Acronym hedjak
Name heptadiminished jak,
square || hin
Circumradius sqrt(2/3) = 0.816497
Lace city
in approx. ASCII-art
   4  
      
h     H   -- hin

        \
         +-- tedhin
wobei 
4 = {4}
h = hex
H = gyro hex
Coordinates
  1. (1/sqrt(2), 0, 0, 0, 0, -sqrt(3/8))                                     & all permutations, all changes of sign in the first 2 coord.s only
    (top square layer)
  2. (1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 0)    & all even changes of sign in one but last coord.
    (bottom hin layer)
Confer
uniform relative:
jak  
segmentopeta:
dijak   tedjak   hinpy  
general polytopal classes:
segmentopeta  

The icosiheptaheptacontidipeton (jak) could be given as point || hin || tac. Its tridiminishing (tedjak) chops off 3 hinpies, in fact one vertex from the first layer and two opposite ones from the third layer, leaving there the equatorial hex only. That one then will be gyrated to either of the other two hexes, from which hin could be thought of as a segmentoteron. If this remaining hex of the third layer is being thought of as a tegum sum of 2 perp. squares, and the vertices of one such square are being omitted as well, then this polyexon will be derived.


Incidence matrix according to Dynkin symbol

xoo3ooo3oxo xox oxx&#x   → height(1,2) = 0
                           height(1,3) = height(2,3) = sqrt(3/8) = 0.612372
(hin || part. para square)

o..3o..3o.. o.. o..    & | 16 * |  3 1  6  2 0 |  3 18  9  6  2  1  6 0 | 1  8  6 12  3  6  3 1 18  9 | 2  5 3 2  3  8  6 12  3 | 1 1 2  5 3
..o3..o3..o ..o ..o      |  * 4 |  0 0  0  8 2 |  0  0  0 12  8  8 12 1 | 0  0  0  0  0  8 12 8 24 24 | 0  0 0 2  8  8  6 24 12 | 0 2 1  8 6
-------------------------+------+--------------+------------------------+-----------------------------+-------------------------+-----------
x.. ... ... ... ...    & |  2 0 | 24 *  *  * * |  2  4  0  2  0  0  0 0 | 1  4  2  2  0  4  1 0  4  0 | 2  2 1 2  2  4  2  2  0 | 1 1 2  2 1
... ... ... x.. ...    & |  2 0 |  * 8  *  * * |  0  0  6  0  2  0  0 0 | 0  0  0  6  3  0  0 1  0  6 | 0  2 3 0  0  0  0  6  3 | 1 0 0  2 3
oo.3oo.3oo. oo. oo.&#x   |  2 0 |  * * 48  * * |  0  4  2  0  0  0  1 0 | 0  2  2  4  1  0  0 0  4  2 | 1  2 2 0  0  2  2  4  1 | 1 0 1  2 2
o.o3o.o3o.o o.o o.o&#x & |  1 1 |  * *  * 32 * |  0  0  0  3  1  1  3 0 | 0  0  0  0  0  3  3 1  9  6 | 0  0 0 1  3  4  3  9  3 | 0 1 1  4 3
... ... ... ..x ...    & |  0 2 |  * *  *  * 4 |  0  0  0  0  4  4  0 1 | 0  0  0  0  0  0  6 8  0 12 | 0  0 0 0  4  0  0 12 12 | 0 1 0  4 6
-------------------------+------+--------------+------------------------+-----------------------------+-------------------------+-----------
x..3o.. ... ... ...    & |  3 0 |  3 0  0  0 0 | 16  *  *  *  *  *  * * | 1  2  0  0  0  2  0 0  0  0 | 2  1 0 2  1  2  0  0  0 | 1 1 2  1 0
xo. ... ... ... ...&#x & |  3 0 |  1 0  2  0 0 |  * 96  *  *  *  *  * * | 0  1  1  1  0  0  0 0  1  0 | 1  1 1 0  0  1  1  1  0 | 1 0 1  1 1
... ... ... xo. ...&#x & |  3 0 |  0 1  2  0 0 |  *  * 48  *  *  *  * * | 0  0  0  2  1  0  0 0  0  1 | 0  1 2 0  0  0  0  2  1 | 1 0 0  1 2
x.o ... ... ... ...&#x & |  2 1 |  1 0  0  2 0 |  *  *  * 48  *  *  * * | 0  0  0  0  0  2  1 0  2  0 | 0  0 0 1  2  2  1  2  0 | 0 1 1  2 1
... ... ... x.x ...&#x & |  2 2 |  0 1  0  2 1 |  *  *  *  * 16  *  * * | 0  0  0  0  0  0  0 1  0  3 | 0  0 0 0  0  0  0  3  3 | 0 0 0  1 3
... ... ... ... o.x&#x & |  1 2 |  0 0  0  2 1 |  *  *  *  *  * 16  * * | 0  0  0  0  0  0  3 1  0  3 | 0  0 0 0  3  0  0  6  3 | 0 1 0  3 3
ooo3ooo3ooo ooo ooo&#x   |  2 1 |  0 0  1  2 0 |  *  *  *  *  *  * 48 * | 0  0  0  0  0  0  0 0  4  2 | 0  0 0 0  0  2  2  4  1 | 0 0 1  2 2
... ... ... ..x ..x      |  0 4 |  0 0  0  0 4 |  *  *  *  *  *  *  * 1  0  0  0  0  0  0  0 8  0  0 | 0  0 0 0  0  0  0  0 12 | 0 0 0  0 6
-------------------------+------+--------------+------------------------+-----------------------------+-------------------------+-----------
x..3o..3o.. ... ...    &   4 0 |  6 0  0  0 0 |  4  0  0  0  0  0  0 0 | 4  *  *  *  *  *  * *  *  * | 2  0 0 2  0  0  0  0  0 | 1 1 2  0 0
xo.3oo. ... ... ...&#x &   4 0 |  3 0  3  0 0 |  1  3  0  0  0  0  0 0 | * 32  *  *  *  *  * *  *  * | 1  1 0 0  0  1  0  0  0 | 1 0 1  1 0
xo. ... ox. ... ...&#x     4 0 |  2 0  4  0 0 |  0  4  0  0  0  0  0 0 | *  * 24  *  *  *  * *  *  * | 1  0 1 0  0  0  1  0  0 | 1 0 1  0 1
xo. ... ... ... ox.&#x &   4 0 |  1 1  4  0 0 |  0  2  2  0  0  0  0 0 | *  *  * 48  *  *  * *  *  * | 0  1 1 0  0  0  0  1  0 | 1 0 0  1 1
... ... ... xo. ox.&#x     4 0 |  0 2  4  0 0 |  0  0  4  0  0  0  0 0 | *  *  *  * 12  *  * *  *  * | 0  0 2 0  0  0  0  0  1 | 1 0 0  0 2
x.o3o.o ... ... ...&#x &   3 1 |  3 0  0  3 0 |  1  0  0  3  0  0  0 0 | *  *  *  *  * 32  * *  *  * | 0  0 0 1  1  1  0  0  0 | 0 1 1  1 0
x.o ... ... ... o.x&#x &   2 2 |  1 0  0  4 1 |  0  0  0  2  0  2  0 0 | *  *  *  *  *  * 24 *  *  * | 0  0 0 0  2  0  0  2  0 | 0 1 0  2 1
... ... ... x.x o.x&#x &   2 4 |  0 1  0  4 4 |  0  0  0  0  2  2  0 1 | *  *  *  *  *  *  * 8  *  * | 0  0 0 0  0  0  0  0  3 | 0 0 0  0 3
xoo ... ... ... ...&#x &   3 1 |  1 0  2  3 0 |  0  1  0  1  0  0  2 0 | *  *  *  *  *  *  * * 96  * | 0  0 0 0  0  1  1  1  0 | 0 0 1  1 1
... ... ... xox ...&#x &   3 2 |  0 1  2  4 1 |  0  0  1  0  1  1  2 0 | *  *  *  *  *  *  * *  * 48 | 0  0 0 0  0  0  0  2  1 | 0 0 0  1 2
-------------------------+------+--------------+------------------------+-----------------------------+-------------------------+-----------
xo.3oo.3ox. ... ...&#x     8 0 | 12 0 12  0 0 |  8 24  0  0  0  0  0 0 | 2  8  6  0  0  0  0 0  0  0 | 4  * * *  *  *  *  *  * | 1 0 1  0 0
xo.3oo. ... ... ox.&#x &   5 0 |  3 1  6  0 0 |  1  6  3  0  0  0  0 0 | 0  2  0  3  0  0  0 0  0  0 | * 16 * *  *  *  *  *  * | 1 0 0  1 0
xo. ... ox. xo. ox.&#zx    8 0 |  4 4 16  0 0 |  0 16 16  0  0  0  0 0 | 0  0  4  8  4  0  0 0  0  0 | *  * 6 *  *  *  *  *  * | 1 0 0  0 1
x.o3o.o3o.o ... ...&#x &   4 1 |  6 0  0  4 0 |  4  0  0  6  0  0  0 0 | 1  0  0  0  0  4  0 0  0  0 | *  * * 8  *  *  *  *  * | 0 1 1  0 0
x.o3o.o ... ... o.x&#x &   3 2 |  3 0  0  6 1 |  1  0  0  6  0  3  0 0 | 0  0  0  0  0  2  3 0  0  0 | *  * * * 16  *  *  *  * | 0 1 0  1 0
xoo3ooo ... ... ...&#x &   4 1 |  3 0  3  4 0 |  1  3  0  3  0  0  3 0 | 0  1  0  0  0  1  0 0  3  0 | *  * * *  * 32  *  *  * | 0 0 1  1 0
xoo ... oxo ... ...&#x &   4 1 |  2 0  4  4 0 |  0  4  0  2  0  0  4 0 | 0  0  1  0  0  0  0 0  4  0 | *  * * *  *  * 24  *  * | 0 0 1  0 1
xoo ... ... ... oxx&#x &   4 2 |  1 1  4  6 1 |  0  2  2  2  1  2  4 0 | 0  0  0  1  0  0  1 0  2  2 | *  * * *  *  *  * 48  * | 0 0 0  1 1
... ... ... xox oxx&#x     4 4 |  0 2  4  8 4 |  0  0  4  0  4  4  4 1 | 0  0  0  0  1  0  0 2  0  4 | *  * * *  *  *  *  * 12 | 0 0 0  0 2
-------------------------+------+--------------+------------------------+-----------------------------+-------------------------+-----------
xo.3oo.3ox. xo. ox.&#zx   16 0 | 24 8 48  0 0 | 16 96 48  0  0  0  0 0 | 4 32 24 48 12  0  0 0  0  0 | 4 16 6 0  0  0  0  0  0 | 1 * *  * *
x.o3o.o3o.o ... o.x&#x &   4 2 |  6 0  0  8 1 |  4  0  0 12  0  4  0 0 | 1  0  0  0  0  8  6 0  0  0 | 0  0 0 2  4  0  0  0  0 | * 4 *  * *
xoo3ooo3oxo ... ...&#x     8 1 | 12 0 12  8 0 |  8 24  0 12  0  0 12 0 | 2  8  6  0  0  8  0 0 24  0 | 1  0 0 2  0  8  6  0  0 | * * 4  * *
xoo3ooo ... ... oxx&#x &   5 2 |  3 1  6  8 1 |  1  6  3  6  1  3  6 0 | 0  2  0  3  0  2  3 0  6  3 | 0  1 0 0  1  2  0  3  0 | * * * 16 *
xoo ... oxo xox oxx&#x     8 4 |  4 4 16 16 4 |  0 16 16  8  8  8 16 1 | 0  0  4  8  4  0  4 4 16 16 | 0  0 1 0  0  0  4  8  4 | * * *  * 6

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