Acronym hexpy
Name hexadecachoral pyramid,
vertex-first rotunda of triacontiditeron
Circumradius 1/sqrt(2) = 0.707107
Lace city
in approx. ASCII-art
    o    		where:
         		o = o3o4o (point)
o   O   o		O = x3o4o (oct)
  o  		o = o3o3o (point)
     		t = x3o3o (tet)
t   T		T = o3o3x (dual tet)
Coordinates
  1. (0, 0, 0, 0, 1/sqrt(2))
    tip
  2. (1/sqrt(2), 0, 0, 0, 0)   & permutations and changes of sign in all but last coordinate
    hexadecachoron base
Dihedral angles
  • at tet between pen and pen:   arccos(-3/5) = 126.869898°
  • at tet between hex and pen:   arccos(1/sqrt(5)) = 63.434949°
Confer
uniform relative:
tac  
related segmentotera:
octasc  
general polytopal classes:
segmentotera   fundamental lace prisms  

Incidence matrix according to Dynkin symbol

ox3oo3oo4oo&#x   → height = 1/sqrt(2) = 0.707107
(pt || hex)

o.3o.3o.4o.    | 1 *  8  0 | 24  0 | 32  0 | 16 0
.o3.o3.o4.o    | * 8  1  6 |  6 12 | 12  8 |  8 1
---------------+-----+------+-------+-------+-----
oo3oo3oo4oo&#x | 1 1 | 8  *   6  0 | 12  0 |  8 0
.x .. .. ..    | 0 2 | * 24   1  4 |  4  4 |  4 1
---------------+-----+------+-------+-------+-----
ox .. .. ..&#x | 1 2 | 2  1 | 24  * |  4  0 |  4 0
.x3.o .. ..    | 0 3 | 0  3 |  * 32 |  1  2 |  2 1
---------------+-----+------+-------+-------+-----
ox3oo .. ..&#x  1 3 | 3  3 |  3  1 | 32  * |  2 0
.x3.o3.o ..     0 4 | 0  6 |  0  4 |  * 16 |  1 1
---------------+-----+------+-------+-------+-----
ox3oo3oo ..&#x  1 4 | 4  6 |  6  4 |  4  1 | 16 *
.x3.o3.o4.o     0 8 | 0 24 |  0 32 |  0 16 |  * 1

ox3oo3oo *b3oo&#x   → height = 1/sqrt(2) = 0.707107
(pt || hex)

o.3o.3o. *b3o.    | 1 *  8  0 | 24  0 | 32 0 0 | 8 8 0
.o3.o3.o *b3.o    | * 8  1  6 |  6 12 | 12 4 4 | 4 4 1
------------------+-----+------+-------+--------+------
oo3oo3oo *b3oo&#x | 1 1 | 8  *   6  0 | 12 0 0 | 4 4 0
.x .. ..    ..    | 0 2 | * 24   1  4 |  4 2 2 | 2 2 1
------------------+-----+------+-------+--------+------
ox .. ..    ..&#x | 1 2 | 2  1 | 24  * |  4 0 0 | 2 2 0
.x3.o ..    ..    | 0 3 | 0  3 |  * 32 |  1 1 1 | 1 1 1
------------------+-----+------+-------+--------+------
ox3oo ..    ..&#x  1 3 | 3  3 |  3  1 | 32 * * | 1 1 0
.x3.o3.o    ..     0 4 | 0  6 |  0  4 |  * 8 * | 1 0 1
.x3.o .. *b3.o     0 4 | 0  6 |  0  4 |  * * 8 | 0 1 1
------------------+-----+------+-------+--------+------
ox3oo3oo    ..&#x  1 4 | 4  6 |  6  4 |  4 1 0 | 8 * *
ox3oo .. *b3oo&#x  1 4 | 4  6 |  6  4 |  4 0 1 | * 8 *
.x3.o3.o *b3.o     0 8 | 0 24 |  0 32 |  0 8 8 | * * 1

xoo3ooo3oxo&#x   → all pairwise heights = 1/sqrt(2) = 0.707107
( (tet || dual tet) || pt)

o..3o..3o..    | 4 * *  3  3 1 0 0 | 3  6  3 3  3 0 0 | 1 3 3 1 3  6  3 0 0 | 1 1 3 3 1 0
.o.3.o.3.o.    | * 4 *  0  3 0 3 1 | 0  3  6 0  3 3 3 | 0 1 3 3 0  3  6 1 3 | 1 0 1 3 3 1
..o3..o3..o    | * * 1  0  0 4 0 4 | 0  0  0 6 12 0 6 | 0 0 0 0 4 12 12 0 4 | 0 1 4 6 4 1
---------------+-------+------------+------------------+---------------------+------------
x.. ... ...    | 2 0 0 | 6  * * * *  2  2  0 1  0 0 0 | 1 2 1 0 2  2  0 0 0 | 1 1 2 1 0 0
oo.3oo.3oo.&#x | 1 1 0 | * 12 * * *  0  2  2 0  1 0 0 | 0 1 2 1 0  2  2 0 0 | 1 0 1 2 1 0
o.o3o.o3o.o&#x | 1 0 1 | *  * 4 * *  0  0  0 3  3 0 0 | 0 0 0 0 3  6  3 0 0 | 0 1 3 3 1 0
... ... .x.    | 0 2 0 | *  * * 6 *  0  0  2 0  0 2 1 | 0 0 1 2 0  0  2 1 2 | 1 0 0 1 2 1
.oo3.oo3.oo&#x | 0 1 1 | *  * * * 4  0  0  0 0  3 0 3 | 0 0 0 0 0  3  6 0 3 | 0 0 1 3 3 1
---------------+-------+------------+------------------+---------------------+------------
x..3o.. ...    | 3 0 0 | 3  0 0 0 0 | 4  *  * *  * * * | 1 1 0 0 1  0  0 0 0 | 1 1 1 0 0 0
xo. ... ...&#x | 2 1 0 | 1  2 0 0 0 | * 12  * *  * * * | 0 1 1 0 0  1  0 0 0 | 1 0 1 1 0 0
... ... ox.&#x | 1 2 0 | 0  2 0 1 0 | *  * 12 *  * * * | 0 0 1 1 0  0  1 0 0 | 1 0 0 1 1 0
x.o ... ...&#x | 2 0 1 | 1  0 2 0 0 | *  *  * 6  * * * | 0 0 0 0 2  2  0 0 0 | 0 1 2 1 0 0
ooo3ooo3ooo&#x | 1 1 1 | 0  1 1 0 1 | *  *  * * 12 * * | 0 0 0 0 0  2  2 0 0 | 0 0 1 2 1 0
... .o.3.x.    | 0 3 0 | 0  0 0 3 0 | *  *  * *  * 4 * | 0 0 0 1 0  0  0 1 1 | 1 0 0 0 1 1
... ... .xo&#x | 0 2 1 | 0  0 0 1 2 | *  *  * *  * * 6 | 0 0 0 0 0  0  2 0 2 | 0 0 0 1 2 1
---------------+-------+------------+------------------+---------------------+------------
x..3o..3o..     4 0 0 | 6  0 0 0 0 | 4  0  0 0  0 0 0 | 1 * * * *  *  * * * | 1 1 0 0 0 0
xo.3oo. ...&#x  3 1 0 | 3  3 0 0 0 | 1  3  0 0  0 0 0 | * 4 * * *  *  * * * | 1 0 1 0 0 0
xo. ... ox.&#x  2 2 0 | 1  4 0 1 0 | 0  2  2 0  0 0 0 | * * 6 * *  *  * * * | 1 0 0 1 0 0
... oo.3ox.&#x  1 3 0 | 0  3 0 3 0 | 0  0  3 0  0 1 0 | * * * 4 *  *  * * * | 1 0 0 0 1 0
x.o3o.o ...&#x  3 0 1 | 3  0 3 0 0 | 1  0  0 3  0 0 0 | * * * * 4  *  * * * | 0 1 1 0 0 0
xoo ... ...&#x  2 1 1 | 1  2 2 0 1 | 0  1  0 1  2 0 0 | * * * * * 12  * * * | 0 0 1 1 0 0
... ... oxo&#x  1 2 1 | 0  2 1 1 2 | 0  0  1 0  2 0 1 | * * * * *  * 12 * * | 0 0 0 1 1 0
.o.3.o.3.x.     0 4 0 | 0  0 0 6 0 | 0  0  0 0  0 4 0 | * * * * *  *  * 1 * | 0 0 0 0 0 2
... .oo3.xo&#x  0 3 1 | 0  0 0 3 3 | 0  0  0 0  0 1 3 | * * * * *  *  * * 4 | 0 0 0 0 1 1
---------------+-------+------------+------------------+---------------------+------------
xo.3oo.3ox.&#x  4 4 0 | 6 12 0 6 0 | 4 12 12 0  0 4 0 | 1 4 6 4 0  0  0 0 0 | 1 * * * * *
x.o3o.o3o.o&#x  4 0 1 | 6  0 4 0 0 | 4  0  0 6  0 0 0 | 1 0 0 0 4  0  0 0 0 | * 1 * * * *
xoo3ooo ...&#x  3 1 1 | 3  3 3 0 1 | 1  3  0 3  3 0 0 | 0 1 0 0 1  3  0 0 0 | * * 4 * * *
xoo ... oxo&#x  2 2 1 | 1  4 2 1 2 | 0  2  2 1  4 0 1 | 0 0 1 0 0  2  2 0 0 | * * * 6 * *
... ooo3oxo&#x  1 3 1 | 0  3 1 3 3 | 0  0  3 0  3 1 3 | 0 0 0 1 0  0  3 0 1 | * * * * 4 *
.oo3.oo3.xo&#x  0 4 1 | 0  0 0 6 4 | 0  0  0 0  0 4 6 | 0 0 0 0 0  0  0 1 4 | * * * * * 1

oxo oox oxo oox&#x   → height(1,2) = height(1,3) = 1/sqrt(2) = 0.707107
                       height(2,3) = 0
(pt || tegum product of 2 {4})

o.. o.. o.. o..     | 1 * *  4 4 0 0  0 0 0 | 2 2 2 2 16 0 0 0 0 | 8 8 8 8 0 0 0 0 | 4 4 4 4 0
.o. .o. .o. .o.     | * 4 *  1 0 1 1  4 0 0 | 1 1 0 0  4 4 2 4 2 | 4 2 4 2 2 2 2 2 | 2 2 2 2 1
..o ..o ..o ..o     | * * 4  0 1 0 0  4 1 1 | 0 0 1 1  4 2 4 2 4 | 2 4 2 4 2 2 2 2 | 2 2 2 2 1
--------------------+-------+----------------+--------------------+-----------------+----------
oo. oo. oo. oo.&#x  | 1 1 0 | 4 * * *  * * *  1 1 0 0  4 0 0 0 0 | 4 2 4 2 0 0 0 0 | 2 2 2 2 0
o.o o.o o.o o.o&#x  | 1 0 1 | * 4 * *  * * *  0 0 1 1  4 0 0 0 0 | 2 4 2 4 0 0 0 0 | 2 2 2 2 0
.x. ... ... ...     | 0 2 0 | * * 2 *  * * *  1 0 0 0  0 4 0 0 0 | 4 0 0 0 2 2 0 0 | 2 2 0 0 1
... ... .x. ...     | 0 2 0 | * * * 2  * * *  0 1 0 0  0 0 0 4 0 | 0 0 4 0 0 0 2 2 | 0 0 2 2 1
.oo .oo .oo .oo&#x  | 0 1 1 | * * * * 16 * *  0 0 0 0  1 1 1 1 1 | 1 1 1 1 1 1 1 1 | 1 1 1 1 1
... ..x ... ...     | 0 0 2 | * * * *  * 2 *  0 0 1 0  0 0 4 0 0 | 0 4 0 0 2 0 2 0 | 2 0 2 0 1
... ... ... ..x     | 0 0 2 | * * * *  * * 2  0 0 0 1  0 0 0 0 4 | 0 0 0 4 0 2 0 2 | 0 2 0 2 1
--------------------+-------+----------------+--------------------+-----------------+----------
ox. ... ... ...&#x  | 1 2 0 | 2 0 1 0  0 0 0 | 2 * * *  * * * * * | 4 0 0 0 0 0 0 0 | 2 2 0 0 0
... ... ox. ...&#x  | 1 2 0 | 2 0 0 1  0 0 0 | * 2 * *  * * * * * | 0 0 4 0 0 0 0 0 | 0 0 2 2 0
... o.x ... ...&#x  | 1 0 2 | 0 2 0 0  0 1 0 | * * 2 *  * * * * * | 0 4 0 0 0 0 0 0 | 2 0 2 0 0
... ... ... o.x&#x  | 1 0 2 | 0 2 0 0  0 0 1 | * * * 2  * * * * * | 0 0 0 4 0 0 0 0 | 0 2 0 2 0
ooo ooo ooo ooo&#x  | 1 1 1 | 1 1 0 0  1 0 0 | * * * * 16 * * * * | 1 1 1 1 0 0 0 0 | 1 1 1 1 0
.xo ... ... ...&#x  | 0 2 1 | 0 0 1 0  2 0 0 | * * * *  * 8 * * * | 1 0 0 0 1 1 0 0 | 1 1 0 0 1
... .ox ... ...&#x  | 0 1 2 | 0 0 0 0  2 1 0 | * * * *  * * 8 * * | 0 1 0 0 1 0 1 0 | 1 0 1 0 1
... ... .xo ...&#x  | 0 2 1 | 0 0 0 1  2 0 0 | * * * *  * * * 8 * | 0 0 1 0 0 0 1 1 | 0 0 1 1 1
... ... ... .ox&#x  | 0 1 2 | 0 0 0 0  2 0 1 | * * * *  * * * * 8 | 0 0 0 1 0 1 0 1 | 0 1 0 1 1
--------------------+-------+----------------+--------------------+-----------------+----------
oxo ... ... ...&#x   1 2 1 | 2 1 1 0  2 0 0 | 1 0 0 0  2 1 0 0 0 | 8 * * * * * * * | 1 1 0 0 0
... oox ... ...&#x   1 1 2 | 1 2 0 0  2 1 0 | 0 0 1 0  2 0 1 0 0 | * 8 * * * * * * | 1 0 1 0 0
... ... oxo ...&#x   1 2 1 | 2 1 0 1  2 0 0 | 0 1 0 0  2 0 0 1 0 | * * 8 * * * * * | 0 0 1 1 0
... ... ... oox&#x   1 1 2 | 1 2 0 0  2 0 1 | 0 0 0 1  2 0 0 0 1 | * * * 8 * * * * | 0 1 0 1 0
.xo .ox ... ...&#x   0 2 2 | 0 0 1 0  4 1 0 | 0 0 0 0  0 2 2 0 0 | * * * * 4 * * * | 1 0 0 0 1
.xo ... ... .ox&#x   0 2 2 | 0 0 1 0  4 0 1 | 0 0 0 0  0 2 0 0 2 | * * * * * 4 * * | 0 1 0 0 1
... .ox .xo ...&#x   0 2 2 | 0 0 0 1  4 1 0 | 0 0 0 0  0 0 2 2 0 | * * * * * * 4 * | 0 0 1 0 1
... ... .xo .ox&#x   0 2 2 | 0 0 0 1  4 0 1 | 0 0 0 0  0 0 0 2 2 | * * * * * * * 4 | 0 0 0 1 1
--------------------+-------+----------------+--------------------+-----------------+----------
oxo oox ... ...&#x   1 2 2 | 2 2 1 0  4 1 0 | 1 0 1 0  4 2 2 0 0 | 2 2 0 0 1 0 0 0 | 4 * * * *
oxo ... ... oox&#x   1 2 2 | 2 2 1 0  4 0 1 | 1 0 0 1  4 2 0 0 2 | 2 0 0 2 0 1 0 0 | * 4 * * *
... oox oxo ...&#x   1 2 2 | 2 2 0 1  4 1 0 | 0 1 1 0  4 0 2 2 0 | 0 2 2 0 0 0 1 0 | * * 4 * *
... ... oxo oox&#x   1 2 2 | 2 2 0 1  4 0 1 | 0 1 0 1  4 0 0 2 2 | 0 0 2 2 0 0 0 1 | * * * 4 *
.xo .ox .xo .ox&#zx  0 4 4 | 0 0 2 2 16 2 2 | 0 0 0 0  0 8 8 8 8 | 0 0 0 0 4 4 4 4 | * * * * 1

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