Acronym retut (alt.: amtut)
Name rectified/ambiated truncated tetrahedron
 
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Circumradius sqrt(85/12) = 2.661453

Rectification wrt. a non-regular polytope is meant to be the singular instance of truncations on all vertices at such a depth that the hyperplane intersections on the former edges will coincide (provided such a choice exists). Within the specific case of tut as a pre-image these intersection points might differ on its 2 edge types. Therefore tut cannot be rectified (within this stronger sense). Nonetheless the Conway operator of ambification (chosing the former edge centers generally) clearly is applicable. This would result in 2 different edge sizes in the outcome polyhedron. That one here is scaled such so that the shorter one becomes unity. Then the larger edge will have size h=sqrt(3).


Incidence matrix according to Dynkin symbol

xox(do)-3-odd(od)-&#ht   → height(1,2) = height(3,4) = sqrt(2/3) = 0.816497
                           height(2,3) = sqrt(8/3) = 1.632993
                           height(4,5) = 0

o..(..)-3-o..(..)         | 3 * * * * | 2 2 0 0 0 0 0 | 1 2 1 0 0 0 0
.o.(..)-3-.o.(..)         | * 3 * * * | 0 2 2 0 0 0 0 | 0 1 2 1 0 0 0
..o(..)-3-..o(..)         | * * 6 * * | 0 0 1 1 1 1 0 | 0 0 1 1 1 1 0
...(o.)-3-...(o.)         | * * * 3 * | 0 0 0 0 2 0 2 | 0 0 1 0 2 0 1
...(.o)-3-...(.o)         | * * * * 3 | 0 0 0 0 0 2 2 | 0 0 0 0 2 1 1
--------------------------+-----------+---------------+--------------
x..(..)   ...(..)         | 2 0 0 0 0 | 3 * * * * * * | 1 1 0 0 0 0 0  x
oo.(..)-3-oo.(..)-&#h     | 1 1 0 0 0 | * 6 * * * * * | 0 1 1 0 0 0 0  h
.oo(..)-3-.oo(..)-&#h     | 0 1 1 0 0 | * * 6 * * * * | 0 0 1 1 0 0 0  h
..x(..)   ...(..)         | 0 0 2 0 0 | * * * 3 * * * | 0 0 0 1 0 1 0  x
..o(o.)-3-..o(o.)-&#h     | 0 0 1 1 0 | * * * * 6 * * | 0 0 1 0 1 0 0  h
..o(.o)-3-..o(.o)-&#x     | 0 0 1 0 1 | * * * * * 6 * | 0 0 0 0 1 1 0  x
...(oo)-3-...(oo)-&#h     | 0 0 0 1 1 | * * * * * * 6 | 0 0 0 0 1 0 1  h
--------------------------+-----------+---------------+--------------
x..(..)-3-o..(..)         | 3 0 0 0 0 | 3 0 0 0 0 0 0 | 1 * * * * * *  x-{3}
xo.(..)   ...(..)-&#h     | 2 1 0 0 0 | 1 2 0 0 0 0 0 | * 3 * * * * *  xhh
...(..)   odd(o.)-&#ht    | 1 2 2 1 0 | 0 2 2 0 2 0 0 | * * 3 * * * *  h-{6}
.ox(..)   ...(..)-&#h     | 0 1 2 0 0 | 0 0 2 1 0 0 0 | * * * 3 * * *  xhh
..o(oo)-3-..o(oo)-&#(h,x) | 0 0 1 1 1 | 0 0 0 0 1 1 1 | * * * * 6 * *  xhh
..x(.o)   ...(..)-&#x     | 0 0 2 0 1 | 0 0 0 1 0 2 0 | * * * * * 3 *  x-{3}
...(do)-3-...(od)-&#zh    | 0 0 0 3 3 | 0 0 0 0 0 0 6 | * * * * * * 1  h-{6}

xo3od3do&#zh   → height = 0
(tegum sum of x3o3d and d-oct)

o.3o.3o.     | 12 * |  2  2 | 1  2 1
.o3.o3.o     |  * 6 |  0  4 | 0  2 2
-------------+------+-------+-------
x. .. ..     |  2 0 | 12  * | 1  1 0
oo3oo3oo&#h  |  1 1 |  * 24 | 0  1 1
-------------+------+-------+-------
x.3o. ..     |  3 0 |  3  0 | 4  * *  x-{3}
xo .. ..&#h  |  2 1 |  1  2 | * 12 *  xhh
.. od3do&#zh |  3 3 |  0  6 | *  * 4  h-{6}

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