Acronym ... Name tetra-augmented axially-octahedral ursachoron xoBo3ofox3xooo&#xt ` ©` Circumradius ... Coordinates (1/sqrt(2), 0, 0; sqrt(f/2))                            & all permutations in first 3 coord.s & all changes of sign in first 3 coord.s (layer o3x3o in lace tower description) (A, A, A; C)                                                & all even changes of sign in first 3 coord.s (layer B3o3o in lace tower description) (f/sqrt(2), 0, 0; 0)                                       & all permutations in first 3 coord.s & all changes of sign in first 3 coord.s (layer o3f3o in lace tower description) (1/sqrt(2), 1/sqrt(2), 0; -1/sqrt(2f))             & all permutations in first 3 coord.s & all changes of sign in first 3 coord.s (layer x3o3x in lace tower description) where f=(1+sqrt(5))/2, A=(1+f -3/2)/sqrt(8) = 0.525334, B=A sqrt(2) = 0.742934, C=sqrt(f/2)-sqrt(7-3f-f -3/2)/2 = 0.255243 Confer related CRFs: xo.o3of.x3xo.o&#xt   related segmentochora: teddipy   general polytopal classes: ursachora

It shall be noted here that just this alternating tetra-augmentation remains convex. Neighbouring teddies well can be augmented, but neither the dihedral angle at the pentagon then would remain convex (as for tetu), nor the cavity could be bridged by further unit edges (as for the iku).

Incidence matrix according to Dynkin symbol

```xoBo3ofox3xooo&#xt   (for heights and B cf. the coords)
(co || pseudo f-oct || pseudo B-tet || oct)

o...3o...3o...     | 12 * * * |  2  2  2  1  0 0  0  0 | 1 2 1  2  1  2  2  2  0  0 0 0 | 1 1 2 1  1  2 0 0
.o..3.o..3.o..     |  * 6 * * |  0  0  4  0  2 1  0  0 | 0 0 0  2  4  2  0  4  2  0 0 0 | 0 2 1 0  4  2 0 0
..o.3..o.3..o.     |  * * 4 * ♦  0  0  0  3  3 0  3  0 | 0 0 0  0  0  0  3  6  3  3 0 0 | 0 0 0 1  3  3 1 0
...o3...o3...o     |  * * * 6 |  0  0  0  0  0 1  2  4 | 0 0 0  0  4  0  0  0  2  4 2 2 | 0 2 0 0  4  0 2 1
-------------------+----------+------------------------+--------------------------------+------------------
x... .... ....     |  2 0 0 0 | 12  *  *  *  * *  *  * | 1 1 0  1  0  0  0  0  0  0 0 0 | 1 1 1 0  0  0 0 0
.... .... x...     |  2 0 0 0 |  * 12  *  *  * *  *  * | 0 1 1  0  0  1  1  0  0  0 0 0 | 1 0 1 1  0  1 0 0
oo..3oo..3oo..&#x  |  1 1 0 0 |  *  * 24  *  * *  *  * | 0 0 0  1  1  1  0  1  0  0 0 0 | 0 1 1 0  1  1 0 0
o.o.3o.o.3o.o.&#x  |  1 0 1 0 |  *  *  * 12  * *  *  * | 0 0 0  0  0  0  2  2  0  0 0 0 | 0 0 0 1  1  2 0 0
.oo.3.oo.3.oo.&#x  |  0 1 1 0 |  *  *  *  * 12 *  *  * | 0 0 0  0  0  0  0  2  1  0 0 0 | 0 0 0 0  2  1 0 0
.o.o3.o.o3.o.o&#x  |  0 1 0 1 |  *  *  *  *  * 6  *  * | 0 0 0  0  4  0  0  0  2  0 0 0 | 0 2 0 0  4  0 0 0
..oo3..oo3..oo&#x  |  0 0 1 1 |  *  *  *  *  * * 12  * | 0 0 0  0  0  0  0  0  1  2 0 0 | 0 0 0 0  2  0 1 0
.... ...x ....     |  0 0 0 2 |  *  *  *  *  * *  * 12 | 0 0 0  0  1  0  0  0  0  1 1 1 | 0 1 0 0  1  0 1 1
-------------------+----------+------------------------+--------------------------------+------------------
x...3o... ....     |  3 0 0 0 |  3  0  0  0  0 0  0  0 | 4 * *  *  *  *  *  *  *  * * * | 1 1 0 0  0  0 0 0
x... .... x...     |  4 0 0 0 |  2  2  0  0  0 0  0  0 | * 6 *  *  *  *  *  *  *  * * * | 1 0 1 0  0  0 0 0
.... o...3x...     |  3 0 0 0 |  0  3  0  0  0 0  0  0 | * * 4  *  *  *  *  *  *  * * * | 1 0 0 0  0  0 0 0
xo.. .... ....&#x  |  2 1 0 0 |  1  0  2  0  0 0  0  0 | * * * 12  *  *  *  *  *  * * * | 0 1 1 0  0  0 0 0
.... of.x ....&#xt |  1 2 0 2 |  0  0  2  0  0 2  0  1 | * * *  * 12  *  *  *  *  * * * | 0 1 0 0  1  0 0 0
.... .... xo..&#x  |  2 1 0 0 |  0  1  2  0  0 0  0  0 | * * *  *  * 12  *  *  *  * * * | 0 0 1 0  0  1 0 0
.... .... x.o.&#x  |  2 0 1 0 |  0  1  0  2  0 0  0  0 | * * *  *  *  * 12  *  *  * * * | 0 0 0 1  0  1 0 0
ooo.3ooo.3ooo.&#x  |  1 1 1 0 |  0  0  1  1  1 0  0  0 | * * *  *  *  *  * 24  *  * * * | 0 0 0 0  1  1 0 0
.ooo3.ooo3.ooo&#x  |  0 1 1 1 |  0  0  0  0  1 1  1  0 | * * *  *  *  *  *  * 12  * * * | 0 0 0 0  2  0 0 0
.... ..ox ....&#x  |  0 0 1 2 |  0  0  0  0  0 0  2  1 | * * *  *  *  *  *  *  * 12 * * | 0 0 0 0  1  0 1 0
...o3...x ....     |  0 0 0 3 |  0  0  0  0  0 0  0  3 | * * *  *  *  *  *  *  *  * 4 * | 0 1 0 0  0  0 0 1
.... ...x3...o     |  0 0 0 3 |  0  0  0  0  0 0  0  3 | * * *  *  *  *  *  *  *  * * 4 | 0 0 0 0  0  0 1 1
-------------------+----------+------------------------+--------------------------------+------------------
x...3o...3x...     ♦ 12 0 0 0 | 12 12  0  0  0 0  0  0 | 4 6 4  0  0  0  0  0  0  0 0 0 | 1 * * *  *  * * *
xo.o3of.x ....&#xt ♦  3 3 0 3 |  3  0  6  0  0 3  0  3 | 1 0 0  3  3  0  0  0  0  0 1 0 | * 4 * *  *  * * *
xo.. .... xo..&#x  ♦  4 1 0 0 |  2  2  4  0  0 0  0  0 | 0 1 0  2  0  2  0  0  0  0 0 0 | * * 6 *  *  * * *
.... o.o.3x.o.&#x  ♦  3 0 1 0 |  0  3  0  3  0 0  0  0 | 0 0 1  0  0  0  3  0  0  0 0 0 | * * * 4  *  * * *
.... ofox ....&#xr ♦  1 2 1 2 |  0  0  2  1  2 2  2  1 | 0 0 0  0  1  0  0  2  2  1 0 0 | * * * * 12  * * * cycle: (ABDC)
.... .... xoo.&#x  ♦  2 1 1 0 |  0  1  2  2  1 0  0  0 | 0 0 0  0  0  1  1  2  0  0 0 0 | * * * *  * 12 * *
.... ..ox3..oo&#x  ♦  0 0 1 3 |  0  0  0  0  0 0  3  3 | 0 0 0  0  0  0  0  0  0  3 0 1 | * * * *  *  * 4 *
...o3...x3...o     ♦  0 0 0 6 |  0  0  0  0  0 0  0 12 | 0 0 0  0  0  0  0  0  0  0 4 4 | * * * *  *  * * 1
```

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