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Acronym
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hex, K-4.2
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Name
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hexadecachoron,
tetracross (β4),
hemitesseract,
hemioctachoron,
16-cell,
aerochoron,
tetrahedral antiprism
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Segmentochoron display
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Cross sections
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 ©
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Circumradius
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1/sqrt(2) = 0.707107
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Inradius
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1/sqrt(8) = 0.353553
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Vertex figure
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 ©
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Vertex layers
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| x3o3o4o |
x3o3o .
tet first |
x3o . o
{3} first |
x . o4o
edge first |
. o3o4o
vertex first |
o3o3x .
opposite tet |
o3o . q |
o . x4o |
. x3o4o
vertex figure |
| |
o3x . o
opposite {3} |
x . o4o
opposite edge |
. o3o4o
opposite vertex |
| x3o3o *b3o |
x3o3o .
tet first |
x3o . *b3o
tet first |
x . o o
edge first |
. o3o *b3o
vertex first |
o3o3x .
opposite tet |
o3o . *b3x
opposite tet |
o . x x |
. x3o *b3o
vertex figure |
| |
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x . o o
opposite edge |
. o3o *b3o
opposite vertex |
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Lace city
in approx. ASCII-art
|
 ©
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o4o
o4o x4o o4o
o4o
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x3o
o3o o3o
o3x
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Coordinates
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- as orthoplex (tetracross): (1/sqrt(2), 0, 0, 0) & all permutations, all changes of sign
- as hemitesseract: (1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 1/sqrt(8)) & all even permutations, all even changes of sign
- as "the other" (mirrored) hemitesseract: (1/sqrt(8), 1/sqrt(8), 1/sqrt(8), -1/sqrt(8)) & all even permutations, all even changes of sign
- (the compound of those 3 such oriented hexadecachora is {3,4,3}[3{3,3,4}]2{3,4,3}, inscribed in an ico)
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General of army
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(is itself convex)
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Colonel of regiment
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(is itself locally convex
– uniform polychoral members:
)
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Dual
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tes
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Confer
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- Grünbaumian relatives:
-
hex+8oct
2hex+8oct
- general pyramid-antiprisms:
-
n-pyap
- compounds:
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haddet
stico
- general polytopal classes:
-
tetrahedrochora
regular
orthoplex
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External
links
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Incidence matrix according to Dynkin symbol
x3o3o4o
. . . . | 8 ♦ 6 | 12 | 8
--------+---+----+----+---
x . . . | 2 | 24 | 4 | 4
--------+---+----+----+---
x3o . . | 3 | 3 | 32 | 2
--------+---+----+----+---
x3o3o . ♦ 4 | 6 | 4 | 16
x3o3o4/3o
. . . . | 8 ♦ 6 | 12 | 8
----------+---+----+----+---
x . . . | 2 | 24 | 4 | 4
----------+---+----+----+---
x3o . . | 3 | 3 | 32 | 2
----------+---+----+----+---
x3o3o . ♦ 4 | 6 | 4 | 16
x3o3/2o4o
. . . . | 8 ♦ 6 | 12 | 8
----------+---+----+----+---
x . . . | 2 | 24 | 4 | 4
----------+---+----+----+---
x3o . . | 3 | 3 | 32 | 2
----------+---+----+----+---
x3o3/2o . ♦ 4 | 6 | 4 | 16
x3o3/2o4/3o
. . . . | 8 ♦ 6 | 12 | 8
------------+---+----+----+---
x . . . | 2 | 24 | 4 | 4
------------+---+----+----+---
x3o . . | 3 | 3 | 32 | 2
------------+---+----+----+---
x3o3/2o . ♦ 4 | 6 | 4 | 16
x3/2o3o4o
. . . . | 8 ♦ 6 | 12 | 8
----------+---+----+----+---
x . . . | 2 | 24 | 4 | 4
----------+---+----+----+---
x3/2o . . | 3 | 3 | 32 | 2
----------+---+----+----+---
x3/2o3o . ♦ 4 | 6 | 4 | 16
x3/2o3o4/3o
. . . . | 8 ♦ 6 | 12 | 8
------------+---+----+----+---
x . . . | 2 | 24 | 4 | 4
------------+---+----+----+---
x3/2o . . | 3 | 3 | 32 | 2
------------+---+----+----+---
x3/2o3o . ♦ 4 | 6 | 4 | 16
x3/2o3/2o4o
. . . . | 8 ♦ 6 | 12 | 8
------------+---+----+----+---
x . . . | 2 | 24 | 4 | 4
------------+---+----+----+---
x3/2o . . | 3 | 3 | 32 | 2
------------+---+----+----+---
x3/2o3/2o . ♦ 4 | 6 | 4 | 16
x3/2o3/2o4/3o
. . . . | 8 ♦ 6 | 12 | 8
--------------+---+----+----+---
x . . . | 2 | 24 | 4 | 4
--------------+---+----+----+---
x3/2o . . | 3 | 3 | 32 | 2
--------------+---+----+----+---
x3/2o3/2o . ♦ 4 | 6 | 4 | 16
x3o3o *b3o
. . . . | 8 ♦ 6 | 12 | 4 4
-----------+---+----+----+----
x . . . | 2 | 24 | 4 | 2 2
-----------+---+----+----+----
x3o . . | 3 | 3 | 32 | 1 1
-----------+---+----+----+----
x3o3o . ♦ 4 | 6 | 4 | 8 *
x3o . *b3o ♦ 4 | 6 | 4 | * 8
x3o3o *b3/2o
. . . . | 8 ♦ 6 | 12 | 4 4
-------------+---+----+----+----
x . . . | 2 | 24 | 4 | 2 2
-------------+---+----+----+----
x3o . . | 3 | 3 | 32 | 1 1
-------------+---+----+----+----
x3o3o . ♦ 4 | 6 | 4 | 8 *
x3o . *b3/2o ♦ 4 | 6 | 4 | * 8
x3o3/2o *b3/2o
. . . . | 8 ♦ 6 | 12 | 4 4
---------------+---+----+----+----
x . . . | 2 | 24 | 4 | 2 2
---------------+---+----+----+----
x3o . . | 3 | 3 | 32 | 1 1
---------------+---+----+----+----
x3o3/2o . ♦ 4 | 6 | 4 | 8 *
x3o . *b3/2o ♦ 4 | 6 | 4 | * 8
x3/2o3o *b3o
. . . . | 8 ♦ 6 | 12 | 4 4
-------------+---+----+----+----
x . . . | 2 | 24 | 4 | 2 2
-------------+---+----+----+----
x3/2o . . | 3 | 3 | 32 | 1 1
-------------+---+----+----+----
x3/2o3o . ♦ 4 | 6 | 4 | 8 *
x3/2o . *b3o ♦ 4 | 6 | 4 | * 8
x3/2o3o *b3/2o
. . . . | 8 ♦ 6 | 12 | 4 4
---------------+---+----+----+----
x . . . | 2 | 24 | 4 | 2 2
---------------+---+----+----+----
x3/2o . . | 3 | 3 | 32 | 1 1
---------------+---+----+----+----
x3/2o3o . ♦ 4 | 6 | 4 | 8 *
x3/2o . *b3/2o ♦ 4 | 6 | 4 | * 8
x3/2o3/2o *b3/2o
. . . . | 8 ♦ 6 | 12 | 4 4
-----------------+---+----+----+----
x . . . | 2 | 24 | 4 | 2 2
-----------------+---+----+----+----
x3/2o . . | 3 | 3 | 32 | 1 1
-----------------+---+----+----+----
x3/2o3/2o . ♦ 4 | 6 | 4 | 8 *
x3/2o . *b3/2o ♦ 4 | 6 | 4 | * 8
s4o3o3o
demi( . . . . ) | 8 ♦ 6 | 12 | 4 4
----------------+---+----+----+----
s4o . . ♦ 2 | 24 | 4 | 2 2
----------------+---+----+----+----
sefa( s4o3o . ) | 3 | 3 | 32 | 1 1
----------------+---+----+----+----
s4o3o . ♦ 4 | 6 | 4 | 8 *
sefa( s4o3o3o ) ♦ 4 | 6 | 4 | * 8
s2s4o3o
demi( . . . . ) | 8 ♦ 3 3 | 9 3 | 3 1 4
----------------+---+-------+------+------
s2s . . ♦ 2 | 12 * | 4 0 | 2 0 2
. s4o . ♦ 2 | * 12 | 2 2 | 1 1 2
----------------+---+-------+------+------
sefa( s2s4o . ) | 3 | 2 1 | 24 * | 1 0 1
sefa( . s4o3o ) | 3 | 0 3 | * 8 | 0 1 1
----------------+---+-------+------+------
s2s4o . ♦ 4 | 4 2 | 4 0 | 6 * *
. s4o3o ♦ 4 | 0 6 | 0 4 | * 2 *
sefa( s2s4o3o ) ♦ 4 | 3 3 | 3 1 | * * 8
s4o2s4o
demi( . . . . ) | 8 ♦ 1 4 1 | 6 6 | 2 2 4
----------------+---+--------+-------+------
s4o . . ♦ 2 | 4 * * | 4 0 | 2 0 2
s 2 s . ♦ 2 | * 16 * | 2 2 | 1 1 2
. . s4o ♦ 2 | * * 4 | 0 4 | 0 2 2
----------------+---+--------+-------+------
sefa( s4o2s . ) | 3 | 1 2 0 | 16 * | 1 1 0
sefa( s 2 s4o ) | 3 | 0 2 1 | * 16 | 0 1 1
----------------+---+--------+-------+------
s4o2s . ♦ 4 | 2 4 0 | 4 0 | 4 * *
s 2 s4o ♦ 4 | 0 4 2 | 0 4 | * 4 *
sefa( s4o2s4o ) ♦ 4 | 1 4 1 | 2 2 | * * 8
or
demi( . . . . ) | 8 ♦ 2 4 | 12 | 4 4
-------------------+---+------+----+----
s4o . . & ♦ 2 | 8 * | 4 | 2 2
s 2 s . ♦ 2 | * 16 | 4 | 2 2
-------------------+---+------+----+----
sefa( s4o2s . ) & | 3 | 1 2 | 32 | 1 1
-------------------+---+------+----+----
s4o2s . & ♦ 4 | 2 4 | 4 | 8 *
sefa( s4o2s4o ) ♦ 4 | 2 4 | 4 | * 8
s2s2s4o
demi( . . . . ) | 8 ♦ 1 2 2 1 | 6 3 3 | 2 1 1 4
----------------+---+---------+--------+--------
s2s . . ♦ 2 | 4 * * * | 4 0 0 | 2 0 0 2
s 2 s . ♦ 2 | * 8 * * | 2 2 0 | 1 1 0 2
. s2s . ♦ 2 | * * 8 * | 2 0 2 | 1 0 1 2
. . s4o ♦ 2 | * * * 4 | 0 2 2 | 0 1 1 2
----------------+---+---------+--------+--------
sefa( s2s2s . ) | 3 | 1 1 1 0 | 16 * * | 1 0 0 1
sefa( s 2 s4o ) | 3 | 0 2 0 1 | * 8 * | 0 1 0 1
sefa( . s2s4o ) | 3 | 0 0 2 1 | * * 8 | 0 0 1 1
----------------+---+---------+--------+--------
s2s2s . ♦ 4 | 2 2 2 0 | 4 0 0 | 4 * * *
s 2 s4o ♦ 4 | 0 4 0 2 | 0 4 0 | * 2 * *
. s2s4o ♦ 4 | 0 0 4 2 | 0 0 4 | * * 2 *
sefa( s2s2s4o ) ♦ 4 | 1 2 2 1 | 2 1 1 | * * * 8
s2s2s2s
demi( . . . . ) | 8 ♦ 1 1 1 1 1 1 | 3 3 3 3 | 1 1 1 1 4
-----------------+---+-------------+---------+----------
s2s . . ♦ 2 | 4 * * * * * | 2 2 0 0 | 1 1 0 0 2
s 2 s . ♦ 2 | * 4 * * * * | 2 0 2 0 | 1 0 1 0 2
s . . s2*a ♦ 2 | * * 4 * * * | 0 2 2 0 | 0 1 1 0 2
. s2s . ♦ 2 | * * * 4 * * | 2 0 0 2 | 1 0 0 1 2
. s 2 s ♦ 2 | * * * * 4 * | 0 2 0 2 | 0 1 0 1 2
. . s2s ♦ 2 | * * * * * 4 | 0 0 2 2 | 0 0 1 1 2
-----------------+---+-------------+---------+----------
sefa( s2s2s . ) | 3 | 1 1 0 1 0 0 | 8 * * * | 1 0 0 0 1
sefa( s2s 2 s ) | 3 | 1 0 1 0 1 0 | * 8 * * | 0 1 0 0 1
sefa( s 2 s2s ) | 3 | 0 1 1 0 0 1 | * * 8 * | 0 0 1 0 1
sefa( . s2s2s ) | 3 | 0 0 0 1 1 1 | * * * 8 | 0 0 0 1 1
-----------------+---+-------------+---------+----------
s2s2s . ♦ 4 | 2 2 0 2 0 0 | 4 0 0 0 | 2 * * * *
s2s 2 s ♦ 4 | 2 0 2 0 2 0 | 0 4 0 0 | * 2 * * *
s 2 s2s ♦ 4 | 0 2 2 0 0 2 | 0 0 4 0 | * * 2 * *
. s2s2s ♦ 4 | 0 0 0 2 2 2 | 0 0 0 4 | * * * 2 *
sefa( s2s2s2s ) ♦ 4 | 1 1 1 1 1 1 | 1 1 1 1 | * * * * 8
xo3oo3ox&#x → height = 1/sqrt(2) = 0.707107
(tet || dual tet)
o.3o.3o. | 4 * ♦ 3 3 0 | 3 6 3 0 | 1 3 3 1 0
.o3.o3.o | * 4 ♦ 0 3 3 | 0 3 6 3 | 0 1 3 3 1
------------+-----+--------+-----------+----------
x. .. .. | 2 0 | 6 * * | 2 2 0 0 | 1 2 1 0 0
oo3oo3oo&#x | 1 1 | * 12 * | 0 2 2 0 | 0 1 2 1 0
.. .. .x | 0 2 | * * 6 | 0 0 2 2 | 0 0 1 2 1
------------+-----+--------+-----------+----------
x.3o. .. | 3 0 | 3 0 0 | 4 * * * | 1 1 0 0 0
xo .. ..&#x | 2 1 | 1 2 0 | * 12 * * | 0 1 1 0 0
.. .. ox&#x | 1 2 | 0 2 1 | * * 12 * | 0 0 1 1 0
.. .o3.x | 0 3 | 0 0 3 | * * * 4 | 0 0 0 1 1
------------+-----+--------+-----------+----------
x.3o.3o. ♦ 4 0 | 6 0 0 | 4 0 0 0 | 1 * * * *
xo3oo ..&#x ♦ 3 1 | 3 3 0 | 1 3 0 0 | * 4 * * *
xo .. ox&#x ♦ 2 2 | 1 4 1 | 0 2 2 0 | * * 6 * *
.. oo3ox&#x ♦ 1 3 | 0 3 3 | 0 0 3 1 | * * * 4 *
.o3.o3.x ♦ 0 4 | 0 0 6 | 0 0 0 4 | * * * * 1
or
o.3o.3o. & | 8 ♦ 3 3 | 3 9 | 1 4 3
--------------+---+-------+------+------
x. .. .. & | 2 | 12 * | 2 2 | 1 2 1
oo3oo3oo&#x | 2 | * 12 | 0 4 | 0 2 2
--------------+---+-------+------+------
x.3o. .. & | 3 | 3 0 | 8 * | 1 1 0
xo .. ..&#x & | 3 | 1 2 | * 24 | 0 1 1
--------------+---+-------+------+------
x.3o.3o. & ♦ 4 | 6 0 | 4 0 | 2 * *
xo3oo ..&#x & ♦ 4 | 3 3 | 1 3 | * 8 *
xo .. ox&#x ♦ 4 | 2 4 | 0 4 | * * 6
oxo3ooo4ooo&#xt → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo oct || pt)
o..3o..4o.. | 1 * * ♦ 6 0 0 | 12 0 0 | 8 0
.o.3.o.4.o. | * 6 * ♦ 1 4 1 | 4 4 4 | 4 4
..o3..o4..o | * * 1 ♦ 0 0 6 | 0 0 12 | 0 8
---------------+-------+--------+---------+----
oo.3oo.4oo.&#x | 1 1 0 | 6 * * | 4 0 0 | 4 0
.x. ... ... | 0 2 0 | * 12 * | 1 2 1 | 2 2
.oo3.oo4.oo&#x | 0 1 1 | * * 6 | 0 0 4 | 0 4
---------------+-------+--------+---------+----
ox. ... ...&#x | 1 2 0 | 2 1 0 | 12 * * | 2 0
.x.3.o. ... | 0 3 0 | 0 3 0 | * 8 * | 1 1
.xo ... ...&#x | 0 2 1 | 0 1 2 | * * 12 | 0 2
---------------+-------+--------+---------+----
ox.3oo. ...&#x ♦ 1 3 0 | 3 3 0 | 3 1 0 | 8 *
.xo3.oo ...&#x ♦ 0 3 1 | 0 3 3 | 0 1 3 | * 8
or
o..3o..4o.. & | 2 * ♦ 6 0 | 12 0 | 8
.o.3.o.4.o. | * 6 ♦ 2 4 | 8 4 | 8
-----------------+-----+-------+------+---
oo.3oo.4oo.&#x & | 1 1 | 12 * | 4 0 | 4
.x. ... ... | 0 2 | * 12 | 2 2 | 4
-----------------+-----+-------+------+---
ox. ... ...&#x & | 1 2 | 2 1 | 24 * | 2
.x.3.o. ... | 0 3 | 0 3 | * 8 | 2
-----------------+-----+-------+------+---
ox.3oo. ...&#x & ♦ 1 3 | 3 3 | 3 1 | 16
ooo3oxo3ooo&#xt → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo oct || pt)
o..3o..3o.. | 1 * * ♦ 6 0 0 | 12 0 0 0 | 4 4 0 0
.o.3.o.3.o. | * 6 * ♦ 1 4 1 | 4 2 2 4 | 2 2 2 2
..o3..o3..o | * * 1 ♦ 0 0 6 | 0 0 0 12 | 0 0 4 4
---------------+-------+--------+-----------+--------
oo.3oo.3oo.&#x | 1 1 0 | 6 * * | 4 0 0 0 | 2 2 0 0
... .x. ... | 0 2 0 | * 12 * | 1 1 1 1 | 1 1 1 1
.oo3.oo3.oo&#x | 0 1 1 | * * 6 | 0 0 0 4 | 0 0 2 2
---------------+-------+--------+-----------+--------
... ox. ...&#x | 1 2 0 | 2 1 0 | 12 * * * | 1 1 0 0
.o.3.x. ... | 0 3 0 | 0 3 0 | * 4 * * | 1 0 1 0
... .x.3.o. | 0 3 0 | 0 3 0 | * * 4 * | 0 1 0 1
... .xo ...&#x | 0 2 1 | 0 1 2 | * * * 12 | 0 0 1 1
---------------+-------+--------+-----------+--------
oo.3ox. ...&#x ♦ 1 3 0 | 3 3 0 | 3 1 0 0 | 4 * * *
... ox.3oo.&#x ♦ 1 3 0 | 3 3 0 | 3 0 1 0 | * 4 * *
.oo3.xo ...&#x ♦ 0 3 1 | 0 3 3 | 0 1 0 3 | * * 4 *
... .xo3.oo&#x ♦ 0 3 1 | 0 3 3 | 0 0 1 3 | * * * 4
or
o..3o..3o.. & | 2 * ♦ 6 0 | 12 0 0 | 4 4
.o.3.o.3.o. | * 6 ♦ 2 4 | 8 2 2 | 4 4
-----------------+-----+-------+--------+----
oo.3oo.3oo.&#x & | 1 1 | 12 * | 4 0 0 | 2 2
... .x. ... | 0 2 | * 12 | 2 1 1 | 2 2
-----------------+-----+-------+--------+----
... ox. ...&#x & | 1 2 | 2 1 | 24 * * | 1 1
.o.3.x. ... | 0 3 | 0 3 | * 4 * | 2 0
... .x.3.o. | 0 3 | 0 3 | * * 4 | 0 2
-----------------+-----+-------+--------+----
oo.3ox. ...&#x & ♦ 1 3 | 3 3 | 3 1 0 | 8 *
... ox.3oo.&#x & ♦ 1 3 | 3 3 | 3 0 1 | * 8
xox oxo4ooo&#xt → both heights = 1/2
(line || ortho pseudo {4} || line)
o.. o..4o.. | 2 * * ♦ 1 4 1 0 0 0 | 4 4 4 0 0 | 4 4 0
.o. .o.4.o. | * 4 * ♦ 0 2 0 2 2 0 | 1 4 2 1 4 | 2 4 2
..o ..o4..o | * * 2 ♦ 0 0 1 0 4 1 | 0 0 4 4 4 | 0 4 4
---------------+-------+-------------+-----------+------
x.. ... ... | 2 0 0 | 1 * * * * * | 4 0 0 0 0 | 4 0 0
oo. oo.4oo.&#x | 1 1 0 | * 8 * * * * | 1 2 1 0 0 | 2 2 0
o.o o.o4o.o&#x | 1 0 1 | * * 2 * * * | 0 0 4 0 0 | 0 4 0
... .x. ... | 0 2 0 | * * * 4 * * | 0 2 0 0 2 | 1 2 1
.oo .oo4.oo&#x | 0 1 1 | * * * * 8 * | 0 0 1 1 2 | 0 2 2
..x ... ... | 0 0 2 | * * * * * 1 | 0 0 0 4 0 | 0 0 4
---------------+-------+-------------+-----------+------
xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 0 | 4 * * * * | 2 0 0
... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 | * 8 * * * | 1 1 0
ooo ooo4ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * 8 * * | 0 2 0
.ox ... ...&#x | 0 1 2 | 0 0 0 0 2 1 | * * * 4 * | 0 0 2
... .xo ...&#x | 0 2 1 | 0 0 0 1 2 0 | * * * * 8 | 0 1 1
---------------+-------+-------------+-----------+------
xo. ox. ...&#x ♦ 2 2 0 | 1 4 0 1 0 0 | 2 2 0 0 0 | 4 * *
... oxo ...&#x ♦ 1 2 1 | 0 2 1 1 2 0 | 0 1 2 0 1 | * 8 *
.ox .xo ...&#x ♦ 0 2 2 | 0 0 0 1 4 1 | 0 0 0 2 2 | * * 4
or
o.. o..4o.. & | 4 * ♦ 1 4 1 0 | 4 4 4 | 4 4
.o. .o.4.o. | * 4 ♦ 0 4 0 2 | 2 8 2 | 4 4
-----------------+-----+----------+--------+----
x.. ... ... & | 2 0 | 2 * * * | 4 0 0 | 4 0
oo. oo.4oo.&#x & | 1 1 | * 16 * * | 1 2 1 | 2 2
o.o o.o4o.o&#x | 2 0 | * * 2 * | 0 0 4 | 0 4
... .x. ... | 0 2 | * * * 4 | 0 4 0 | 2 2
-----------------+-----+----------+--------+----
xo. ... ...&#x & | 2 1 | 1 2 0 0 | 8 * * | 2 0
... ox. ...&#x & | 1 2 | 0 2 0 1 | * 16 * | 1 1
ooo ooo4ooo&#x | 2 1 | 0 2 1 0 | * * 8 | 0 2
-----------------+-----+----------+--------+----
xo. ox. ...&#x & ♦ 2 2 | 1 4 0 1 | 2 2 0 | 8 *
... oxo ...&#x ♦ 2 2 | 0 4 1 1 | 0 2 2 | * 8
xox oxo oxo&#xt → both heights = 1/2
(line || ortho pseudo {4} || line)
o.. o.. o.. | 2 * * ♦ 1 4 1 0 0 0 0 | 4 2 2 4 0 0 0 | 2 2 2 2 0 0
.o. .o. .o. | * 4 * ♦ 0 2 0 1 1 2 0 | 1 2 2 2 1 2 2 | 1 1 2 2 1 1
..o ..o ..o | * * 2 ♦ 0 0 1 0 0 4 1 | 0 0 0 4 4 2 2 | 0 0 2 2 2 2
---------------+-------+---------------+---------------+------------
x.. ... ... | 2 0 0 | 1 * * * * * * | 4 0 0 0 0 0 0 | 2 2 0 0 0 0
oo. oo. oo.&#x | 1 1 0 | * 8 * * * * * | 1 1 1 1 0 0 0 | 1 1 1 1 0 0
o.o o.o o.o&#x | 1 0 1 | * * 2 * * * * | 0 0 0 4 0 0 0 | 0 0 2 2 0 0
... .x. ... | 0 2 0 | * * * 2 * * * | 0 2 0 0 0 2 0 | 1 0 2 0 1 0
... ... .x. | 0 2 0 | * * * * 2 * * | 0 0 2 0 0 0 2 | 0 1 0 2 0 1
.oo .oo .oo&#x | 0 1 1 | * * * * * 8 * | 0 0 0 1 1 1 1 | 0 0 1 1 1 1
..x ... ... | 0 0 2 | * * * * * * 1 | 0 0 0 0 4 0 0 | 0 0 0 0 2 2
---------------+-------+---------------+---------------+------------
xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 0 0 | 4 * * * * * * | 1 1 0 0 0 0
... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 0 | * 4 * * * * * | 1 0 1 0 0 0
... ... ox.&#x | 1 2 0 | 0 2 0 0 1 0 0 | * * 4 * * * * | 0 1 0 1 0 0
ooo ooo ooo&#x | 1 1 1 | 0 1 1 0 0 1 0 | * * * 8 * * * | 0 0 1 1 0 0
.ox ... ...&#x | 0 1 2 | 0 0 0 0 0 2 1 | * * * * 4 * * | 0 0 0 0 1 1
... .xo ...&#x | 0 2 1 | 0 0 0 1 0 2 0 | * * * * * 4 * | 0 0 1 0 1 0
... ... .xo&#x | 0 2 1 | 0 0 0 0 1 2 0 | * * * * * * 4 | 0 0 0 1 0 1
---------------+-------+---------------+---------------+------------
xo. ox. ...&#x ♦ 2 2 0 | 1 4 0 1 0 0 0 | 2 2 0 0 0 0 0 | 2 * * * * *
xo. ... ox.&#x ♦ 2 2 0 | 1 4 0 0 1 0 0 | 2 0 2 0 0 0 0 | * 2 * * * *
... oxo ...&#x ♦ 1 2 1 | 0 2 1 1 0 2 0 | 0 1 0 2 0 1 0 | * * 4 * * *
... ... oxo&#x ♦ 1 2 1 | 0 2 1 0 1 2 0 | 0 0 1 2 0 0 1 | * * * 4 * *
.ox .xo ...&#x ♦ 0 2 2 | 0 0 0 1 0 4 1 | 0 0 0 0 2 2 0 | * * * * 2 *
.ox ... .xo&#x ♦ 0 2 2 | 0 0 0 0 1 4 1 | 0 0 0 0 2 0 2 | * * * * * 2
or
o.. o.. o.. & | 4 * ♦ 1 4 1 0 0 | 4 2 2 4 | 2 2 2 2
.o. .o. .o. | * 4 ♦ 0 4 0 1 1 | 2 4 4 2 | 2 2 2 2
-----------------+-----+------------+---------+--------
x.. ... ... & | 2 0 | 2 * * * * | 4 0 0 0 | 2 2 0 0
oo. oo. oo.&#x & | 1 1 | * 16 * * * | 1 1 1 1 | 1 1 1 1
o.o o.o o.o&#x | 2 0 | * * 2 * * | 0 0 0 4 | 0 0 2 2
... .x. ... | 0 2 | * * * 2 * | 0 4 0 0 | 2 0 2 0
... ... .x. | 0 2 | * * * * 2 | 0 0 4 0 | 0 2 0 2
-----------------+-----+------------+---------+--------
xo. ... ...&#x & | 2 1 | 1 2 0 0 0 | 8 * * * | 1 1 0 0
... ox. ...&#x & | 1 2 | 0 2 0 1 0 | * 8 * * | 1 0 1 0
... ... ox.&#x & | 1 2 | 0 2 0 0 1 | * * 8 * | 0 1 0 1
ooo ooo ooo&#x | 2 1 | 0 2 1 0 0 | * * * 8 | 0 0 1 1
-----------------+-----+------------+---------+--------
xo. ox. ...&#x & ♦ 2 2 | 1 4 0 1 0 | 2 2 0 0 | 4 * * *
xo. ... ox.&#x & ♦ 2 2 | 1 4 0 0 1 | 2 0 2 0 | * 4 * *
... oxo ...&#x ♦ 2 2 | 0 4 1 1 0 | 0 2 0 2 | * * 4 *
... ... oxo&#x ♦ 2 2 | 0 4 1 0 1 | 0 0 2 2 | * * * 4
xoo3oox oqo&#xt → both heights = 1/sqrt(6) = 0.408248
({3} || ortho q-line || dual {3})
o..3o.. o.. | 3 * * ♦ 2 2 2 0 0 | 1 4 2 1 4 0 0 | 2 4 2 0
.o.3.o. .o. | * 2 * ♦ 0 3 0 3 0 | 0 3 0 0 6 3 0 | 1 3 3 1
..o3..o ..o | * * 3 ♦ 0 0 2 2 2 | 0 0 1 2 4 4 1 | 0 2 4 2
----------------+-------+-----------+----------------+--------
x.. ... ... | 2 0 0 | 3 * * * * | 1 2 1 0 0 0 0 | 2 2 0 0
oo.3oo. oo.&#x | 1 1 0 | * 6 * * * | 0 2 0 0 2 0 0 | 1 2 1 0
o.o3o.o o.o&#x | 1 0 1 | * * 6 * * | 0 0 1 1 2 0 0 | 0 2 2 0
.oo3.oo .oo&#x | 0 1 1 | * * * 6 * | 0 0 0 0 2 2 0 | 0 1 2 1
... ..x ... | 0 0 2 | * * * * 3 | 0 0 0 1 0 2 1 | 0 0 2 2
----------------+-------+-----------+----------------+--------
x..3o.. ... | 3 0 0 | 3 0 0 0 0 | 1 * * * * * * | 2 0 0 0
xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 | * 6 * * * * * | 1 1 0 0
x.o ... ...&#x | 2 0 1 | 1 0 2 0 0 | * * 3 * * * * | 0 2 0 0
... o.x ...&#x | 1 0 2 | 0 0 2 0 1 | * * * 3 * * * | 0 0 2 0
ooo3ooo ooo&#xt | 1 1 1 | 0 1 1 1 0 | * * * * 12 * * | 0 1 1 0
... .ox ...&#x | 0 1 2 | 0 0 0 2 1 | * * * * * 6 * | 0 0 1 1
..o3..x ... | 0 0 3 | 0 0 0 0 3 | * * * * * * 1 | 0 0 0 2
----------------+-------+-----------+----------------+--------
xo.3oo. ...&#x ♦ 3 1 0 | 3 3 0 0 0 | 1 3 0 0 0 0 0 | 2 * * *
xoo ... ...&#xt ♦ 2 1 1 | 1 2 2 1 0 | 0 1 1 0 2 0 0 | * 6 * *
... oox ...&#xt ♦ 1 1 2 | 0 1 2 2 1 | 0 0 0 1 2 1 0 | * * 6 *
.oo3.ox ...&#x ♦ 0 1 3 | 0 0 0 3 3 | 0 0 0 0 0 3 1 | * * * 2
or
o..3o.. o.. & | 6 * ♦ 2 2 2 | 1 4 3 4 | 2 6
.o.3.o. .o. | * 2 ♦ 0 6 0 | 0 6 0 6 | 2 6
-------------------+-----+--------+-----------+-----
x.. ... ... & | 2 0 | 6 * * | 1 2 1 0 | 2 2
oo.3oo. oo.&#x & | 1 1 | * 12 * | 0 2 0 2 | 1 3
o.o3o.o o.o&#x | 2 0 | * * 6 | 0 0 2 2 | 0 4
-------------------+-----+--------+-----------+-----
x..3o.. ... & | 3 0 | 3 0 0 | 2 * * * | 2 0
xo. ... ...&#x & | 2 1 | 1 2 0 | * 12 * * | 1 1
x.o ... ...&#x & | 3 0 | 1 0 2 | * * 6 * | 0 2
ooo3ooo ooo&#xt | 2 1 | 0 2 1 | * * * 12 | 0 2
-------------------+-----+--------+-----------+-----
xo.3oo. ...&#x & ♦ 3 1 | 3 3 0 | 1 3 0 0 | 4 *
xoo ... ...&#xt & ♦ 3 1 | 1 3 2 | 0 1 1 2 | * 12
oxoo3ooox&#xr → all heights = sqrt(2/3) = 0.816497
(pt || ({3} || inv {3}) || pt)
o...3o... & | 2 * ♦ 3 3 0 0 | 3 3 6 0 0 | 1 1 3 3
.o..3.o.. & | * 6 ♦ 1 1 2 2 | 2 2 4 1 3 | 1 1 3 3
----------------+-----+---------+------------+--------
oo..3oo..&#x & | 1 1 | 6 * * * | 2 0 2 0 0 | 1 0 2 1
o.o.3o.o.&#x & | 1 1 | * 6 * * | 0 2 2 0 0 | 0 1 1 2
.x.. .... & | 0 2 | * * 6 * | 1 1 0 1 1 | 1 1 1 1
.oo.3.oo.&#x & | 0 2 | * * * 6 | 0 0 2 0 2 | 0 0 2 2
----------------+-----+---------+------------+--------
ox.. ....&#x & | 1 2 | 2 0 1 0 | 6 * * * * | 1 0 1 0
.... o.x.&#x & | 1 2 | 0 2 1 0 | * 6 * * * | 0 1 0 1
ooo.3ooo.&#x & | 1 2 | 1 1 0 1 | * * 12 * * | 0 0 1 1
.x..3.o.. & | 0 3 | 0 0 3 0 | * * * 2 * | 1 1 0 0
.xo. .... & | 0 3 | 0 0 1 2 | * * * * 6 | 0 0 1 1
----------------+-----+---------+------------+--------
ox..3oo..&#x & ♦ 1 3 | 3 0 3 0 | 3 0 0 1 0 | 2 * * *
o.o.3o.x.&#x & ♦ 1 3 | 0 3 3 0 | 0 3 0 1 0 | * 2 * *
oxo. ....&#x & ♦ 1 3 | 2 1 1 2 | 1 0 2 0 1 | * * 6 *
.... oox.&#x & ♦ 1 3 | 1 2 1 2 | 0 1 2 0 1 | * * * 6