Acronym hop (old: hep)
Name heptapeton,
6D simplex (α6),
pyroex(id),
hexateral pyramid,
vertex figure of oca,
Gosset polytope 05
|,>,O device line pyramid pyramid pyramid pyramid pyramid = |>>>>>
Circumradius sqrt(3/7) = 0.654654
Inradius 1/sqrt(84) = 0.109109
Dual (selfdual, in different orientation)
Lace city
in approx. ASCII-art 
o3o3o3o   o3o3o3o
                 
                 
                 
     x3o3o3o
Lace hyper city
in approx. ASCII-art 
 ©  

             
             
    x3o3o
tet 
  o3o3o      
             
        o3o3o
             
  o3o3o
perp {3}
Volume sqrt(7)/5760 = 0.00045933
Surface 7 sqrt(3)/480 = 0.025259
Rel. Roundness 5 π3 sqrt(7)/21609 = 1.898165 %
Dihedral angles
  • at pen between hix and hix:   arccos(1/6) = 80.405932°
Face vector 7, 21, 35, 35, 21, 7
Confer
compounds:
stef  
ambification:
ril  
general polytopal classes:
Wythoffian polypeta   simplex   noble polytopes   scalene   tettene   pennene   segmentopeta   fundamental lace prisms   lace simplices   Coxeter-Elte-Gosset polytopes  
analogs:
regular simplex Sn  
External
links 		hedrondude   wikipedia   polytopewiki   mathworld

This polypeton also can be obtained as 7-2-3 step prism. Whereat an n-p-q step prism is obtained from the n-trioprism using simultanuously its k-th vertex on the first n-gon, the pk-th vertex on the next orthogonal n-gon and the qk-th vertex on the final orthogonal n-gon: i.e. using the convex hull of those n points when k=1,...,n.


Incidence matrix according to Dynkin symbol

x3o3o3o3o3o

. . . . . . | 7 ♦  6 | 15 | 20 | 15 | 6
------------+---+----+----+----+----+--
x . . . . . | 2 | 21 ♦  5 | 10 | 10 | 5
------------+---+----+----+----+----+--
x3o . . . . | 3 |  3 | 35 ♦  4 |  6 | 4
------------+---+----+----+----+----+--
x3o3o . . . ♦ 4 |  6 |  4 | 35 |  3 | 3
------------+---+----+----+----+----+--
x3o3o3o . . ♦ 5 | 10 | 10 |  5 | 21 | 2
------------+---+----+----+----+----+--
x3o3o3o3o . ♦ 6 | 15 | 20 | 15 |  6 | 7


ox3oo3oo3oo3oo&#x   → height = sqrt(7/12) = 0.763763
(pt || hix)

o.3o.3o.3o.3o.    | 1 * ♦ 6  0 | 15  0 | 20  0 | 15 0 | 6 0
.o3.o3.o3.o3.o    | * 6 ♦ 1  5 |  5 10 | 10 10 | 10 5 | 5 1
------------------+-----+------+-------+-------+------+----
oo3oo3oo3oo3oo&#x | 1 1 | 6  * ♦  5  0 | 10  0 | 10 0 | 5 0
.x .. .. .. ..    | 0 2 | * 15 ♦  1  4 |  4  6 |  6 4 | 4 1
------------------+-----+------+-------+-------+------+----
ox .. .. .. ..&#x | 1 2 | 2  1 | 15  * ♦  4  0 |  6 0 | 4 0
.x3.o .. .. ..    | 0 3 | 0  3 |  * 20 ♦  1  3 |  3 3 | 3 1
------------------+-----+------+-------+-------+------+----
ox3oo .. .. ..&#x ♦ 1 3 | 3  3 |  3  1 | 20  * |  3 0 | 3 0
.x3.o3.o .. ..    ♦ 0 4 | 0  6 |  0  4 |  * 15 |  1 2 | 2 1
------------------+-----+------+-------+-------+------+----
ox3oo3oo .. ..&#x ♦ 1 4 | 4  6 |  6  4 |  4  1 | 15 * | 2 0
.x3.o3.o3.o ..    ♦ 0 5 | 0 10 |  0 10 |  0  5 |  * 6 | 1 1
------------------+-----+------+-------+-------+------+----
ox3oo3oo3oo ..&#x ♦ 1 5 | 5 10 | 10 10 | 10  5 |  5 1 | 6 *
.x3.o3.o3.o3.o    ♦ 0 6 | 0 15 |  0 20 |  0 15 |  0 6 | * 1


xo ox3oo3oo3oo&#x   → height = sqrt(7/20) = 0.591608
(line || perp pen)

o. o.3o.3o.3o.    | 2 * ♦ 1  5  0 | 5 10  0 | 10 10 0 | 10  5 0 | 5 1
.o .o3.o3.o3.o    | * 5 ♦ 0  2  4 | 1  8  6 |  4 12 4 |  6  8 1 | 4 2
------------------+-----+---------+---------+---------+---------+----
x. .. .. .. ..    | 2 0 | 1  *  * ♦ 5  0  0 | 10  0 0 | 10  0 0 | 5 0
oo oo3oo3oo3oo&#x | 1 1 | * 10  * ♦ 1  4  0 |  4  6 0 |  6  4 0 | 4 1
.. .x .. .. ..    | 0 2 | *  * 10 ♦ 0  2  3 |  1  6 3 |  3  6 1 | 3 2
------------------+-----+---------+---------+---------+---------+----
xo .. .. .. ..&#x | 2 1 | 1  2  0 ♦ 5  *  * |  4  0 0 |  6  0 0 | 4 0
.. ox .. .. ..&#x | 1 2 | 0  2  1 ♦ * 20  * |  1  3 0 |  3  3 0 | 3 1
.. .x3.o .. ..    | 0 3 | 0  0  3 ♦ *  * 10 |  0  2 2 |  1  4 1 | 2 2
------------------+-----+---------+---------+---------+---------+----
xo ox .. .. ..&#x ♦ 2 2 | 1  4  1 | 2  2  0 | 10  * * |  3  0 0 | 3 0
.. ox3oo .. ..&#x ♦ 1 3 | 0  3  3 | 0  3  1 |  * 20 * |  1  2 0 | 2 1
.. .x3.o3.o ..    ♦ 0 4 | 0  0  6 | 0  0  4 |  *  * 5 |  0  2 1 | 1 2
------------------+-----+---------+---------+---------+---------+----
xo ox3oo .. ..&#x ♦ 2 3 | 1  6  3 | 3  6  1 |  3  2 0 | 10  * * | 2 0
.. ox3oo3oo ..&#x ♦ 1 4 | 0  4  6 | 0  6  4 |  0  4 1 |  * 10 * | 1 1
.. .x3.o3.o3.o    ♦ 0 5 | 0  0 10 | 0  0 10 |  0  0 5 |  *  * 1 | 0 2
------------------+-----+---------+---------+---------+---------+----
xo ox3oo3oo ..&#x ♦ 2 4 | 1  8  6 | 4 12  4 |  6  8 1 |  4  2 0 | 5 *
.. ox3oo3oo3oo&#x ♦ 1 5 | 0  5 10 | 0 10 10 |  0 10 5 |  0  5 1 | * 2


xo3oo ox3oo3oo&#x   → height = sqrt(7/24) = 0.540062
({3} || perp tet)

o.3o. o.3o.3o.    | 3 * ♦ 2  4 0 | 1  8  6 0 | 4 12  4 0 | 6  8 1 | 4 2
.o3.o .o3.o3.o    | * 4 ♦ 0  3 3 | 0  3  9 3 | 1  9  9 1 | 3  9 3 | 3 3
------------------+-----+--------+-----------+-----------+--------+----
x. .. .. .. ..    | 2 0 | 3  * * ♦ 1  4  0 0 | 4  6  0 0 | 6  4 0 | 4 1
oo3oo oo3oo3oo&#x | 1 1 | * 12 * ♦ 0  2  3 0 | 1  6  3 0 | 3  6 1 | 3 2
.. .. .x .. ..    | 0 2 | *  * 6 ♦ 0  0  3 2 | 0  3  6 1 | 1  6 3 | 2 3
------------------+-----+--------+-----------+-----------+--------+----
x.3o. .. .. ..    | 3 0 | 3  0 0 | 1  *  * * ♦ 4  0  0 0 | 6  0 0 | 4 0
xo .. .. .. ..&#x | 2 1 | 1  2 0 | * 12  * * ♦ 1  3  0 0 | 3  3 0 | 3 1
.. .. ox .. ..&#x | 1 2 | 0  2 1 | *  * 18 * ♦ 0  2  2 0 | 1  4 1 | 2 2
.. .. .x3.o ..    | 0 3 | 0  0 3 | *  *  * 4 ♦ 0  0  3 1 | 0  3 3 | 1 3
------------------+-----+--------+-----------+-----------+--------+----
xo3oo .. .. ..&#x ♦ 3 1 | 3  3 0 | 1  3  0 0 | 4  *  * * | 3  0 0 | 3 0
xo .. ox .. ..&#x ♦ 2 2 | 1  4 1 | 0  2  2 0 | * 18  * * | 1  2 0 | 2 1
.. .. ox3oo ..&#x ♦ 1 3 | 0  3 3 | 0  0  3 1 | *  * 12 * | 0  2 1 | 1 2
.. .. .x3.o3.o    ♦ 0 4 | 0  0 6 | 0  0  0 4 | *  *  * 1 | 0  0 3 | 0 3
------------------+-----+--------+-----------+-----------+--------+----
xo3oo ox .. ..&#x ♦ 3 2 | 3  6 1 | 1  6  3 0 | 2  3  0 0 | 6  * * | 2 0
xo .. ox3oo ..&#x ♦ 2 3 | 1  6 3 | 0  3  6 1 | 0  3  2 0 | * 12 * | 1 1
.. .. ox3oo3oo&#x ♦ 1 4 | 0  4 6 | 0  0  6 4 | 0  0  4 1 | *  * 3 | 0 2
------------------+-----+--------+-----------+-----------+--------+----
xo3oo ox3oo ..&#x ♦ 3 3 | 3  9 3 | 1  9  9 1 | 3  9  3 0 | 3  3 0 | 4 *
xo .. ox3oo3oo&#x ♦ 2 4 | 1  8 6 | 0  4 12 4 | 0  6  8 1 | 0  4 2 | * 3


oxo3ooo3ooo3ooo&#x   → height(1,2) = height (2,3) = sqrt(3/5) = 0.774597
                       height(1,3) = 1
( (pt || pen) || pt)

o..3o..3o..3o..    | 1 * * ♦ 5 1  0 0 | 10 5  0  0 | 10 10 0  0 | 5 10 0 0 | 1 5 0
.o.3.o.3.o.3.o.    | * 5 * ♦ 1 0  4 1 |  4 1  6  4 |  6  4 4  6 | 4  6 1 4 | 1 4 1
..o3..o3..o3..o    | * * 1 ♦ 0 1  0 5 |  0 5  0 10 |  0 10 0 10 | 0 10 0 5 | 0 5 1
-------------------+-------+----------+------------+------------+----------+------
oo.3oo.3oo.3oo.&#x | 1 1 0 | 5 *  * * ♦  4 1  0  0 |  6  4 0  0 | 4  6 0 0 | 1 4 0
o.o3o.o3o.o3o.o&#x | 1 0 1 | * 1  * * ♦  0 5  0  0 |  0 10 0  0 | 0 10 0 0 | 0 5 0
.x. ... ... ...    | 0 2 0 | * * 10 * ♦  1 0  3  1 |  3  1 3  3 | 3  3 1 3 | 1 3 1
.oo3.oo3.oo3.oo&#x | 0 1 1 | * *  * 5 ♦  0 1  0  4 |  0  4 0  6 | 0  6 0 4 | 0 4 1
-------------------+-------+----------+------------+------------+----------+------
ox. ... ... ...&#x | 1 2 0 | 2 0  1 0 | 10 *  *  * ♦  3  1 0  0 | 3  3 0 0 | 1 3 0
ooo3ooo3ooo3ooo&#x | 1 1 1 | 1 1  0 1 |  * 5  *  * ♦  0  4 0  0 | 0  6 0 0 | 0 4 0
.x.3.o. ... ...    | 0 3 0 | 0 0  3 0 |  * * 10  * ♦  1  0 2  1 | 2  1 1 2 | 1 2 1
.xo ... ... ...&#x | 0 2 1 | 0 0  1 2 |  * *  * 10 ♦  0  1 0  3 | 0  3 0 3 | 0 3 1
-------------------+-------+----------+------------+------------+----------+------
ox.3oo. ... ...&#x ♦ 1 3 0 | 3 0  3 0 |  3 0  1  0 | 10  * *  * | 2  1 0 0 | 1 2 0
oxo ... ... ...&#x ♦ 1 2 1 | 2 1  1 2 |  1 2  0  1 |  * 10 *  * | 0  3 0 0 | 0 3 0
.x.3.o.3.o. ...    ♦ 0 4 0 | 0 0  6 0 |  0 0  4  0 |  *  * 5  * | 1  0 1 1 | 1 1 1
.xo3.oo ... ...&#x ♦ 0 3 1 | 0 0  3 3 |  0 0  1  3 |  *  * * 10 | 0  1 0 2 | 0 2 1
-------------------+-------+----------+------------+------------+----------+------
ox.3oo.3oo. ...&#x ♦ 1 4 0 | 4 0  6 0 |  6 0  4  0 |  4  0 1  0 | 5  * * * | 1 1 0
oxo3ooo ... ...&#x ♦ 1 3 1 | 3 1  3 3 |  3 3  1  3 |  1  3 0  1 | * 10 * * | 0 2 0
.x.3.o.3.o.3.o.    ♦ 0 5 0 | 0 0 10 0 |  0 0 10  0 |  0  0 5  0 | *  * 1 * | 1 0 1
.xo3.oo3.oo ...&#x ♦ 0 4 1 | 0 0  6 4 |  0 0  4  6 |  0  0 1  4 | *  * * 5 | 0 1 1
-------------------+-------+----------+------------+------------+----------+------
ox.3oo.3oo.3oo.&#x ♦ 1 5 0 | 5 0 10 0 | 10 0 10  0 | 10  0 5  0 | 5  0 1 0 | 1 * *
oxo3ooo3ooo ...&#x ♦ 1 4 1 | 4 1  6 4 |  6 4  4  6 |  4  6 1  4 | 1  4 0 1 | * 5 *
.xo3.oo3.oo3.oo&#x ♦ 0 5 1 | 0 0 10 5 |  0 0 10 10 |  0  0 5 10 | 0  0 1 5 | * * 1

or
o..3o..3o..3o..     & | 2 * ♦  5 1  0 | 10 5  0 | 10 10 0 |  5 10 0 | 1 5
.o.3.o.3.o.3.o.       | * 5 ♦  2 0  4 |  8 1  6 | 12  4 4 |  8  6 1 | 2 4
----------------------+-----+---------+---------+---------+---------+----
oo.3oo.3oo.3oo.&#x  & | 1 1 | 10 *  * ♦  4 1  0 |  6  4 0 |  4  6 0 | 1 4
o.o3o.o3o.o3o.o&#x    | 2 0 |  * 1  * ♦  0 5  0 |  0 10 0 |  0 10 0 | 0 5
.x. ... ... ...       | 0 2 |  * * 10 ♦  2 0  3 |  6  1 3 |  6  3 1 | 2 3
----------------------+-----+---------+---------+---------+---------+----
ox. ... ... ...&#x  & | 1 2 |  2 0  1 | 20 *  * ♦  3  1 0 |  3  3 0 | 1 3
ooo3ooo3ooo3ooo&#x    | 2 1 |  2 1  0 |  * 5  * ♦  0  4 0 |  0  6 0 | 0 4
.x.3.o. ... ...       | 0 3 |  0 0  3 |  * * 10 ♦  2  0 2 |  4  1 1 | 2 2
----------------------+-----+---------+---------+---------+---------+----
ox.3oo. ... ...&#x  & ♦ 1 3 |  3 0  3 |  3 0  1 | 20  * * |  2  1 0 | 1 2
oxo ... ... ...&#x    ♦ 2 2 |  4 1  1 |  2 2  0 |  * 10 * |  0  3 0 | 0 3
.x.3.o.3.o. ...       ♦ 0 4 |  0 0  6 |  0 0  4 |  *  * 5 |  2  0 1 | 2 1
----------------------+-----+---------+---------+---------+---------+----
ox.3oo.3oo. ...&#x  & ♦ 1 4 |  4 0  6 |  6 0  4 |  4  0 1 | 10  * * | 1 1
oxo3ooo ... ...&#x    ♦ 2 3 |  6 1  3 |  6 3  1 |  2  3 0 |  * 10 * | 0 2
.x.3.o.3.o.3.o.       ♦ 0 5 |  0 0 10 |  0 0 10 |  0  0 5 |  *  * 1 | 2 0
----------------------+-----+---------+---------+---------+---------+----
ox.3oo.3oo.3oo.&#x  & ♦ 1 5 |  5 0 10 | 10 0 10 | 10  0 5 |  5  0 1 | 2 *
oxo3ooo3ooo ...&#x    ♦ 2 4 |  8 1  6 | 12 4  4 |  8  6 1 |  2  4 0 | * 5


oxo3ooo3ooo oox&#x   → height(1,2) = sqrt(5/8) = 0.790569
                   height(1,3) = sqrt(3)/2 = 0.866025
                   height(2,3) = sqrt(3/8) = 0.612372
( (pt || tet) || line)

o..3o..3o.. o..    | 1 * * ♦ 4 2 0 0 0 | 6 1 8 0  0 0 | 4 12 4 0 0 0 | 1 8 6 0 0 | 2 4 0
.o.3.o.3.o. .o.    | * 4 * ♦ 1 0 3 2 0 | 3 0 2 3  6 1 | 3  6 1 1 6 3 | 1 6 3 2 3 | 2 3 1
..o3..o3..o ..o    | * * 2 ♦ 0 1 0 4 1 | 0 1 4 0  6 4 | 0  6 4 0 4 6 | 0 4 6 1 4 | 1 4 1
-------------------+-------+-----------+--------------+--------------+-----------+------
oo.3oo.3oo. oo.&#x | 1 1 0 | 4 * * * * ♦ 3 0 2 0  0 0 | 3  6 1 0 0 0 | 1 6 3 0 0 | 2 3 0
o.o3o.o3o.o o.o&#x | 1 0 1 | * 2 * * * ♦ 0 1 4 0  0 0 | 0  6 4 0 0 0 | 0 4 6 0 0 | 1 4 0
.x. ... ... ...    | 0 2 0 | * * 6 * * ♦ 1 0 0 2  2 0 | 2  2 0 1 4 1 | 1 4 1 2 2 | 2 2 1
.oo3.oo3.oo .oo&#x | 0 1 1 | * * * 8 * ♦ 0 0 1 0  3 1 | 0  3 1 0 3 3 | 0 3 3 1 3 | 1 3 1
... ... ... ..x    | 0 0 2 | * * * * 1 ♦ 0 1 0 0  0 4 | 0  0 4 0 0 6 | 0 0 6 0 4 | 0 4 1
-------------------+-------+-----------+--------------+--------------+-----------+------
ox. ... ... ...&#x | 1 2 0 | 2 0 1 0 0 | 6 * * *  * * ♦ 2  2 0 0 0 0 | 1 4 1 0 0 | 2 2 0
... ... ... o.x&#x | 1 0 2 | 0 2 0 0 1 | * 1 * *  * * ♦ 0  0 4 0 0 0 | 0 0 6 0 0 | 0 4 0
ooo3ooo3ooo ooo&#x | 1 1 1 | 1 1 0 1 0 | * * 8 *  * * ♦ 0  3 1 0 0 0 | 0 3 3 0 0 | 1 3 0
.x.3.o. ... ...    | 0 3 0 | 0 0 3 0 0 | * * * 4  * * ♦ 1  0 0 1 2 0 | 1 2 0 2 1 | 2 1 1
.xo ... ... ...&#x | 0 2 1 | 0 0 1 2 0 | * * * * 12 * ♦ 0  1 0 0 2 1 | 0 2 1 1 2 | 1 2 1
... ... ... .ox&#x | 0 1 2 | 0 0 0 2 1 | * * * *  * 4 ♦ 0  0 1 0 0 3 | 0 0 3 0 3 | 0 3 1
-------------------+-------+-----------+--------------+--------------+-----------+------
ox.3oo. ... ...&#x ♦ 1 3 0 | 3 0 3 0 0 | 3 0 0 1  0 0 | 4  * * * * * | 1 2 0 0 0 | 2 1 0
oxo ... ... ...&#x ♦ 1 2 1 | 2 1 1 2 0 | 1 0 2 0  1 0 | * 12 * * * * | 0 2 1 0 0 | 1 2 0
... ... ... oox&#x ♦ 1 1 2 | 1 2 0 2 1 | 0 1 2 0  0 1 | *  * 4 * * * | 0 0 3 0 0 | 0 3 0
.x.3.o.3.o. ...    ♦ 0 4 0 | 0 0 6 0 0 | 0 0 0 4  0 0 | *  * * 1 * * | 1 0 0 2 0 | 2 0 1
.xo3.oo ... ...&#x ♦ 0 3 1 | 0 0 3 3 0 | 0 0 0 1  3 0 | *  * * * 8 * | 0 1 0 1 1 | 1 1 1
.xo ... ... .ox&#x ♦ 0 2 2 | 0 0 1 4 1 | 0 0 0 0  2 2 | *  * * * * 6 | 0 0 1 0 2 | 0 2 1
-------------------+-------+-----------+--------------+--------------+-----------+------
ox.3oo.3oo. ...&#x ♦ 1 4 0 | 4 0 6 0 0 | 6 0 0 4  0 0 | 4  0 0 1 0 0 | 1 * * * * | 2 0 0
oxo3ooo ... ...&#x ♦ 1 3 1 | 3 1 3 3 0 | 3 0 3 1  3 0 | 1  3 0 0 1 0 | * 8 * * * | 1 1 0
oxo ... ... oox&#x ♦ 1 2 2 | 2 2 1 4 1 | 1 1 4 0  2 2 | 0  2 2 0 0 1 | * * 6 * * | 0 2 0
.xo3.oo3.oo ...&#x ♦ 0 4 1 | 0 0 6 4 0 | 0 0 0 4  6 0 | 0  0 0 1 4 0 | * * * 2 * | 1 0 1
.xo3.oo ... .ox&#x ♦ 0 3 2 | 0 0 3 6 1 | 0 0 0 1  6 3 | 0  0 0 0 2 3 | * * * * 4 | 0 1 1
-------------------+-------+-----------+--------------+--------------+-----------+------
oxo3ooo3ooo ...&#x ♦ 1 4 1 | 4 1 6 4 0 | 6 0 4 4  6 0 | 4  6 0 1 4 0 | 1 4 0 1 0 | 2 * *
oxo3ooo ... oox&#x ♦ 1 3 2 | 3 2 3 6 1 | 3 1 6 1  6 3 | 1  6 3 0 2 3 | 0 2 3 0 1 | * 4 *
.xo3.oo3.oo .ox&#x ♦ 0 4 2 | 0 0 6 8 1 | 0 0 0 4 12 4 | 0  0 0 1 8 6 | 0 0 0 2 4 | * * 1


oxo3ooo oox3ooo&#x   → height(1,2) = height(1,3) = sqrt(2/3) = 0.816497
                   height(2,3) = 1/sqrt(3) = 0.577350
(pt || ({3} || perp {3}) )

o..3o.. o..3o..    | 1 * * ♦ 3 3 0 0 0 | 3 3 9 0 0 0 0 | 1 1 9 9 0 0 0 | 3 9 3 0 0 | 3 3 0
.o.3.o. .o.3.o.    | * 3 * ♦ 1 0 2 3 0 | 2 0 3 1 6 3 0 | 1 0 6 3 1 6 3 | 3 6 1 3 2 | 3 2 1
..o3..o ..o3..o    | * * 3 ♦ 0 1 0 3 2 | 0 2 3 0 3 6 1 | 0 1 3 6 1 6 3 | 1 6 3 2 3 | 2 3 1
-------------------+-------+-----------+---------------+---------------+-----------+------
oo.3oo. oo.3oo.&#x | 1 1 0 | 3 * * * * ♦ 2 0 3 0 0 0 0 | 1 0 6 3 0 0 0 | 3 6 1 0 0 | 3 2 0
o.o3o.o o.o3o.o&#x | 1 0 1 | * 3 * * * ♦ 0 2 3 0 0 0 0 | 0 1 3 6 0 0 0 | 1 6 3 0 0 | 2 3 0
.x. ... ... ...    | 0 2 0 | * * 3 * * ♦ 1 0 0 1 3 0 0 | 1 0 3 0 3 3 0 | 3 3 0 3 1 | 3 1 1
.oo3.oo .oo3.oo&#x | 0 1 1 | * * * 9 * ♦ 0 0 1 0 2 2 0 | 0 0 2 2 1 4 1 | 1 4 1 2 2 | 2 2 1
... ... ..x ...    | 0 0 2 | * * * * 3 ♦ 0 1 0 0 0 3 1 | 0 1 0 3 0 3 3 | 0 3 3 1 3 | 1 3 1
-------------------+-------+-----------+---------------+---------------+-----------+------
ox. ... ... ...&#x | 1 2 0 | 2 0 1 0 0 | 3 * * * * * * ♦ 1 0 3 0 0 0 0 | 3 3 0 0 0 | 3 1 0
... ... o.x ...&#x | 1 0 2 | 0 2 0 0 1 | * 3 * * * * * ♦ 0 1 0 3 0 0 0 | 0 3 3 0 0 | 1 3 0
ooo3ooo ooo3ooo&#x | 1 1 1 | 1 1 0 1 0 | * * 9 * * * * ♦ 0 0 2 2 0 0 0 | 1 4 1 0 0 | 2 2 0
.x.3.o. ... ...    | 0 3 0 | 0 0 3 0 0 | * * * 1 * * * ♦ 1 0 0 0 3 0 0 | 3 0 0 3 0 | 3 0 1
.xo ... ... ...&#x | 0 2 1 | 0 0 1 2 0 | * * * * 9 * * ♦ 0 0 1 0 1 2 0 | 1 2 0 2 1 | 2 1 1
... ... .ox ...&#x | 0 1 2 | 0 0 0 2 1 | * * * * * 9 * ♦ 0 0 0 1 0 2 1 | 0 2 1 1 2 | 1 2 1
... ... ..x3..o    | 0 0 3 | 0 0 0 0 3 | * * * * * * 1 ♦ 0 1 0 0 0 0 3 | 0 0 3 0 3 | 0 3 1
-------------------+-------+-----------+---------------+---------------+-----------+------
ox.3oo. ... ...&#x ♦ 1 3 0 | 3 0 3 0 0 | 3 0 0 1 0 0 0 | 1 * * * * * * | 3 0 0 0 0 | 3 0 0
... ... o.x3o.o&#x ♦ 1 0 3 | 0 3 0 0 3 | 0 3 0 0 0 0 1 | * 1 * * * * * | 0 0 3 0 0 | 0 3 0
oxo ... ... ...&#x ♦ 1 2 1 | 2 1 1 2 0 | 1 0 2 0 1 0 0 | * * 9 * * * * | 1 2 0 0 0 | 2 1 0
... ... oox ...&#x ♦ 1 1 2 | 1 2 0 2 1 | 0 1 2 0 0 1 0 | * * * 9 * * * | 0 2 1 0 0 | 1 2 0
.xo3.oo ... ...&#x ♦ 0 3 1 | 0 0 3 3 0 | 0 0 0 1 3 0 0 | * * * * 3 * * | 1 0 0 2 0 | 2 0 1
.xo ... .ox ...&#x ♦ 0 2 2 | 0 0 1 4 1 | 0 0 0 0 2 2 0 | * * * * * 9 * | 0 1 0 1 1 | 1 1 1
... ... .ox3.oo&#x ♦ 0 1 3 | 0 0 0 3 3 | 0 0 0 0 0 3 1 | * * * * * * 3 | 0 0 1 0 2 | 0 2 1
-------------------+-------+-----------+---------------+---------------+-----------+------
oxo3ooo ... ...&#x ♦ 1 3 1 | 3 1 3 3 0 | 3 0 3 1 3 0 0 | 1 0 3 0 1 0 0 | 3 * * * * | 2 0 0
oxo ... oox ...&#x ♦ 1 2 2 | 2 2 1 4 1 | 1 1 4 0 2 2 0 | 0 0 2 2 0 1 0 | * 9 * * * | 1 1 0
... ... oox3ooo&#x ♦ 1 1 3 | 1 3 0 3 3 | 0 3 3 0 0 3 1 | 0 1 0 3 0 0 1 | * * 3 * * | 0 2 0
.xo3.oo .ox ...&#x ♦ 0 3 2 | 0 0 3 6 1 | 0 0 0 1 6 3 0 | 0 0 0 0 2 3 0 | * * * 3 * | 1 0 1
.xo ... .ox3.oo&#x ♦ 0 2 3 | 0 0 1 6 3 | 0 0 0 0 3 6 1 | 0 0 0 0 0 3 2 | * * * * 3 | 0 1 1
-------------------+-------+-----------+---------------+---------------+-----------+------
oxo3ooo oox ...&#x ♦ 1 3 2 | 3 2 3 6 1 | 3 1 6 1 6 3 0 | 1 0 6 3 2 3 0 | 2 3 0 1 0 | 3 * *
oxo ... oox3ooo&#x ♦ 1 2 3 | 2 3 1 6 3 | 1 3 6 0 3 6 1 | 0 1 3 6 0 3 2 | 0 3 2 0 1 | * 3 *
.xo3.oo .ox3.oo&#x ♦ 0 3 3 | 0 0 3 9 3 | 0 0 0 1 9 9 1 | 0 0 0 0 3 9 3 | 0 0 0 3 3 | * * 1

or
o..3o.. o..3o..       | 1 * ♦ 6 0 0 | 6 9 0  0 | 2 18 0 0 | 6 9 0 | 6 0
.o.3.o. .o.3.o.     & | * 6 ♦ 1 2 3 | 2 3 1  9 | 1  9 4 6 | 4 6 5 | 5 1
----------------------+-----+-------+----------+----------+-------+----
oo.3oo. oo.3oo.&#x  & | 1 1 | 6 * * ♦ 2 3 0  0 | 1  9 0 0 | 4 6 0 | 5 0
.x. ... ... ...     & | 0 2 | * 6 * ♦ 1 0 1  3 | 1  3 3 3 | 3 3 4 | 4 1
.oo3.oo .oo3.oo&#x    | 0 2 | * * 9 ♦ 0 1 0  4 | 0  4 2 4 | 2 4 4 | 4 1
----------------------+-----+-------+----------+----------+-------+----
ox. ... ... ...&#x  & | 1 2 | 2 1 0 | 6 * *  * ♦ 1  3 0 0 | 3 3 0 | 4 0
ooo3ooo ooo3ooo&#x    | 1 2 | 2 0 1 | * 9 *  * ♦ 0  4 0 0 | 2 4 0 | 4 0
.x.3.o. ... ...     & | 0 3 | 0 3 0 | * * 2  * ♦ 1  0 3 0 | 3 0 3 | 3 1
.xo ... ... ...&#x  & | 0 3 | 0 1 2 | * * * 18 ♦ 0  1 1 2 | 1 2 3 | 3 1
----------------------+-----+-------+----------+----------+-------+----
ox.3oo. ... ...&#x  & ♦ 1 3 | 3 3 0 | 3 0 1  0 | 2  * * * | 3 0 0 | 3 0
oxo ... ... ...&#x  & ♦ 1 3 | 3 1 2 | 1 2 0  1 | * 18 * * | 1 2 0 | 3 0
.xo3.oo ... ...&#x  & ♦ 0 4 | 0 3 3 | 0 0 1  3 | *  * 6 * | 1 0 2 | 2 1
.xo ... .ox ...&#x    ♦ 0 4 | 0 2 4 | 0 0 0  4 | *  * * 9 | 0 1 2 | 2 1
----------------------+-----+-------+----------+----------+-------+----
oxo3ooo ... ...&#x  & ♦ 1 4 | 4 3 3 | 3 3 1  3 | 1  3 1 0 | 6 * * | 2 0
oxo ... oox ...&#x    ♦ 1 4 | 4 2 4 | 2 4 0  4 | 0  4 0 1 | * 9 * | 2 0
.xo3.oo .ox ...&#x  & ♦ 0 5 | 0 4 6 | 0 0 1  9 | 0  0 2 3 | * * 6 | 1 1
----------------------+-----+-------+----------+----------+-------+----
oxo3ooo oox ...&#x  & ♦ 1 5 | 5 4 6 | 4 6 1  9 | 1  9 2 3 | 2 3 1 | 6 *
.xo3.oo .ox3.oo&#x    ♦ 0 6 | 0 6 9 | 0 0 2 18 | 0  0 6 9 | 0 0 6 | * 1


xoo oxo3ooo oox&#x   → height(1,2) = height(2,3) = sqrt(5/12) = 0.645497
                       height(1,3) = 1/sqrt(2) = 0.707107
( (line || perp line) || perp {3})

o.. o..3o.. o..    | 2 * * ♦ 1 3 2 0 0 0 | 3 3 2 1  6 0 0 0 | 6 3 1  6 1 3 0 0 | 6 1 3 2 3 0 | 2 3 1
.o. .o.3.o. .o.    | * 3 * ♦ 0 2 0 2 2 0 | 1 4 0 0  4 1 4 1 | 2 2 0  8 2 2 2 2 | 4 1 1 4 4 1 | 2 2 2
..o ..o3..o ..o    | * * 2 ♦ 0 0 2 0 3 1 | 0 0 1 2  6 0 3 3 | 3 0 1  6 0 6 1 3 | 3 0 3 2 6 1 | 1 3 2
-------------------+-------+-------------+------------------+------------------+-------------+------
x.. ... ... ...    | 2 0 0 | 1 * * * * * ♦ 3 0 2 0  0 0 0 0 | 6 3 1  0 0 0 0 0 | 6 1 3 0 0 0 | 2 3 0
oo. oo.3oo. oo.&#x | 1 1 0 | * 6 * * * * ♦ 1 2 0 0  2 0 0 0 | 2 2 0  4 1 1 0 0 | 4 1 1 2 2 0 | 2 2 1
o.o o.o3o.o o.o&#x | 1 0 1 | * * 4 * * * ♦ 0 0 1 1  3 0 0 0 | 3 0 1  3 0 3 0 0 | 3 0 3 1 3 0 | 1 3 1
... .x. ... ...    | 0 2 0 | * * * 3 * * ♦ 0 2 0 0  0 1 2 0 | 0 1 0  4 2 0 2 1 | 2 1 0 4 2 1 | 2 1 2
.oo .oo3.oo .oo&#x | 0 1 1 | * * * * 6 * ♦ 0 0 0 0  2 0 2 1 | 1 0 0  4 0 2 1 2 | 2 0 1 2 4 1 | 1 2 2
... ... ... ..x    | 0 0 2 | * * * * * 1 ♦ 0 0 0 2  0 0 0 3 | 0 0 1  0 0 6 0 3 | 0 0 3 0 6 1 | 0 3 2
-------------------+-------+-------------+------------------+------------------+-------------+------
xo. ... ... ...&#x | 2 1 0 | 1 2 0 0 0 0 | 3 * * *  * * * * ♦ 2 2 0  0 0 0 0 0 | 4 1 1 0 0 0 | 2 2 0
... ox. ... ...&#x | 1 2 0 | 0 2 0 1 0 0 | * 6 * *  * * * * ♦ 0 1 0  2 1 0 0 0 | 2 1 0 2 1 0 | 2 1 1
x.o ... ... ...&#x | 2 0 1 | 1 0 2 0 0 0 | * * 2 *  * * * * ♦ 3 0 1  0 0 0 0 0 | 3 0 3 0 0 0 | 1 3 0
... ... ... o.x&#x | 1 0 2 | 0 0 2 0 0 1 | * * * 2  * * * * ♦ 0 0 1  0 0 3 0 0 | 0 0 3 0 3 0 | 0 3 1
ooo ooo3ooo ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * * * 12 * * * ♦ 1 0 0  2 0 1 0 0 | 2 0 1 1 2 0 | 1 2 1
... .x.3.o. ...    | 0 3 0 | 0 0 0 3 0 0 | * * * *  * 1 * * ♦ 0 0 0  0 2 0 2 0 | 0 1 0 4 0 1 | 2 0 2
... .xo ... ...&#x | 0 2 1 | 0 0 0 1 2 0 | * * * *  * * 6 * ♦ 0 0 0  2 0 0 1 1 | 1 0 0 2 2 1 | 1 1 2
... ... ... .ox&#x | 0 1 2 | 0 0 0 0 2 1 | * * * *  * * * 3 ♦ 0 0 0  0 0 2 0 2 | 0 0 1 0 4 1 | 0 2 2
-------------------+-------+-------------+------------------+------------------+-------------+------
xoo ... ... ...&#x ♦ 2 1 1 | 1 2 2 0 1 0 | 1 0 1 0  2 0 0 0 | 6 * *  * * * * * | 2 0 1 0 0 0 | 1 2 0
xo. ox. ... ...&#x ♦ 2 2 0 | 1 4 0 1 0 0 | 2 2 0 0  0 0 0 0 | * 3 *  * * * * * | 2 1 0 0 0 0 | 2 1 0
x.o ... ... o.x&#x ♦ 2 0 2 | 1 0 4 0 0 1 | 0 0 2 2  0 0 0 0 | * * 1  * * * * * | 0 0 3 0 0 0 | 0 3 0
... oxo ... ...&#x ♦ 1 2 1 | 0 2 1 1 2 0 | 0 1 0 0  2 0 1 0 | * * * 12 * * * * | 1 0 0 1 1 0 | 1 1 1
... ox.3oo. ...&#x ♦ 1 3 0 | 0 3 0 3 0 0 | 0 3 0 0  0 1 0 0 | * * *  * 2 * * * | 0 1 0 2 0 0 | 2 0 1
... ... ... oox&#x ♦ 1 1 2 | 0 1 2 0 2 1 | 0 0 0 1  2 0 0 1 | * * *  * * 6 * * | 0 0 1 0 2 0 | 0 2 1
... .xo3.oo ...&#x ♦ 0 3 1 | 0 0 0 3 3 0 | 0 0 0 0  0 1 3 0 | * * *  * * * 2 * | 0 0 0 2 0 1 | 1 0 2
... .xo ... .ox&#x ♦ 0 2 2 | 0 0 0 1 4 1 | 0 0 0 0  0 0 2 2 | * * *  * * * * 3 | 0 0 0 0 2 1 | 0 1 2
-------------------+-------+-------------+------------------+------------------+-------------+------
xoo oxo ... ...&#x ♦ 2 2 1 | 1 4 2 1 2 0 | 2 2 1 0  4 0 1 0 | 2 1 0  2 0 0 0 0 | 6 * * * * * | 1 1 0
xo. ox.3oo. ...&#x ♦ 2 3 0 | 1 6 0 3 0 0 | 3 6 0 0  0 1 0 0 | 0 3 0  0 2 0 0 0 | * 1 * * * * | 2 0 0
xoo ... ... oox&#x ♦ 2 1 2 | 1 2 4 0 2 1 | 1 0 2 2  4 0 0 1 | 2 0 1  0 0 2 0 0 | * * 3 * * * | 0 2 0
... oxo3ooo ...&#x ♦ 1 3 1 | 0 3 1 3 3 0 | 0 3 0 0  3 1 3 0 | 0 0 0  3 1 0 1 0 | * * * 4 * * | 1 0 1
... oxo ... oox&#x ♦ 1 2 2 | 0 2 2 1 4 1 | 0 1 0 1  4 0 2 2 | 0 0 0  2 0 2 0 1 | * * * * 6 * | 0 1 1
... .xo3.oo .ox&#x ♦ 0 3 2 | 0 0 0 3 6 1 | 0 0 0 0  0 1 6 3 | 0 0 0  0 0 0 2 3 | * * * * * 1 | 0 0 2
-------------------+-------+-------------+------------------+------------------+-------------+------
xoo oxo3ooo ...&#x ♦ 2 3 1 | 1 6 2 3 3 0 | 3 6 1 0  6 1 3 0 | 3 3 0  6 2 0 1 0 | 3 1 0 2 0 0 | 2 * *
xoo oxo ... oox&#x ♦ 2 2 2 | 1 4 4 1 4 1 | 2 2 2 2  8 0 2 2 | 4 1 1  4 0 4 0 1 | 2 0 2 0 2 0 | * 3 *
... oxo3ooo oox&#x ♦ 1 3 2 | 0 3 2 3 6 1 | 0 3 0 1  6 1 6 3 | 0 0 0  6 1 3 2 3 | 0 0 0 2 3 1 | * * 2

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