 0  1 0  0  0 0 0 0
....... ..o....3..x....     | 0  0  3  0 0  0 0 |  0  0  0  0  0  3  0  0  0  0  0  0  0  0  0 0 | * *  *  *  * *  *  *  * 4  *  *  *  *  * *  *  *  * * *  * * | 0 0 0 1 0  0 1  0  0 0 0 0
....... ..ox... .......&#x  | 0  0  1  2 0  0 0 |  0  0  0  0  0  0  2  0  0  1  0  0  0  0  0 0 | * *  *  *  * *  *  *  * * 12  *  *  *  * *  *  *  * * *  * * | 0 0 0 0 0  1 0  1  0 0 0 0
....... ....... ..xfo..&#xt | 0  0  2  2 1  0 0 |  0  0  0  0  0  1  2  0  0  0  2  0  0  0  0 0 | * *  *  *  * *  *  *  * *  * 12  *  *  * *  *  *  * * *  * * | 0 0 1 0 0  0 0  0  1 0 0 0
....... ..o..x. .......&#x  | 0  0  1  0 0  2 0 |  0  0  0  0  0  0  0  2  0  0  0  0  0  1  0 0 | * *  *  *  * *  *  *  * *  *  * 12  *  * *  *  *  * * *  * * | 0 0 0 0 0  0 1  1  0 0 0 0
....... ....... ..x..o.&#x  | 0  0  2  0 0  1 0 |  0  0  0  0  0  1  0  2  0  0  0  0  0  0  0 0 | * *  *  *  * *  *  *  * *  *  *  * 12  * *  *  *  * * *  * * | 0 0 0 0 0  0 1  0  1 0 0 0
..oo.o.3..oo.o.3..oo.o.&#x  | 0  0  1  1 0  1 0 |  0  0  0  0  0  0  1  1  0  0  0  1  0  0  0 0 | * *  *  *  * *  *  *  * *  *  *  *  * 24 *  *  *  * * *  * * | 0 0 0 0 0  0 0  1  1 0 0 0
...x...3...x... .......     | 0  0  0  6 0  0 0 |  0  0  0  0  0  0  0  0  3  3  0  0  0  0  0 0 | * *  *  *  * *  *  *  * *  *  *  *  *  * 4  *  *  * * *  * * | 0 0 0 0 1  0 0  0  0 1 0 0
...xo.. ....... .......&#x  | 0  0  0  2 1  0 0 |  0  0  0  0  0  0  0  0  1  0  2  0  0  0  0 0 | * *  *  *  * *  *  *  * *  *  *  *  *  * * 12  *  * * *  * * | 0 0 1 0 0  0 0  0  0 1 0 0
....... ...x.x. .......&#x  | 0  0  0  2 0  2 0 |  0  0  0  0  0  0  0  0  0  1  0  2  0  1  0 0 | * *  *  *  * *  *  *  * *  *  *  *  *  * *  * 12  * * *  * * | 0 0 0 0 0  0 0  1  0 1 0 0
...ooo.3...ooo.3...ooo.&#x  | 0  0  0  1 1  1 0 |  0  0  0  0  0  0  0  0  0  0  1  1  1  0  0 0 | * *  *  *  * *  *  *  * *  *  *  *  *  * *  *  * 24 * *  * * | 0 0 0 0 0  0 0  0  1 1 0 0
....ofx ....... .......&#xt | 0  0  0  0 1  2 2 |  0  0  0  0  0  0  0  0  0  0  0  0  2  0  2 1 | * *  *  *  * *  *  *  * *  *  *  *  *  * *  *  *  * 6 *  * * | 0 0 0 0 0  0 0  0  0 2 0 0
....... .....x.3.....o.     | 0  0  0  0 0  3 0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  3  0 0 | * *  *  *  * *  *  *  * *  *  *  *  *  * *  *  *  * * 4  * * | 0 0 0 0 0  0 1  0  0 0 1 0
....... .....xo .......&#x  | 0  0  0  0 0  2 1 |  0  0  0  0  0  0  0  0  0  0  0  0  0  1  2 0 | * *  *  *  * *  *  *  * *  *  *  *  *  * *  *  *  * * * 12 * | 0 0 0 0 0  0 0  0  0 1 1 0
......x3......o .......     | 0  0  0  0 0  0 3 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 3 | * *  *  *  * *  *  *  * *  *  *  *  *  * *  *  *  * * *  * 4 | 0 0 0 0 0  0 0  0  0 1 0 1
----------------------------+-------------------+------------------------------------------------+--------------------------------------------------------------+---------------------------
o......3x......3o......     ♦ 6  0  0  0 0  0 0 | 12  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 | 1 * * * *  * *  *  * * * *
ox.....3xo..... .......&#x  ♦ 3  3  0  0 0  0 0 |  3  6  3  0  0  0  0  0  0  0  0  0  0  0  0 0 | 1 0  3  3  0 1  0  0  0 0  0  0  0  0  0 0  0  0  0 0 0  0 0 | * 4 * * *  * *  *  * * * *
oxFxo.. ....... ofxfo..&#xt ♦ 1  4  4  4 1  0 0 |  0  4  2  4  4  2  4  0  2  0  4  0  0  0  0 0 | 0 0  2  0  2 0  2  0  4 0  0  2  0  0  0 0  2  0  0 0 0  0 0 | * * 6 * *  * *  *  * * * *
....... xoo....3ofx....&#xt ♦ 3  3  3  0 0  0 0 |  3  6  0  3  0  3  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 | * * * 4 *  * *  *  * * * *
.x.x...3.o.x... .......&#x  ♦ 0  3  0  6 0  0 0 |  0  0  3  0  6  0  0  0  3  3  0  0  0  0  0 0 | 0 0  0  0  0 1  3  3  0 0  0  0  0  0  0 1  0  0  0 0 0  0 0 | * * * * 4  * *  *  * * * *
....... .oox... .......&#x  ♦ 0  1  1  2 0  0 0 |  0  0  0  1  2  0  2  0  0  1  0  0  0  0  0 0 | 0 0  0  0  0 0  0  1  2 0  1  0  0  0  0 0  0  0  0 0 0  0 0 | * * * * * 12 *  *  * * * *
....... ..o..x.3..x..o.&#x  ♦ 0  0  3  0 0  3 0 |  0  0  0  0  0  3  0  6  0  0  0  0  0  3  0 0 | 0 0  0  0  0 0  0  0  0 1  0  0  3  3  0 0  0  0  0 0 1  0 0 | * * * * *  * 4  *  * * * *
....... ..ox.x. .......&#x  ♦ 0  0  1  2 0  2 0 |  0  0  0  0  0  0  2  2  0  1  0  2  0  1  0 0 | 0 0  0  0  0 0  0  0  0 0  1  0  1  0  2 0  0  1  0 0 0  0 0 | * * * * *  * * 12  * * * *
....... ....... ..xfoo.&#xr ♦ 0  0  2  2 1  1 0 |  0  0  0  0  0  1  2  2  0  0  2  2  1  0  0 0 | 0 0  0  0  0 0  0  0  0 0  0  1  0  1  2 0  0  0  2 0 0  0 0 | * * * * *  * *  * 12 * * *
...xofx3...xFxo .......&#xt ♦ 0  0  0  6 3  6 3 |  0  0  0  0  0  0  0  0  3  3  6  6  6  3  6 3 | 0 0  0  0  0 0  0  0  0 0  0  0  0  0  0 1  3  3  6 3 0  3 1 | * * * * *  * *  *  * 4 * *
....... .....xo3.....oo&#x  ♦ 0  0  0  0 0  3 1 |  0  0  0  0  0  0  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  0  0 0 1  3 0 | * * * * *  * *  *  * * 4 *
......x3......o3......o     ♦ 0  0  0  0 0  0 4 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 6 | 0 0  0  0  0 0  0  0  0 0  0  0  0  0  0 0  0  0  0 0 0  0 4 | * * * * *  * *  *  * * * 1