Acronym tritra ope
Name (triangle, dual triangle, trip, gyro trip) lace simplex
Circumradius sqrt(3)/2 = 0.866025
Lace city
in approx. ASCII-art 
x3o x                     
(1)                       
                     x3o o
                     (3)  
                          
               o3x o      
               (4)        
   o3x x                  
   (2)
Face vector 18, 78, 138, 121, 55, 12
Confer
more general:
(N-g, dual N-g, para N-p, gyro N-p) lace simplex  
general polytopal classes:
segmentopeta   lace simplices  

Incidence matrix according to Dynkin symbol

xoxo3oxox xxoo&#x   → height(1,2) = height(3,4) = sqrt(2/3) = 0.816497
                      height(1,3) = height(2,4) = sqrt(3)/2 = 0.866025
                      height(1,4) = height(2,3) = sqrt(5/12) = 0.645497

o...3o... o...    & | 12 * |  2 1  2  1  2 0 0 | 1 2  3 2  2 1  2  1  2  4  2 0 0 | 1 1 3 1 2 1 2 1  3  3  4  2  1  2  2 0 | 1 1 1 2 3 3 1 2 1  3 2 | 2 1 1 3
..o.3..o. ..o.    & |  * 6 |  0 0  0  2  4 2 2 | 0 0  0 0  4 1  2  4  2  4  8 1 3 | 0 0 0 2 2 2 1 2  4  2  2  6  6  4  4 1 | 0 1 1 2 2 1 4 3 3  6 2 | 1 2 2 3
--------------------+------+-------------------+----------------------------------+----------------------------------------+------------------------+--------
x... .... ....    & |  2 0 | 12 *  *  *  * * * | 1 1  1 0  1 0  1  0  0  0  0 0 0 | 1 1 1 1 1 1 1 0  1  1  0  1  0  0  0 0 | 1 1 1 2 1 1 1 1 0  1 0 | 2 1 1 1
.... .... x...    & |  2 0 |  * 6  *  *  * * * | 0 2  0 2  0 1  0  0  2  0  0 0 0 | 1 0 3 0 2 0 2 1  0  0  4  0  0  2  0 0 | 1 1 1 0 3 3 0 2 1  0 2 | 2 1 0 3
oo..3oo.. oo..&#x   |  2 0 |  * * 12  *  * * * | 0 0  2 1  0 0  0  0  0  2  0 0 0 | 0 1 2 0 0 0 0 0  2  2  2  0  0  0  1 0 | 1 0 0 2 2 2 0 0 0  2 1 | 2 0 1 2
o.o.3o.o. o.o.&#x & |  1 1 |  * *  * 12  * * * | 0 0  0 0  2 1  0  0  0  2  2 0 0 | 0 0 0 1 2 0 0 0  2  1  2  2  1  2  2 0 | 0 1 0 1 2 1 1 2 1  3 2 | 1 1 1 3
o..o3o..o o..o&#x & |  1 1 |  * *  *  * 24 * * | 0 0  0 0  0 0  1  1  1  1  1 0 0 | 0 0 0 0 0 1 1 1  1  1  1  1  1  1  1 0 | 0 0 1 1 1 1 1 1 1  2 1 | 1 1 1 2
..x. .... ....    & |  0 2 |  * *  *  *  * 6 * | 0 0  0 0  2 0  0  2  0  0  0 1 1 | 0 0 0 2 1 2 0 1  2  0  0  2  2  0  0 1 | 0 1 1 2 1 0 4 1 1  2 0 | 1 2 2 1
..oo3..oo ..oo&#x   |  0 2 |  * *  *  *  * * 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  4  4  2  2 1 | 0 0 0 0 0 0 4 2 2  4 1 | 0 2 2 2
--------------------+------+-------------------+----------------------------------+----------------------------------------+------------------------+--------
x...3o... ....    & |  3 0 |  3 0  0  0  0 0 0 | 4 *  * *  * *  *  *  *  *  * * * | 1 1 0 1 0 1 0 0  0  0  0  0  0  0  0 0 | 1 1 1 2 0 0 1 0 0  0 0 | 2 1 1 0
x... .... x...    & |  4 0 |  2 2  0  0  0 0 0 | * 6  * *  * *  *  *  *  *  * * * | 1 0 1 0 1 0 1 0  0  0  0  0  0  0  0 0 | 1 1 1 0 1 1 0 1 0  0 0 | 2 1 0 1
xo.. .... ....&#x & |  3 0 |  1 0  2  0  0 0 0 | * * 12 *  * *  *  *  *  *  * * * | 0 1 1 0 0 0 0 0  1  1  0  0  0  0  0 0 | 1 0 0 2 1 1 0 0 0  1 0 | 2 0 1 1
.... .... xx..&#x   |  4 0 |  0 2  2  0  0 0 0 | * *  * 6  * *  *  *  *  *  * * * | 0 0 2 0 0 0 0 0  0  0  2  0  0  0  0 0 | 1 0 0 0 2 2 0 0 0  0 1 | 2 0 0 2
x.x. .... ....&#x & |  2 2 |  1 0  0  2  0 1 0 | * *  * * 12 *  *  *  *  *  * * * | 0 0 0 1 1 0 0 0  1  0  0  1  0  0  0 0 | 0 1 0 1 1 0 1 1 0  1 0 | 1 1 1 1
.... .... x.o.&#x & |  2 1 |  0 1  0  2  0 0 0 | * *  * *  * 6  *  *  *  *  * * * | 0 0 0 0 2 0 0 0  0  0  2  0  0  2  0 0 | 0 1 0 0 2 1 0 2 1  0 2 | 1 1 0 3
x..o .... ....&#x & |  2 1 |  1 0  0  0  2 0 0 | * *  * *  * * 12  *  *  *  * * * | 0 0 0 0 0 1 1 0  0  1  0  1  0  0  0 0 | 0 0 1 1 0 1 1 1 0  1 0 | 1 1 1 1
.... o..x ....&#x & |  1 2 |  0 0  0  0  2 1 0 | * *  * *  * *  * 12  *  *  * * * | 0 0 0 0 0 1 0 1  1  0  0  0  1  0  0 0 | 0 0 1 1 1 0 1 0 1  1 0 | 1 1 1 1
.... .... x..o&#x & |  2 1 |  0 1  0  0  2 0 0 | * *  * *  * *  *  * 12  *  * * * | 0 0 0 0 0 0 1 1  0  0  1  0  0  1  0 0 | 0 0 1 0 1 1 0 1 1  0 1 | 1 1 0 2
ooo.3ooo. ooo.&#x & |  2 1 |  0 0  1  1  1 0 0 | * *  * *  * *  *  *  * 24  * * * | 0 0 0 0 0 0 0 0  1  1  1  0  0  0  1 0 | 0 0 0 1 1 1 0 0 0  2 1 | 1 0 1 2
o.oo3o.oo o.oo&#x & |  1 2 |  0 0  0  1  1 0 1 | * *  * *  * *  *  *  *  * 24 * * | 0 0 0 0 0 0 0 0  0  0  0  1  1  1  1 0 | 0 0 0 0 0 0 1 1 1  2 1 | 0 1 1 2
..x.3..o. ....    & |  0 3 |  0 0  0  0  0 3 0 | * *  * *  * *  *  *  *  *  * 2 * | 0 0 0 2 0 2 0 0  0  0  0  0  0  0  0 1 | 0 1 1 2 0 0 4 0 0  0 0 | 1 2 2 0
..xo .... ....&#x & |  0 3 |  0 0  0  0  0 1 2 | * *  * *  * *  *  *  *  *  * * 6 | 0 0 0 0 0 0 0 0  0  0  0  2  2  0  0 1 | 0 0 0 0 0 0 4 1 1  2 0 | 0 2 2 1
--------------------+------+-------------------+----------------------------------+----------------------------------------+------------------------+--------
x...3o... x...    & ♦  6 0 |  6 3  0  0  0 0 0 | 2 3  0 0  0 0  0  0  0  0  0 0 0 | 2 * * * * * * *  *  *  *  *  *  *  * * | 1 1 1 0 0 0 0 0 0  0 0 | 2 1 0 0
xo..3ox.. ....&#x   ♦  6 0 |  6 0  6  0  0 0 0 | 2 0  6 0  0 0  0  0  0  0  0 0 0 | * 2 * * * * * *  *  *  *  *  *  *  * * | 1 0 0 2 0 0 0 0 0  0 0 | 2 0 1 0
xo.. .... xx..&#x & ♦  6 0 |  2 3  4  0  0 0 0 | 0 1  2 2  0 0  0  0  0  0  0 0 0 | * * 6 * * * * *  *  *  *  *  *  *  * * | 1 0 0 0 1 1 0 0 0  0 0 | 2 0 0 1
x.x.3o.o. ....&#x & ♦  3 3 |  3 0  0  3  0 3 0 | 1 0  0 0  3 0  0  0  0  0  0 1 0 | * * * 4 * * * *  *  *  *  *  *  *  * * | 0 1 0 1 0 0 1 0 0  0 0 | 1 1 1 0
x.x. .... x.o.&#x & ♦  4 2 |  2 2  0  4  0 1 0 | 0 1  0 0  2 2  0  0  0  0  0 0 0 | * * * * 6 * * *  *  *  *  *  *  *  * * | 0 1 0 0 1 0 0 1 0  0 0 | 1 1 0 1
x..o3o..x ....&#x & ♦  3 3 |  3 0  0  0  6 3 0 | 1 0  0 0  0 0  3  3  0  0  0 1 0 | * * * * * 4 * *  *  *  *  *  *  *  * * | 0 0 1 1 0 0 1 0 0  0 0 | 1 1 1 0
x..o .... x..o&#x & ♦  4 1 |  2 2  0  0  4 0 0 | 0 1  0 0  0 0  2  0  2  0  0 0 0 | * * * * * * 6 *  *  *  *  *  *  *  * * | 0 0 1 0 0 1 0 1 0  0 0 | 1 1 0 1
.... o..x x..o&#x & ♦  2 2 |  0 1  0  0  4 1 0 | 0 0  0 0  0 0  0  2  2  0  0 0 0 | * * * * * * * 6  *  *  *  *  *  *  * * | 0 0 1 0 1 0 0 0 1  0 0 | 1 1 0 1
xox. .... ....&#x & ♦  3 2 |  1 0  2  2  2 1 0 | 0 0  1 0  1 0  0  1  0  2  0 0 0 | * * * * * * * * 12  *  *  *  *  *  * * | 0 0 0 1 1 0 0 0 0  1 0 | 1 0 1 1
.... oxo. ....&#x & ♦  3 1 |  1 0  2  1  2 0 0 | 0 0  1 0  0 0  1  0  0  2  0 0 0 | * * * * * * * *  * 12  *  *  *  *  * * | 0 0 0 1 0 1 0 0 0  1 0 | 1 0 1 1
.... .... xxo.&#x & ♦  4 1 |  0 2  2  2  2 0 0 | 0 0  0 1  0 1  0  0  1  2  0 0 0 | * * * * * * * *  *  * 12  *  *  *  * * | 0 0 0 0 1 1 0 0 0  0 1 | 1 0 0 2
x.xo .... ....&#x & ♦  2 3 |  1 0  0  2  2 1 2 | 0 0  0 0  1 0  1  0  0  0  2 0 1 | * * * * * * * *  *  *  * 12  *  *  * * | 0 0 0 0 0 0 1 1 0  1 0 | 0 1 1 1
.... o.ox ....&#x & ♦  1 3 |  0 0  0  1  2 1 2 | 0 0  0 0  0 0  0  1  0  0  2 0 1 | * * * * * * * *  *  *  *  * 12  *  * * | 0 0 0 0 0 0 1 0 1  1 0 | 0 1 1 1
.... .... x.oo&#x & ♦  2 2 |  0 1  0  2  2 0 1 | 0 0  0 0  0 1  0  0  1  0  2 0 0 | * * * * * * * *  *  *  *  *  * 12  * * | 0 0 0 0 0 0 0 1 1  0 1 | 0 1 0 2
oooo3oooo oooo&#x   ♦  2 2 |  0 0  1  2  2 0 1 | 0 0  0 0  0 0  0  0  0  2  2 0 0 | * * * * * * * *  *  *  *  *  *  * 12 * | 0 0 0 0 0 0 0 0 0  2 1 | 0 0 1 2
..xo3..ox ....&#x   ♦  0 6 |  0 0  0  0  0 6 6 | 0 0  0 0  0 0  0  0  0  0  0 2 6 | * * * * * * * *  *  *  *  *  *  *  * 1 | 0 0 0 0 0 0 4 0 0  0 0 | 0 2 2 0
--------------------+------+-------------------+----------------------------------+----------------------------------------+------------------------+--------
xo..3ox.. xx..&#x   ♦ 12 0 | 12 6 12  0  0 0 0 | 4 6 12 6  0 0  0  0  0  0  0 0 0 | 2 2 6 0 0 0 0 0  0  0  0  0  0  0  0 0 | 1 * * * * * * * *  * * | 2 0 0 0
x.x.3o.o. x.o.&#x & ♦  6 3 |  6 3  0  6  0 3 0 | 2 3  0 0  6 3  0  0  0  0  0 1 0 | 1 0 0 2 3 0 0 0  0  0  0  0  0  0  0 0 | * 2 * * * * * * *  * * | 1 1 0 0
x..o3o..x x..o&#x & ♦  6 3 |  6 3  0  0 12 3 0 | 2 3  0 0  0 0  6  6  6  0  0 1 0 | 1 0 0 0 0 2 3 3  0  0  0  0  0  0  0 0 | * * 2 * * * * * *  * * | 1 1 0 0
xox.3oxo. ....&#x & ♦  6 3 |  6 0  6  3  6 3 0 | 2 0  6 0  3 0  3  3  0  6  0 1 0 | 0 1 0 1 0 1 0 0  3  3  0  0  0  0  0 0 | * * * 4 * * * * *  * * | 1 0 1 0
xox. .... xxo.&#x & ♦  6 2 |  2 3  4  4  4 1 0 | 0 1  2 2  2 2  0  2  2  4  0 0 0 | 0 0 1 0 1 0 0 1  2  0  2  0  0  0  0 0 | * * * * 6 * * * *  * * | 1 0 0 1
.... oxo. xxo.&#x & ♦  6 1 |  2 3  4  2  4 0 0 | 0 1  2 2  0 1  2  0  2  4  0 0 0 | 0 0 1 0 0 0 1 0  0  2  2  0  0  0  0 0 | * * * * * 6 * * *  * * | 1 0 0 1
x.xo3o.ox ....&#x & ♦  3 6 |  3 0  0  3  6 6 6 | 1 0  0 0  3 0  3  3  0  0  6 2 6 | 0 0 0 1 0 1 0 0  0  0  0  3  3  0  0 1 | * * * * * * 4 * *  * * | 0 1 1 0
x.xo .... x.oo&#x & ♦  4 3 |  2 2  0  4  4 1 2 | 0 1  0 0  2 2  2  0  2  0  4 0 1 | 0 0 0 0 1 0 1 0  0  0  0  2  0  2  0 0 | * * * * * * * 6 *  * * | 0 1 0 1
.... o.ox x.oo&#x & ♦  2 3 |  0 1  0  2  4 1 2 | 0 0  0 0  0 1  0  2  2  0  4 0 1 | 0 0 0 0 0 0 0 1  0  0  0  0  2  2  0 0 | * * * * * * * * 6  * * | 0 1 0 1
xoxo .... ....&#x & ♦  3 3 |  1 0  2  3  4 1 2 | 0 0  1 0  1 0  1  1  0  4  4 0 1 | 0 0 0 0 0 0 0 0  1  1  0  1  1  0  2 0 | * * * * * * * * * 12 * | 0 0 1 1
.... .... xxoo&#x   ♦  4 2 |  0 2  2  4  4 0 1 | 0 0  0 1  0 2  0  0  2  4  4 0 0 | 0 0 0 0 0 0 0 0  0  0  2  0  0  2  2 0 | * * * * * * * * *  * 6 | 0 0 0 2
--------------------+------+-------------------+----------------------------------+----------------------------------------+------------------------+--------
xox.3oxo. xxo.&#x & ♦ 12 3 | 12 6 12  6 12 3 0 | 4 6 12 6  6 3  6  6  6 12  0 1 0 | 2 2 6 2 3 2 3 3  6  6  6  0  0  0  0 0 | 1 1 1 2 3 3 0 0 0  0 0 | 2 * * *
x.xo3o.ox x.oo&#x & ♦  6 6 |  6 3  0  6 12 6 6 | 2 3  0 0  6 3  6  6  6  0 12 2 6 | 1 0 0 2 3 2 3 3  0  0  0  6  6  6  0 1 | 0 1 1 0 0 0 2 3 3  0 0 | * 2 * *
xoxo3oxox ....&#x   ♦  6 6 |  6 0  6  6 12 6 6 | 2 0  6 0  6 0  6  6  0 12 12 2 6 | 0 1 0 2 0 2 0 0  6  6  0  6  6  0  6 1 | 0 0 0 2 0 0 2 0 0  6 0 | * * 2 *
xoxo .... xxoo&#x & ♦  6 3 |  2 3  4  6  8 1 2 | 0 1  2 2  2 3  2  2  4  8  8 0 1 | 0 0 1 0 1 0 1 1  2  2  4  2  2  4  4 0 | 0 0 0 0 1 1 0 1 1  2 2 | * * * 6

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