Acronym ...
Name vertex figure of small prismated icositetrachoron
 
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Circumradius sqrt[(2-sqrt(2))/2] = 0.541196
Confer
uniform variant:
squap  

Incidence matrix according to Dynkin symbol

xo4ox&#q   → height = sqrt[1+1/sqrt(2)] = 1.306563
({4} || dual {4})

o.4o.    | 4 * | 2 2 0 | 1 2 1 0
.o4.o    | * 4 | 0 2 2 | 0 1 2 1
---------+-----+-------+--------
x. ..    | 2 0 | 4 * * | 1 1 0 0  x
oo4oo&#q | 1 1 | * 8 * | 0 1 1 0  q
.. .x    | 0 2 | * * 4 | 0 0 1 1  x
---------+-----+-------+--------
x.4o.    | 4 0 | 4 0 0 | 1 * * *
xo ..&#q | 2 1 | 1 2 0 | * 4 * *
.. ox&#q | 1 2 | 0 2 1 | * * 4 *
.o4.x    | 0 4 | 0 0 4 | * * * 1
or
o.4o.    & | 8 | 2 2 | 1 3
-----------+---+-----+----
x. ..    & | 2 | 8 * | 1 1  x
oo4oo&#q   | 2 | * 8 | 0 2  q
-----------+---+-----+----
x.4o.    & | 4 | 4 0 | 2 *
xo ..&#q & | 3 | 1 2 | * 8

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