Acronym tuta (alt.: pabdirit, tutcup, tutaltut), tut || inv tut, K-4.55 Name truncated tetrahedral alterprism,truncated tetrahedral cupoliprism,runcic snub cubic hosochoron,truncated tetrahedron atop inverted truncated tetrahedron,truncated tetrahedron atop alternate truncated tetrahedron,tetrahedrally medial part of rectified tesseract,parabidiminished rectified tesseract ` ©` Segmentochoron display Circumradius sqrt(3/2) = 1.224745 Lace cityin approx. ASCII-art ``` x3x o3x u3o o3u x3o x3x ``` Coordinates (3, 1, 1, 1)/sqrt(8)       all permutations in first 3 coords, even changes of sign in all coords General of army (is itself convex) Colonel of regiment (is itself locally convex) Dihedral angles at {3} between tet and tricu:   120° at {6} between tricu and tut:   120° at {4} between tricu and tricu:   90° at {3} between tricu and tut:   60° Confer uniform relative: rit   segmentochora: tet || tut   related CRFs: mibdirit   general polytopal classes: scaliform   segmentochora   lace simplices Externallinks

This polychoron also can be derived as equatorial stratos of rit, if that one will be considered with respect to an axial tetrahedral symmetry.

Klitzing in automn of 2000 both found this very polychoron and also obtained therefrom the precise concept of some weakening of uniformity, which shortly thereafter became known as scaliformity. Thus tutcup truely was the first known scaliform polytope!

Incidence matrix according to Dynkin symbol

```xo3xx3ox&#x   → height = 1/sqrt(2) = 0.707107
(tut || inv tut)

o.3o.3o.    | 12  * | 1  2  2  0 0 | 2 1  2  2  1 0 0 | 1 2 1 1 0
.o3.o3.o    |  * 12 | 0  0  2  2 1 | 0 0  1  2  2 1 2 | 0 1 1 2 1
------------+-------+--------------+------------------+----------
x. .. ..    |  2  0 | 6  *  *  * * | 2 0  2  0  0 0 0 | 1 2 1 0 0
.. x. ..    |  2  0 | * 12  *  * * | 1 1  0  1  0 0 0 | 1 1 0 0 0
oo3oo3oo&#x |  1  1 | *  * 24  * * | 0 0  1  1  1 0 0 | 0 1 1 1 0
.. .x ..    |  0  2 | *  *  * 12 * | 0 0  0  1  0 1 1 | 0 1 0 1 1
.. .. .x    |  0  2 | *  *  *  * 6 | 0 0  0  0  2 0 2 | 0 0 1 2 1
------------+-------+--------------+------------------+----------
x.3x. ..&#x |  6  0 | 3  3  0  0 0 | 4 *  *  *  * * * | 1 1 0 0 0
.. x.3o.    |  3  0 | 0  3  0  0 0 | * 4  *  *  * * * | 1 0 0 1 0
xo .. ..&#x |  2  1 | 1  0  2  0 0 | * * 12  *  * * * | 0 1 1 0 0
.. xx ..&#x |  2  2 | 0  1  2  1 0 | * *  * 12  * * * | 0 1 0 1 0
.. .. ox&#x |  1  2 | 0  0  2  0 1 | * *  *  * 12 * * | 0 0 1 1 0
.o3.x ..    |  0  3 | 0  0  0  3 0 | * *  *  *  * 4 * | 0 1 0 0 1
.. .x3.x&#x |  0  6 | 0  0  0  3 3 | * *  *  *  * * 4 | 0 0 0 1 1
------------+-------+--------------+------------------+----------
x.3x.3o.    ♦ 12  0 | 6 12  0  0 0 | 4 4  0  0  0 0 0 | 1 * * * *
xo3xx ..&#x ♦  6  3 | 3  3  6  3 0 | 1 0  3  3  0 1 0 | * 4 * * *
xo .. ox&#x ♦  2  2 | 1  0  4  0 1 | 0 0  2  0  2 0 0 | * * 6 * *
.. xx3ox&#x ♦  3  6 | 0  3  6  3 3 | 0 1  0  3  3 0 1 | * * * 4 *
.o3.x3.x    ♦  0 12 | 0  0  0 12 6 | 0 0  0  0  0 4 4 | * * * * 1
```
```or
o.3o.3o.    & | 24 |  1  2  2 | 2 1  3  2 | 1 3 1
--------------+----+----------+-----------+------
x. .. ..    & |  2 | 12  *  * | 0 0  2  0 | 1 2 1
.. x. ..    & |  2 |  * 24  * | 1 1  0  1 | 1 2 0
oo3oo3oo&#x   |  2 |  *  * 24 | 0 0  2  1 | 0 2 1
--------------+----+----------+-----------+------
x.3x. ..&#x & |  6 |  3  3  0 | 8 *  *  * | 1 1 0
.. x.3o.    & |  3 |  0  3  0 | * 8  *  * | 1 1 0
xo .. ..&#x & |  3 |  1  0  2 | * * 24  * | 0 1 1
.. xx ..&#x   |  4 |  0  2  2 | * *  * 12 | 0 2 0
--------------+----+----------+-----------+------
x.3x.3o.    & ♦ 12 |  6 12  0 | 4 4  0  0 | 2 * *
xo3xx ..&#x & ♦  9 |  3  6  6 | 3 1  3  3 | * 8 *
xo .. ox&#x   ♦  4 |  2  0  4 | 0 0  4  0 | * * 6
```

```s4o3x2s

demi( . . . . ) | 24 |  2  1  2 | 1  2 2  3 | 1 1 3
----------------+----+----------+-----------+------
demi( . . x . ) |  2 | 24  *  * | 1  1 1  0 | 1 0 2
s4o . .   |  2 |  * 12  * | 0  0 2  2 | 1 1 2
s . 2 s   |  2 |  *  * 24 | 0  1 0  2 | 0 1 2
----------------+----+----------+-----------+------
demi( . o3x . ) |  3 |  3  0  0 | 8  * *  * | 1 0 1
s 2 x2s   |  4 |  2  0  2 | * 12 *  * | 0 0 2
sefa( s4o3x . ) |  6 |  3  3  0 | *  * 8  * | 1 0 1
sefa( s4o 2 s ) |  3 |  0  1  2 | *  * * 24 | 0 1 1
----------------+----+----------+-----------+------
s4o3x .   ♦ 12 | 12  6  0 | 4  0 4  0 | 2 * *
s4o 2 s   ♦  4 |  0  2  4 | 0  0 0  4 | * 6 *
sefa( s4o3x2s ) ♦  9 |  6  3  6 | 1  3 1  3 | * * 8

starting figure: x4o3x x
```

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