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 |
| ||||
Volume | sqrt(7)/5760 = 0.00045933 | ||||
Surface | 7 sqrt(3)/480 = 0.025259 | ||||
Rel. Roundness | 5 π3 sqrt(7)/21609 = 1.898165 % | ||||
Dihedral angles | |||||
Face vector | 7, 21, 35, 35, 21, 7 | ||||
Confer |
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External links |
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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