Acronym ...
Name point || hex || gyro hex,
Dutour polyteron type A
Lace city
in approx. ASCII-art
o        	where:
    T    	T = x3o3o (tet)
O       o	t = o3o3x (dual tet)
    t    	O = o3x3o (oct)
o        	o = o3o3o (point)
         
|   |   +	o3o3o *b3o (point)
|   +----	x3o3o *b3o (hex)
+--------	o3o3o *b3x (gyro hex)
Coordinates
  1. (0, 0, 0, 0, 1/sqrt(2))
    (right layer of single vertex)
  2. (1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 0)    & all even changes of sign in one but last coord.
    (medial hex layer)
  3. (1/sqrt(2), 0, 0, 0, -1/sqrt(2))                         & all permutations, all changes of sign in one but last coord.
    (left hex layer)
Dihedral angles
(at margins)
  • at equatorial tet between hex and pen:   arccos[-2/sqrt(5)] = 153.434949°
  • at tet between pen and tete:   arccos(-3/5) = 126.869898°
  • at tet between hex and tete:   arccos[-1/sqrt(5)] = 116.565051°
  • at non-equatorial tet between hex and pen:   arccos[-1/sqrt(5)] = 116.565051°
  • at tet between hex and hex:   90°
Confer
uniform relative:
hin  
related segmentotera:
hexpy  
general polytopal classes:
Dutour's Delone cells  

This polyteron is nothing but an external blend of hin and hexpy, adjoining at an hex. There it just happens that the dihedral angle between some of the pens becomes flat, i.e. they join into a tete each.


Incidence matrix according to Dynkin symbol

oxo3ooo3ooo *b3oox&#xt both heights = 1/sqrt(2) = 0.707107

o..3o..3o.. *b3o..     | 1 * *  8  0  0  0 | 24  0  0  0  0 | 32 0  0  0  0 0 0 | 8 8 0 0 0
.o.3.o.3.o. *b3.o.     | * 8 * | 1  6  4  0 |  6 12 12  6  0 | 12 4 12  6  4 0 0 | 4 4 4 1 0
..o3..o3..o *b3..o     | * * 8 | 0  0  4  6 |  0  0  6 12 12 |  0 0  4  6 12 4 4 | 1 0 4 4 1
-----------------------+-------+------------+----------------+-------------------+----------
oo.3oo.3oo. *b3oo.&#x  | 1 1 0 | 8  *  *  *   6  0  0  0  0 | 12 0  0  0  0 0 0 | 4 4 0 0 0
.x. ... ...    ...     | 0 2 0 | * 24  *  * |  1  4  2  0  0 |  4 2  4  1  0 0 0 | 2 2 2 0 0
.oo3.oo3.oo *b3.oo&#x  | 0 1 1 | *  * 32  * |  0  0  3  3  0 |  0 0  3  3  3 0 0 | 1 0 3 1 0
... ... ...    ..x     | 0 0 2 | *  *  * 24 |  0  0  0  2  4 |  0 0  0  1  4 2 2 | 0 0 2 2 1
-----------------------+-------+------------+----------------+-------------------+----------
ox. ... ...    ...&#x  | 1 2 0 | 2  1  0  0 | 24  *  *  *  * |  4 0  0  0  0 0 0 | 2 2 0 0 0
.x.3.o. ...    ...     | 0 3 0 | 0  3  0  0 |  * 32  *  *  * |  1 1  1  0  0 0 0 | 1 1 1 0 0
.xo ... ...    ...&#x  | 0 2 1 | 0  1  2  0 |  *  * 48  *  * |  0 0  2  1  0 0 0 | 1 0 2 0 0
... ... ...    .ox&#x  | 0 1 2 | 0  0  2  1 |  *  *  * 48  * |  0 0  0  1  2 0 0 | 0 0 2 1 0
... ..o ... *b3..x     | 0 0 3 | 0  0  0  3 |  *  *  *  * 32 |  0 0  0  0  1 1 1 | 0 0 1 1 1
-----------------------+-------+------------+----------------+-------------------+----------
ox.3oo. ...    ...&#x   1 3 0 | 3  3  0  0 |  3  1  0  0  0 | 32 *  *  *  * * * | 1 1 0 0 0
.x.3.o. ... *b3.o.      0 4 0 | 0  6  0  0 |  0  4  0  0  0 |  * 8  *  *  * * * | 0 1 1 0 0
.xo3.oo ...    ...&#x   0 3 1 | 0  3  3  0 |  0  1  3  0  0 |  * * 32  *  * * * | 1 0 1 0 0
.xo ... ...    .ox&#x   0 2 2 | 0  1  4  1 |  0  0  2  2  0 |  * *  * 24  * * * | 0 0 2 0 0
... .oo ... *b3.ox&#x   0 1 3 | 0  0  3  3 |  0  0  0  3  1 |  * *  *  * 32 * * | 0 0 1 1 0
..o3..o ... *b3..x      0 0 4 | 0  0  0  6 |  0  0  0  0  4 |  * *  *  *  * 8 * | 0 0 1 0 1
... ..o3..o *b3..x      0 0 4 | 0  0  0  6 |  0  0  0  0  4 |  * *  *  *  * * 8 | 0 0 0 1 1
-----------------------+-------+------------+----------------+-------------------+----------
oxo3ooo3ooo    ...&#xt  1 4 1 | 4  6  4  0 |  6  4  6  0  0 |  4 0  4  0  0 0 0 | 8 * * * *
ox.3oo. ... *b3oo.&#x   1 4 0 | 4  6  0  0 |  6  4  0  0  0 |  4 1  0  0  0 0 0 | * 8 * * *
.xo3.oo ... *b3.ox&#x   0 4 4 | 0  6 12  6 |  0  4 12 12  4 |  0 1  4  6  4 1 0 | * * 8 * *
... .oo3.oo *b3.ox&#x   0 1 4 | 0  0  4  6 |  0  0  0  6  4 |  0 0  0  0  4 0 1 | * * * 8 *
..o3..o3..o *b3..x      0 0 8 | 0  0  0 24 |  0  0  0  0 32 |  0 0  0  0  0 8 8 | * * * * 1

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