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Acronym
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hexpy
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Name
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hexadecachoral pyramid,
vertex-first rotunda of triacontiditeron
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Circumradius
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1/sqrt(2) = 0.707107
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Lace city
in approx. ASCII-art
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o where:
o = o3o4o (point)
o O o O = x3o4o (oct)
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o o = o3o3o (point)
t = x3o3o (tet)
t T T = o3o3x (dual tet)
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Coordinates
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-
(0, 0, 0, 0, 1/sqrt(2))
tip
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(1/sqrt(2), 0, 0, 0, 0) & permutations and changes of sign in all but last coordinate
hexadecachoron base
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Dihedral angles
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- at tet between pen and pen: arccos(-3/5) = 126.869898°
- at tet between hex and pen: arccos(1/sqrt(5)) = 63.434949°
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Face vector
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9, 32, 56, 48, 17
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Confer
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- uniform relative:
-
tac
- related segmentotera:
-
octasc
- ambification:
-
hexaico
- general polytopal classes:
-
segmentotera
fundamental lace prisms
lace simplices
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External
links
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Because all faces of this polyteron are regular triangles it allows for some of the same operations as regulars do:
its truncation results in hextum and
its rectification results in hexaico.
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
o(qo) o(ox)3o(oo)4o(oo)&#x → height = 1/sqrt(2) = 0.707107
(pt || hex)
o(..) o(..)3o(..)4o(..) | 1 * * ♦ 2 6 0 0 | 12 12 0 0 | 8 24 0 | 16 0
.(o.) .(o.)3.(o.)4.(o.) | * 2 * ♦ 1 0 6 0 | 0 6 12 0 | 0 12 8 | 8 1
.(.o) .(.o)3.(.o)4.(.o) | * * 6 ♦ 0 1 2 4 | 4 2 8 4 | 4 8 8 | 8 1
----------------------------+-------+-----------+------------+---------+-----
o(o.) o(o.)3o(o.)4o(o.)&#x | 1 1 0 | 2 * * * ♦ 0 6 0 0 | 0 12 0 | 8 0
o(.o) o(.o)3o(.o)4o(.o)&#x | 1 0 1 | * 6 * * ♦ 4 2 0 0 | 4 8 0 | 8 0
.(oo) .(oo)3.(oo)4.(oo)&#x | 0 1 1 | * * 12 * ♦ 0 1 4 0 | 0 4 4 | 4 1
.(..) .(.x) .(..) .(..) | 0 0 2 | * * * 12 ♦ 1 0 2 2 | 2 2 4 | 4 1
----------------------------+-------+-----------+------------+---------+-----
.(..) o(.x) .(..) .(..)&#x | 1 0 2 | 0 2 0 1 | 12 * * * | 2 2 0 | 4 0
o(oo) o(oo)3o(oo)4o(oo)&#x | 1 1 1 | 1 1 1 0 | * 12 * * | 0 4 0 | 4 0
.(..) .(ox) .(..) .(..)&#x | 0 1 2 | 0 0 2 1 | * * 24 * | 0 1 2 | 2 1
.(..) .(.x)3.(.o) .(..) | 0 0 3 | 0 0 0 3 | * * * 8 | 1 0 2 | 2 1
----------------------------+-------+-----------+------------+---------+-----
.(..) o(.x)3o(.o) .(..)&#x ♦ 1 0 3 | 0 3 0 3 | 3 0 0 1 | 8 * * | 2 0
.(..) o(ox) .(..) .(..)&#x ♦ 1 1 2 | 1 2 2 1 | 1 2 1 0 | * 24 * | 2 0
.(..) .(ox)3.(oo) .(..)&#x ♦ 0 1 3 | 0 0 3 3 | 0 0 3 1 | * * 16 | 1 1
----------------------------+-------+-----------+------------+---------+-----
.(..) o(ox)3o(oo) .(..)&#x ♦ 1 1 3 | 1 3 3 3 | 3 3 3 1 | 1 3 1 | 16 *
.(qo) .(ox)3.(oo)4.(oo)&#zx ♦ 0 2 6 | 0 0 12 12 | 0 0 24 8 | 0 0 16 | * 1
o(xo) o(ox) o(xo) o(ox)&#x → height(1,2) = height(1,3) = height(1,23) = 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
----------------------------+-------+----------------+--------------------+-----------------+----------
o(o.) o(o.) o(o.) o(o.)&#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
----------------------------+-------+----------------+--------------------+-----------------+----------
o(x.) .(..) .(..) .(..)&#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
.(..) .(..) o(x.) .(..)&#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
o(oo) o(oo) o(oo) o(oo)&#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
----------------------------+-------+----------------+--------------------+-----------------+----------
o(xo) .(..) .(..) .(..)&#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
.(..) o(ox) .(..) .(..)&#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
.(..) .(..) o(xo) .(..)&#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
.(..) .(..) .(..) o(ox)&#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
----------------------------+-------+----------------+--------------------+-----------------+----------
o(xo) o(ox) .(..) .(..)&#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 * * * *
o(xo) .(..) .(..) o(ox)&#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 * * *
.(..) o(ox) o(xo) .(..)&#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 * *
.(..) .(..) o(xo) o(ox)&#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
o(xoxo) o(oxox)&#xrt → height(1234,5) = 1/sqrt(2) = 0.707107
(pt || hex)
o(....) o(....) | 1 * ♦ 8 0 0 0 | 4 16 4 0 0 | 16 16 0 0 0 | 4 8 4 0
.(o...) .(o...) & | * 8 ♦ 1 1 4 1 | 1 4 1 6 6 | 6 6 2 4 2 | 2 4 2 1
----------------------+-----+----------+--------------+-------------+--------
o(o...) o(o...)&#x & | 1 1 | 8 * * * ♦ 1 4 1 0 0 | 6 6 0 0 0 | 2 4 2 0
.(x...) .(....) & | 0 2 | * 4 * * ♦ 1 0 0 4 0 | 4 0 2 2 0 | 2 2 0 1
.(oo..) .(oo..)&#x & | 0 2 | * * 16 * ♦ 0 1 0 2 2 | 2 2 1 2 1 | 1 2 1 1
.(o.o.) .(o.o.)&#x & | 0 2 | * * * 4 ♦ 0 0 1 0 4 | 0 4 0 2 2 | 0 2 2 1
----------------------+-----+----------+--------------+-------------+--------
o(x...) .(....)&#x & | 1 2 | 2 1 0 0 | 4 * * * * | 4 0 0 0 0 | 2 2 0 0
o(oo..) o(oo..)&#x & | 1 2 | 2 0 1 0 | * 16 * * * | 2 2 0 0 0 | 1 2 1 0
o(o.o.) o(o.o.)&#x & | 1 2 | 2 0 0 1 | * * 4 * * | 0 4 0 0 0 | 0 2 2 0
.(xo..) .(....)&#x & | 0 3 | 0 1 2 0 | * * * 16 * | 1 0 1 1 0 | 1 1 0 1
.(ooo.) .(ooo.)&#x & | 0 3 | 0 0 2 1 | * * * * 16 | 0 1 0 1 1 | 0 1 1 1
----------------------+-----+----------+--------------+-------------+--------
o(xo..) .(....)&#x & ♦ 1 3 | 3 1 2 0 | 1 2 0 1 0 | 16 * * * * | 1 1 0 0
o(ooo.) o(ooo.)&#x & ♦ 1 3 | 3 0 2 1 | 0 2 1 0 1 | * 16 * * * | 0 1 1 0
.(xo..) .(ox..)&#x & ♦ 0 4 | 0 2 4 0 | 0 0 0 4 0 | * * 4 * * | 1 0 0 1
.(xo.o) .(....)&#x & ♦ 0 4 | 0 1 4 1 | 0 0 0 2 2 | * * * 8 * | 0 1 0 1
.(oooo) .(oooo)&#x ♦ 0 4 | 0 0 4 2 | 0 0 0 0 4 | * * * * 4 | 0 0 1 1
----------------------+-----+----------+--------------+-------------+--------
o(xo..) o(ox..)&#x & ♦ 1 4 | 4 2 4 0 | 2 4 0 4 0 | 4 0 1 0 0 | 4 * * *
o(xo.o) .(....)&#x & ♦ 1 4 | 4 1 4 1 | 1 4 1 2 2 | 2 2 0 1 0 | * 8 * *
o(oooo) o(oooo)&#x ♦ 1 4 | 4 0 4 2 | 0 4 2 0 4 | 0 4 0 0 1 | * * 4 *
.(xoxo) .(oxox)&#xr ♦ 0 8 | 0 4 16 4 | 0 0 0 16 16 | 0 0 4 8 4 | * * * 1