Acronym squasc (old: squippypy, alt: edihex), 4pyr-pyr, K-4.4
Name square-scalene,
square-pyramidal pyramid,
line atop fully orthogonal square,
1/4-luna of hex,
4 tet rosette,
edge-diminished hex
|,>,O device line prism pyramid pyramid = ||>>
Circumradius 1/sqrt(2) = 0.707107
Lace city
in approx. ASCII-art
    o4o
       
o4o x4o
      x o
         
         
x o   o x
Coordinates
    • (0, 0, 0, 1/sqrt(2))
      first tip
    • (0, 0, 1/sqrt(2), 0)
      second tip
    • (1/sqrt(2), 0, 0, 0)   & permutations and changes of sign in first 2 coordinates
      square base
    • (1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 1/sqrt(8))
      tip
    • (1/sqrt(8), 1/sqrt(8), -1/sqrt(8), -1/sqrt(8))   & all permutations except that with both minus in first 2 coordinates
      squippy base
General of army (is itself convex)
Colonel of regiment (is itself locally convex)
Dual (topologically selfdual, in different orientation)
Dihedral angles
Confer
uniform relative:
hex  
related segmentochora:
line || perp {n} (n=4)   n-af (n=2)   octpy  
general polytopal classes:
luna   segmentochora  

Incidence matrix according to Dynkin symbol

xo ox4oo&#x   → height = 1/2
(line || perp {4})

o. o.4o.    | 2 *  1 4 0 | 4 4 0 | 4 1
.o .o4.o    | * 4 | 0 2 2 | 1 4 1 | 2 2
------------+-----+-------+-------+----
x. .. ..    | 2 0 | 1 * * | 4 0 0 | 4 0
oo oo4oo&#x | 1 1 | * 8 * | 1 2 0 | 2 1
.. .x ..    | 0 2 | * * 4 | 0 2 1 | 1 2
------------+-----+-------+-------+----
xo .. ..&#x | 2 1 | 1 2 0 | 4 * * | 2 0
.. ox ..&#x | 1 2 | 0 2 1 | * 8 * | 1 1
.. .x4.o    | 0 4 | 0 0 4 | * * 1 | 0 2
------------+-----+-------+-------+----
xo ox ..&#x  2 2 | 1 4 1 | 2 2 0 | 4 *
.. ox4oo&#x  1 4 | 0 4 4 | 0 4 1 | * 2

xo ox ox&#x   → height = 1/2
(line || perp {4})

o. o.4o.    | 2 *  1 4 0 0 | 4 2 2 0 | 2 2 1
.o .o4.o    | * 4 | 0 2 1 1 | 1 2 2 1 | 1 1 2
------------+-----+---------+---------+------
x. .. ..    | 2 0 | 1 * * * | 4 0 0 0 | 2 2 0
oo oo oo&#x | 1 1 | * 8 * * | 1 1 1 0 | 1 1 1
.. .x ..    | 0 2 | * * 2 * | 0 2 0 1 | 1 0 2
.. .. .x    | 0 2 | * * * 2 | 0 0 2 1 | 0 1 2
------------+-----+---------+---------+------
xo .. ..&#x | 2 1 | 1 2 0 0 | 4 * * * | 1 1 0
.. ox ..&#x | 1 2 | 0 2 1 0 | * 4 * * | 1 0 1
.. .. ox&#x | 1 2 | 0 2 0 1 | * * 4 * | 0 1 1
.. .x .x    | 0 4 | 0 0 2 2 | * * * 1 | 0 0 2
------------+-----+---------+---------+------
xo ox ..&#x  2 2 | 1 4 1 0 | 2 2 0 0 | 2 * *
xo .. ox&#x  2 2 | 1 4 0 1 | 2 0 2 0 | * 2 *
.. ox ox&#x  1 4 | 0 4 2 2 | 0 2 2 1 | * * 2

oox4ooo&#x   → height(1,2) = 1
               height(1,3) = height(2,3) = 1/sqrt(2) = 0.707107

o.. o..    | 1 * *  1 4 0 0 | 4 4 0 0 | 4 1 0
.o. .o.    | * 1 *  1 0 4 0 | 4 0 4 0 | 4 0 1
..o ..o    | * * 4 | 0 1 1 2 | 1 2 2 1 | 2 1 1
-----------+-------+---------+---------+------
oo.4oo.&#x | 1 1 0 | 1 * * * | 4 0 0 0 | 4 0 0
o.o4o.o&#x | 1 0 1 | * 4 * * | 1 2 0 0 | 2 1 0
.oo4.oo&#x | 0 1 1 | * * 4 * | 1 0 2 0 | 2 0 1
..x ...    | 0 0 2 | * * * 4 | 0 1 1 1 | 1 1 1
-----------+-------+---------+---------+------
ooo4ooo&#x | 1 1 1 | 1 1 1 0 | 4 * * * | 2 0 0
o.x ...&#x | 1 0 2 | 0 2 0 1 | * 4 * * | 1 1 0
.ox ...&#x | 0 1 2 | 0 0 2 1 | * * 4 * | 1 0 1
..x4..o    | 0 0 4 | 0 0 0 4 | * * * 1 | 0 1 1
-----------+-------+---------+---------+------
oox ...&#x  1 1 2 | 1 2 2 1 | 2 1 1 0 | 4 * *
o.x4o.o&#x  1 0 4 | 0 4 0 4 | 0 4 0 1 | * 1 *
.ox4.oo&#x  0 1 4 | 0 0 4 4 | 0 0 4 1 | * * 1
or
o.. o..    & | 2 *  1 4 0 | 4 4 0 | 4 1
..o ..o      | * 4 | 0 2 2 | 1 4 1 | 2 2
-------------+-----+-------+-------+----
oo.4oo.&#x   | 2 0 | 1 * * | 4 0 0 | 4 0
o.o4o.o&#x & | 1 1 | * 8 * | 1 2 0 | 2 1
..x ...      | 0 2 | * * 4 | 0 2 1 | 1 2
-------------+-----+-------+-------+----
ooo4ooo&#x   | 2 1 | 1 2 0 | 4 * * | 2 0
o.x ...&#x & | 1 2 | 0 2 1 | * 8 * | 1 1
..x4..o      | 0 4 | 0 0 4 | * * 1 | 0 2
-------------+-----+-------+-------+----
oox ...&#x    2 2 | 1 4 1 | 2 2 0 | 4 *
o.x4o.o&#x &  1 4 | 0 4 4 | 0 4 1 | * 2

oox oox&#x   → height(1,2) = 1
               height(1,3) = height(2,3) = 1/sqrt(2) = 0.707107

o.. o..    | 1 * *  1 4 0 0 0 | 4 2 2 0 0 0 | 2 2 1 0
.o. .o.    | * 1 *  1 0 4 0 0 | 4 0 0 2 2 0 | 2 2 0 1
..o ..o    | * * 4 | 0 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1
-----------+-------+-----------+-------------+--------
oo. oo.&#x | 1 1 0 | 1 * * * * | 4 0 0 0 0 0 | 2 2 0 0
o.o o.o&#x | 1 0 1 | * 4 * * * | 1 1 1 0 0 0 | 1 1 1 0
.oo .oo&#x | 0 1 1 | * * 4 * * | 1 0 0 1 1 0 | 1 1 0 1
..x ...    | 0 0 2 | * * * 2 * | 0 1 0 1 0 1 | 1 0 1 1
... ..x    | 0 0 2 | * * * * 2 | 0 0 1 0 1 1 | 0 1 1 1
-----------+-------+-----------+-------------+--------
ooo ooo&#x | 1 1 1 | 1 1 1 0 0 | 4 * * * * * | 1 1 0 0
o.x ...&#x | 1 0 2 | 0 2 0 1 0 | * 2 * * * * | 1 0 1 0
... o.x&#x | 1 0 2 | 0 2 0 0 1 | * * 2 * * * | 0 1 1 0
.ox ...&#x | 0 1 2 | 0 0 2 1 0 | * * * 2 * * | 1 0 0 1
... .ox&#x | 0 1 2 | 0 0 2 0 1 | * * * * 2 * | 0 1 0 1
..x ..x    | 0 0 4 | 0 0 0 2 2 | * * * * * 1 | 0 0 1 1
-----------+-------+-----------+-------------+--------
oox ...&#x  1 1 2 | 1 2 2 1 0 | 2 1 0 1 0 0 | 2 * * *
... oox&#x  1 1 2 | 1 2 2 0 1 | 2 0 1 0 1 0 | * 2 * *
o.x o.x&#x  1 0 4 | 0 4 0 4 0 | 0 2 2 0 0 1 | * * 1 *
.ox .ox&#x  0 1 4 | 0 0 4 4 0 | 0 0 0 2 2 1 | * * * 1
or
o.. o..    & | 2 *  1 4 0 | 4 4 0 | 4 1
..o ..o      | * 4 | 0 2 2 | 1 4 1 | 2 2
-------------+-----+-------+-------+----
oo. oo.&#x   | 2 0 | 1 * * | 4 0 0 | 4 0
o.o o.o&#x & | 1 1 | * 8 * | 1 2 0 | 2 1
..x ...    & | 0 2 | * * 4 | 0 2 1 | 1 2
-------------+-----+-------+-------+----
ooo ooo&#x   | 2 1 | 1 2 0 | 4 * * | 2 0
o.x ...&#x & | 1 2 | 0 2 1 | * 8 * | 1 1
..x ..x      | 0 4 | 0 0 4 | * * 1 | 0 2
-------------+-----+-------+-------+----
oox ...&#x &  2 2 | 1 4 1 | 2 2 0 | 4 *
o.x o.x&#x &  1 4 | 0 4 4 | 0 4 1 | * 2

pt || squippy   → height = 1/sqrt(2) = 0.707107

o.. o..    | 1 * *  1 4 0 0 | 4 4 0 0 | 4 1 0
.o. .o.    | * 1 *  1 0 4 0 | 4 0 4 0 | 4 0 1
..o ..o    | * * 4 | 0 1 1 2 | 1 2 2 1 | 2 1 1
-----------+-------+---------+---------+------
oo.4oo.&#x | 1 1 0 | 1 * * * | 4 0 0 0 | 4 0 0
o.o4o.o&#x | 1 0 1 | * 4 * * | 1 2 0 0 | 2 1 0
.oo4.oo&#x | 0 1 1 | * * 4 * | 1 0 2 0 | 2 0 1
..x ...    | 0 0 2 | * * * 4 | 0 1 1 1 | 1 1 1
-----------+-------+---------+---------+------
ooo4ooo&#x | 1 1 1 | 1 1 1 0 | 4 * * * | 2 0 0
o.x ...&#x | 1 0 2 | 0 2 0 1 | * 4 * * | 1 1 0
.ox ...&#x | 0 1 2 | 0 0 2 1 | * * 4 * | 1 0 1
..x4..o    | 0 0 4 | 0 0 0 4 | * * * 1 | 0 1 1
-----------+-------+---------+---------+------
oox ...&#x  1 1 2 | 1 2 2 1 | 2 1 1 0 | 4 * *
o.x4o.o&#x  1 0 4 | 0 4 0 4 | 0 4 0 1 | * 1 *
.ox4.oo&#x  0 1 4 | 0 0 4 4 | 0 0 4 1 | * * 1

xoo oxx&#x   → height(1,2) = height(1,3) = 1/sqrt(2) = 0.707107
               height(2,3) = 1

o.. o..    | 2 * *  1 2 2 0 0 0 | 2 1 2 1 2 0 | 1 1 2 1
.o. .o.    | * 2 * | 0 2 0 1 1 0 | 1 2 0 0 2 1 | 1 0 1 2
..o ..o    | * * 2 | 0 0 2 0 1 1 | 0 0 1 2 2 1 | 0 1 1 2
-----------+-------+-------------+-------------+--------
x.. ...    | 2 0 0 | 1 * * * * * | 2 0 2 0 0 0 | 1 1 2 0
oo. oo.&#x | 1 1 0 | * 4 * * * * | 1 1 0 0 1 0 | 1 0 1 1
o.o o.o&#x | 1 0 1 | * * 4 * * * | 0 0 1 1 1 0 | 0 1 1 1
... .x.    | 0 2 0 | * * * 1 * * | 0 2 0 0 0 1 | 1 0 0 2
.oo .oo&#x | 0 1 1 | * * * * 2 * | 0 0 0 0 2 1 | 0 0 1 2
... ..x    | 0 0 2 | * * * * * 1 | 0 0 0 2 0 1 | 0 1 0 2
-----------+-------+-------------+-------------+--------
xo. ...&#x | 2 1 0 | 1 2 0 0 0 0 | 2 * * * * * | 1 0 1 0
... ox.&#x | 1 2 0 | 0 2 0 1 0 0 | * 2 * * * * | 1 0 0 1
x.o ...&#x | 2 0 1 | 1 0 2 0 0 0 | * * 2 * * * | 0 1 1 0
... o.x&#x | 1 0 2 | 0 0 2 0 0 1 | * * * 2 * * | 0 1 0 1
ooo ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * * * 4 * | 0 0 1 1
... .xx&#x | 0 2 2 | 0 0 0 1 2 1 | * * * * * 1 | 0 0 0 2
-----------+-------+-------------+-------------+--------
xo. ox.&#x  2 2 0 | 1 4 0 1 0 0 | 2 2 0 0 0 0 | 1 * * *
x.o o.x&#x  2 0 2 | 1 0 4 0 0 1 | 0 0 2 2 0 0 | * 1 * *
xoo ...&#x  2 1 1 | 1 2 2 0 1 0 | 1 0 1 0 2 0 | * * 2 *
... oxx&#x  1 2 2 | 0 2 2 1 2 1 | 0 1 0 1 2 1 | * * * 2
or
o.. o..      | 2 *  1 4 0 0 | 4 2 2 0 | 2 2 1
.o. .o.    & | * 4 | 0 2 1 1 | 1 2 2 1 | 1 1 2
-------------+-----+---------+---------+------
x.. ...      | 2 0 | 1 * * * | 4 0 0 0 | 2 2 0
oo. oo.&#x & | 1 1 | * 8 * * | 1 1 1 0 | 1 1 1
... .x.    & | 0 2 | * * 2 * | 0 2 0 1 | 1 0 2
.oo .oo&#x   | 0 2 | * * * 2 | 0 0 2 1 | 0 1 2
-------------+-----+---------+---------+------
xo. ...&#x & | 2 1 | 1 2 0 0 | 4 * * * | 1 1 0
... ox.&#x & | 1 2 | 0 2 1 0 | * 4 * * | 1 0 1
ooo ooo&#x   | 1 2 | 0 2 0 1 | * * 4 * | 0 1 1
... .xx&#x   | 0 4 | 0 0 2 2 | * * * 1 | 0 0 2
-------------+-----+---------+---------+------
xo. ox.&#x &  2 2 | 1 4 1 0 | 2 2 0 0 | 2 * *
xoo ...&#x    2 2 | 1 4 0 1 | 2 0 2 0 | * 2 *
... oxx&#x    1 4 | 0 4 2 2 | 0 2 2 1 | * * 2

xoox&#x   → height(1,2) = height(1,3) = height(2,4) = height(3,4) = sqrt(3)/2 = 0.866025
            height(1,4) = height(2,3) = 1

o...    | 2 * * * | 1 1 1 1 0 0 0 0 | 1 1 1 1 1 1 0 0 0 | 1 1 1 1 0
.o..    | * 1 * *  0 2 0 0 1 2 0 0 | 1 0 0 2 2 0 2 1 0 | 1 1 0 2 1
..o.    | * * 1 *  0 0 2 0 1 0 2 0 | 0 1 0 2 0 2 2 0 1 | 1 0 1 2 1
...o    | * * * 2 | 0 0 0 1 0 1 1 1 | 0 0 1 0 1 1 1 1 1 | 0 1 1 1 1
--------+---------+-----------------+-------------------+----------
x...    | 2 0 0 0 | 1 * * * * * * * | 1 1 1 0 0 0 0 0 0 | 1 1 1 0 0
oo..&#x | 1 1 0 0 | * 2 * * * * * * | 1 0 0 1 1 0 0 0 0 | 1 1 0 1 0
o.o.&#x | 1 0 1 0 | * * 2 * * * * * | 0 1 0 1 0 1 0 0 0 | 1 0 1 1 0
o..o&#x | 1 0 0 1 | * * * 2 * * * * | 0 0 1 0 1 1 0 0 0 | 0 1 1 1 0
.oo.&#x | 0 1 1 0 | * * * * 1 * * * | 0 0 0 2 0 0 2 0 0 | 1 0 0 2 1
.o.o&#x | 0 1 0 1 | * * * * * 2 * * | 0 0 0 0 1 0 1 1 0 | 0 1 0 1 1
..oo&#x | 0 0 1 1 | * * * * * * 2 * | 0 0 0 0 0 1 1 0 1 | 0 0 1 1 1
...x    | 0 0 0 2 | * * * * * * * 1 | 0 0 1 0 0 0 0 1 1 | 0 1 1 0 1
--------+---------+-----------------+-------------------+----------
xo..&#x | 2 1 0 0 | 1 2 0 0 0 0 0 0 | 1 * * * * * * * * | 1 1 0 0 0
x.o.&#x | 2 0 1 0 | 1 0 2 0 0 0 0 0 | * 1 * * * * * * * | 1 0 1 0 0
x..x&#x | 2 0 0 2 | 1 0 0 2 0 0 0 1 | * * 1 * * * * * * | 0 1 1 0 0
ooo.&#x | 1 1 1 0 | 0 1 1 0 1 0 0 0 | * * * 2 * * * * * | 1 0 0 1 0
oo.o&#x | 1 1 0 1 | 0 1 0 1 0 1 0 0 | * * * * 2 * * * * | 0 1 0 1 0
o.oo&#x | 1 0 1 1 | 0 0 1 1 0 0 1 0 | * * * * * 2 * * * | 0 0 1 1 0
.ooo&#x | 0 1 1 1 | 0 0 0 0 1 1 1 0 | * * * * * * 2 * * | 0 0 0 1 1
.o.x&#x | 0 1 0 2 | 0 0 0 0 0 2 0 1 | * * * * * * * 1 * | 0 1 0 0 1
..ox&#x | 0 0 1 2 | 0 0 0 0 0 0 2 1 | * * * * * * * * 1 | 0 0 1 0 1
--------+---------+-----------------+-------------------+----------
xoo.&#x  2 1 1 0 | 1 2 2 0 1 0 0 0 | 1 1 0 2 0 0 0 0 0 | 1 * * * *
xo.x&#x  2 1 0 2 | 1 2 0 2 0 2 0 1 | 1 0 1 0 2 0 0 1 0 | * 1 * * *
x.ox&#x  2 0 1 2 | 1 0 2 2 0 0 2 1 | 0 1 1 0 0 2 0 0 1 | * * 1 * *
oooo&#x  1 1 1 1 | 0 1 1 1 1 1 1 0 | 0 0 0 1 1 1 1 0 0 | * * * 2 *
.oox&#x  0 1 1 2 | 0 0 0 0 1 2 2 1 | 0 0 0 0 0 0 2 1 1 | * * * * 1

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