Acronym | ope, K-4.11 (alt: dytpuf) | |||||||||||||||
Name |
octahedron prism, vertex figure of rat, dyadic tegmipucofastegium, equatorial cross-section of tet-first dot | |||||||||||||||
Segmentochoron display | ||||||||||||||||
Cross sections |
© | |||||||||||||||
Circumradius | sqrt(3)/2 = 0.866025 | |||||||||||||||
Lace city in approx. ASCII-art |
x3o x3o -- x x3o (trip) o3x o3x -- x o3x (gyro trip) | | | +-- s2s3s (oct) +-------- s2s3s (oct) | |||||||||||||||
Coordinates | (1/sqrt(2), 0, 0, 1/2) & all permutations in all but last coord., all changes of sign | |||||||||||||||
Volume | sqrt(2)/3 = 0.471405 | |||||||||||||||
General of army | (is itself convex) | |||||||||||||||
Colonel of regiment |
(is itself locally convex – uniform polychoral members:
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Dual | cute | |||||||||||||||
Dihedral angles | ||||||||||||||||
Face vector | 12, 30, 28, 10 | |||||||||||||||
Confer |
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External links |
It should be mentioned that into ope a q-tet could be vertex-inscribed: within either base an opposite vertex-pair has distance q each; if those pairs would be chosen mutually orthogonal in either base, then the distance from such a vertex to the non-orthogonal one within that base clearly is x (an edge length) and the distance from that point to the opposite base too is x, thus resulting totally in q again. – In fact this best can be seen in the representation of ope as x o o || x x x || x o o where that q-tet is easily given by the alternate vertices of the equatorial cube. (An according tetradiminishing however necessarily yields q-edges in its hull.)
Incidence matrix according to Dynkin symbol
x x3o4o . . . . | 12 ♦ 1 4 | 4 4 | 4 1 --------+----+------+-------+---- x . . . | 2 | 6 * | 4 0 | 4 0 . x . . | 2 | * 24 | 1 2 | 2 1 --------+----+------+-------+---- x x . . | 4 | 2 2 | 12 * | 2 0 . x3o . | 3 | 0 3 | * 16 | 1 1 --------+----+------+-------+---- x x3o . ♦ 6 | 3 6 | 3 2 | 8 * . x3o4o ♦ 6 | 0 12 | 0 8 | * 2
x x3o4/3o . . . . | 12 ♦ 1 4 | 4 4 | 4 1 ----------+----+------+-------+---- x . . . | 2 | 6 * | 4 0 | 4 0 . x . . | 2 | * 24 | 1 2 | 2 1 ----------+----+------+-------+---- x x . . | 4 | 2 2 | 12 * | 2 0 . x3o . | 3 | 0 3 | * 16 | 1 1 ----------+----+------+-------+---- x x3o . ♦ 6 | 3 6 | 3 2 | 8 * . x3o4/3o ♦ 6 | 0 12 | 0 8 | * 2
x x3/2o4o . . . . | 12 ♦ 1 4 | 4 4 | 4 1 ----------+----+------+-------+---- x . . . | 2 | 6 * | 4 0 | 4 0 . x . . | 2 | * 24 | 1 2 | 2 1 ----------+----+------+-------+---- x x . . | 4 | 2 2 | 12 * | 2 0 . x3/2o . | 3 | 0 3 | * 16 | 1 1 ----------+----+------+-------+---- x x3/2o . ♦ 6 | 3 6 | 3 2 | 8 * . x3/2o4o ♦ 6 | 0 12 | 0 8 | * 2
x x3/2o4/3o . . . . | 12 ♦ 1 4 | 4 4 | 4 1 ------------+----+------+-------+---- x . . . | 2 | 6 * | 4 0 | 4 0 . x . . | 2 | * 24 | 1 2 | 2 1 ------------+----+------+-------+---- x x . . | 4 | 2 2 | 12 * | 2 0 . x3/2o . | 3 | 0 3 | * 16 | 1 1 ------------+----+------+-------+---- x x3/2o . ♦ 6 | 3 6 | 3 2 | 8 * . x3/2o4/3o ♦ 6 | 0 12 | 0 8 | * 2
x o3x3o . . . . | 12 ♦ 1 4 | 4 2 2 | 2 2 1 --------+----+------+--------+------ x . . . | 2 | 6 * | 4 0 0 | 2 2 0 . . x . | 2 | * 24 | 1 1 1 | 1 1 1 --------+----+------+--------+------ x . x . | 4 | 2 2 | 12 * * | 1 1 0 . o3x . | 3 | 0 3 | * 8 * | 1 0 1 . . x3o | 3 | 0 3 | * * 8 | 0 1 1 --------+----+------+--------+------ x o3x . ♦ 6 | 3 6 | 3 2 0 | 4 * * x . x3o ♦ 6 | 3 6 | 3 0 2 | * 4 * . o3x3o ♦ 6 | 0 12 | 0 4 4 | * * 2
x o3x3/2o . . . . | 12 ♦ 1 4 | 4 2 2 | 2 2 1 ----------+----+------+--------+------ x . . . | 2 | 6 * | 4 0 0 | 2 2 0 . . x . | 2 | * 24 | 1 1 1 | 1 1 1 ----------+----+------+--------+------ x . x . | 4 | 2 2 | 12 * * | 1 1 0 . o3x . | 3 | 0 3 | * 8 * | 1 0 1 . . x3/2o | 3 | 0 3 | * * 8 | 0 1 1 ----------+----+------+--------+------ x o3x . ♦ 6 | 3 6 | 3 2 0 | 4 * * x . x3/2o ♦ 6 | 3 6 | 3 0 2 | * 4 * . o3x3/2o ♦ 6 | 0 12 | 0 4 4 | * * 2
x o3/2x3/2o . . . . | 12 ♦ 1 4 | 4 2 2 | 2 2 1 ------------+----+------+--------+------ x . . . | 2 | 6 * | 4 0 0 | 2 2 0 . . x . | 2 | * 24 | 1 1 1 | 1 1 1 ------------+----+------+--------+------ x . x . | 4 | 2 2 | 12 * * | 1 1 0 . o3/2x . | 3 | 0 3 | * 8 * | 1 0 1 . . x3/2o | 3 | 0 3 | * * 8 | 0 1 1 ------------+----+------+--------+------ x o3/2x . ♦ 6 | 3 6 | 3 2 0 | 4 * * x . x3/2o ♦ 6 | 3 6 | 3 0 2 | * 4 * . o3/2x3/2o ♦ 6 | 0 12 | 0 4 4 | * * 2
x s2s3s . demi( . . . ) | 12 ♦ 1 1 1 2 | 1 1 1 2 3 | 1 1 3 ----------------+----+----------+------------+------ . s2s . ♦ 2 | 6 * * * | 1 0 0 0 2 | 0 1 2 . s 2 s ♦ 2 | * 6 * * | 0 1 0 0 2 | 0 1 2 x demi( . . . ) | 2 | * * 6 * | 1 1 0 2 0 | 1 0 3 . sefa( . s3s ) | 2 | * * * 12 | 0 0 1 1 1 | 1 1 1 ----------------+----+----------+------------+------ x s2s . | 4 | 2 0 2 0 | 3 * * * * | 0 0 2 x s 2 s | 4 | 0 2 2 0 | * 3 * * * | 0 0 2 . . s3s ♦ 3 | 0 0 0 3 | * * 4 * * | 1 1 0 x sefa( . s3s ) | 4 | 0 0 2 2 | * * * 6 * | 1 0 1 . sefa( s2s3s ) | 3 | 1 1 0 1 | * * * * 12 | 0 1 1 ----------------+----+----------+------------+------ x . s3s ♦ 6 | 0 0 3 6 | 0 0 2 3 0 | 2 * * . s2s3s ♦ 6 | 3 3 0 6 | 0 0 2 0 6 | * 2 * x sefa( s2s3s ) ♦ 6 | 2 2 3 2 | 1 1 0 1 2 | * * 6
x s2s6o . demi( . . . ) | 12 ♦ 2 1 2 | 2 1 2 3 | 1 1 3 ----------------+----+---------+----------+------ . s2s . ♦ 2 | 12 * * | 1 0 0 2 | 0 1 2 x demi( . . . ) | 2 | * 6 * | 2 0 2 0 | 1 0 3 . sefa( . s6o ) | 2 | * * 12 | 0 1 1 1 | 1 1 1 ----------------+----+---------+----------+------ x s2s . | 4 | 2 2 0 | 6 * * * | 0 0 2 . . s6o ♦ 3 | 0 0 3 | * 4 * * | 1 1 0 x sefa( . s6o ) | 4 | 0 2 2 | * * 6 * | 1 0 1 . sefa( s2s6o ) | 3 | 2 0 1 | * * * 12 | 0 1 1 ----------------+----+---------+----------+------ x . s6o ♦ 6 | 0 3 6 | 0 2 3 0 | 2 * * . s2s6o ♦ 6 | 6 0 6 | 0 2 0 6 | * 2 * x sefa( s2s6o ) ♦ 6 | 4 3 2 | 2 0 1 2 | * * 6
xx3oo4oo&#x → height = 1
(oct || oct)
o.3o.4o. | 6 * ♦ 4 1 0 | 4 4 0 | 1 4 0
.o3.o4.o | * 6 ♦ 0 1 4 | 0 4 4 | 0 4 1
------------+-----+---------+--------+------
x. .. .. | 2 0 | 12 * * | 2 1 0 | 1 2 0
oo3oo4oo&#x | 1 1 | * 6 * | 0 4 0 | 0 4 0
.x .. .. | 0 2 | * * 12 | 0 1 2 | 0 2 1
------------+-----+---------+--------+------
x.3o. .. | 3 0 | 3 0 0 | 8 * * | 1 1 0
xx .. ..&#x | 2 2 | 1 2 1 | * 12 * | 0 2 0
.x3.o .. | 0 3 | 0 0 3 | * * 8 | 0 1 1
------------+-----+---------+--------+------
x.3o.4o. ♦ 6 0 | 12 0 0 | 8 0 0 | 1 * *
xx3oo ..&#x ♦ 3 3 | 3 3 3 | 1 3 1 | * 8 *
.x3.o4.o ♦ 0 6 | 0 0 12 | 0 0 8 | * * 1
oo3xx3oo&#x → height = 1
(oct || oct)
o.3o.3o. | 6 * ♦ 4 1 0 | 2 2 4 0 0 | 1 2 2 0
.o3.o3.o | * 6 ♦ 0 1 4 | 0 0 4 2 2 | 0 2 2 1
------------+-----+---------+------------+--------
.. x. .. | 2 0 | 12 * * | 1 1 1 0 0 | 1 1 1 0
oo3oo3oo&#x | 1 1 | * 6 * | 0 0 4 0 0 | 0 2 2 0
.. .x .. | 0 2 | * * 12 | 0 0 1 1 1 | 0 1 1 1
------------+-----+---------+------------+--------
o.3x. .. | 3 0 | 3 0 0 | 4 * * * * | 1 1 0 0
.. x.3o. | 3 0 | 3 0 0 | * 4 * * * | 1 0 1 0
.. xx ..&#x | 2 2 | 1 2 1 | * * 12 * * | 0 1 1 0
.o3.x .. | 0 3 | 0 0 3 | * * * 4 * | 0 1 0 1
.. .x3.o | 0 3 | 0 0 3 | * * * * 4 | 0 0 1 1
------------+-----+---------+------------+--------
o.3x.3o. ♦ 6 0 | 12 0 0 | 4 4 0 0 0 | 1 * * *
oo3xx ..&#x ♦ 3 3 | 3 3 3 | 1 0 3 1 0 | * 4 * *
.. xx3oo&#x ♦ 3 3 | 3 3 3 | 0 1 3 0 1 | * * 4 *
.o3.x3.o ♦ 0 6 | 0 0 12 | 0 0 0 4 4 | * * * 1
xx xo3ox&#x → height = sqrt(2/3) = 0.816497
(trip || gyro trip)
o. o.3o. | 6 * ♦ 1 2 2 0 0 | 2 1 2 2 1 0 0 | 1 2 1 1 0
.o .o3.o | * 6 ♦ 0 0 2 1 2 | 0 0 2 1 2 2 1 | 0 1 2 1 1
------------+-----+------------+---------------+----------
x. .. .. | 2 0 | 3 * * * * | 2 0 2 0 0 0 0 | 1 2 1 0 0
.. x. .. | 2 0 | * 6 * * * | 1 1 0 1 0 0 0 | 1 1 0 1 0
oo oo3oo&#x | 1 1 | * * 12 * * | 0 0 1 1 1 0 0 | 0 1 1 1 0
.x .. .. | 0 2 | * * * 3 * | 0 0 2 0 0 2 0 | 0 1 2 0 1
.. .. .x | 0 2 | * * * * 6 | 0 0 0 0 1 1 1 | 0 0 1 1 1
------------+-----+------------+---------------+----------
x. x. .. | 4 0 | 2 2 0 0 0 | 3 * * * * * * | 1 1 0 0 0
.. x.3o. | 3 0 | 0 3 0 0 0 | * 2 * * * * * | 1 0 0 1 0
xx .. ..&#x | 2 2 | 1 0 2 1 0 | * * 6 * * * * | 0 1 1 0 0
.. xo ..&#x | 2 1 | 0 1 2 0 0 | * * * 6 * * * | 0 1 0 1 0
.. .. ox&#x | 1 2 | 0 0 2 0 1 | * * * * 6 * * | 0 0 1 1 0
.x .. .x | 0 4 | 0 0 0 2 2 | * * * * * 3 * | 0 0 1 0 1
.. .o3.x | 0 3 | 0 0 0 0 3 | * * * * * * 2 | 0 0 0 1 1
------------+-----+------------+---------------+----------
x. x.3o. ♦ 6 0 | 3 6 0 0 0 | 3 2 0 0 0 0 0 | 1 * * * *
xx xo ..&#x ♦ 4 2 | 2 2 4 1 0 | 1 0 2 2 0 0 0 | * 3 * * *
xx .. ox&#x ♦ 2 4 | 1 0 4 2 2 | 0 0 2 0 2 1 0 | * * 3 * *
.. xo3ox&#x ♦ 3 3 | 0 3 6 0 3 | 0 1 0 3 3 0 1 | * * * 2 *
.x .o3.x ♦ 0 6 | 0 0 0 3 6 | 0 0 0 0 0 3 2 | * * * * 1
s2s3s || s2s3s → height = 1 (oct || oct) demi( . . . ) | 6 * ♦ 1 1 2 1 0 0 0 | 1 3 1 1 2 0 0 | 1 1 3 0 demi( . . . ) | * 6 ♦ 0 0 0 1 1 1 2 | 0 0 1 1 2 1 3 | 0 1 3 1 ------------------------------+-----+---------------+---------------+-------- s2s . ♦ 2 0 | 3 * * * * * * | 0 2 1 0 0 0 0 | 1 0 1 0 s 2 s ♦ 2 0 | * 3 * * * * * | 0 2 0 1 0 0 0 | 1 0 1 0 sefa( . s3s ) | 2 0 | * * 6 * * * * | 1 1 0 0 1 0 0 | 1 1 1 0 demi( . . . ) || demi( . . . ) | 1 1 | * * * 6 * * * | 0 0 1 1 2 0 0 | 0 1 3 0 s2s . ♦ 0 2 | * * * * 3 * * | 0 0 1 0 0 0 2 | 0 0 2 1 s 2 s ♦ 0 2 | * * * * * 3 * | 0 0 0 1 0 0 2 | 0 0 2 1 sefa( . s3s ) | 0 2 | * * * * * * 6 | 0 0 0 0 1 1 1 | 0 1 1 1 ------------------------------+-----+---------------+---------------+-------- . s3s ♦ 3 0 | 0 0 3 0 0 0 0 | 2 * * * * * * | 1 1 0 0 sefa( s2s3s ) | 3 0 | 1 1 1 0 0 0 0 | * 6 * * * * * | 1 0 1 0 s2s . || s2s . | 2 2 | 1 0 0 2 1 0 0 | * * 3 * * * * | 0 0 2 0 s 2 s || s 2 s | 2 2 | 0 1 0 2 0 1 0 | * * * 3 * * * | 0 0 2 0 sefa( . s3s ) || sefa( . s3s ) | 2 2 | 0 0 1 2 0 0 1 | * * * * 6 * * | 0 1 1 0 . s3s ♦ 0 3 | 0 0 0 0 0 0 3 | * * * * * 2 * | 0 1 0 1 sefa( s2s3s ) | 0 3 | 0 0 0 0 1 1 1 | * * * * * * 6 | 0 0 1 1 ------------------------------+-----+---------------+---------------+-------- s2s3s ♦ 6 0 | 3 3 6 0 0 0 0 | 2 6 0 0 0 0 0 | 1 * * * . s3s || . s3s ♦ 3 3 | 0 0 3 3 0 0 3 | 1 0 0 0 3 1 0 | * 2 * * sefa( s2s3s ) || sefa( s2s3s ) ♦ 3 3 | 1 1 1 3 1 1 1 | 0 1 1 1 1 0 1 | * * 6 * s2s3s ♦ 0 6 | 0 0 0 0 n n 6 | 0 0 0 0 0 2 6 | * * * 1
s2s6o || s2s6o → height = 1 (oct || oct: triangular appip) demi( . . . ) | 6 * ♦ 2 2 1 0 0 | 1 3 2 2 0 0 | 1 1 3 0 demi( . . . ) | * 6 ♦ 0 0 1 2 2 | 0 0 2 2 1 3 | 0 1 3 1 ------------------------------+-----+-----------+-------------+-------- s2s . ♦ 2 0 | 6 * * * * | 0 2 1 0 0 0 | 1 0 2 0 sefa( . s6o ) | 2 0 | * 6 * * * | 1 1 0 1 0 0 | 1 1 1 0 demi( . . . ) || demi( . . . ) | 1 1 | * * 6 * * | 0 0 2 2 0 0 | 0 1 3 0 s2s . ♦ 0 2 | * * * 6 * | 0 0 1 0 0 2 | 0 0 2 1 sefa( . s6o ) | 0 2 | * * * * 6 | 0 0 0 1 1 1 | 0 1 1 1 ------------------------------+-----+-----------+-------------+-------- . s6o ♦ 3 0 | 0 3 0 0 0 | 2 * * * * * | 1 1 0 0 sefa( s2s6o ) | 3 0 | 2 1 0 0 0 | * 6 * * * * | 1 0 1 0 s2s . || s2s . | 2 2 | 1 0 2 1 0 | * * 6 * * * | 0 0 2 0 sefa( . s6o ) || sefa( . s6o ) | 2 2 | 0 1 2 0 1 | * * * 6 * * | 0 1 1 0 . s6o ♦ 0 3 | 0 0 0 0 3 | * * * * 2 * | 0 1 0 1 sefa( s2s6o ) | 0 3 | 0 0 0 2 1 | * * * * * 6 | 0 0 1 1 ------------------------------+-----+-----------+-------------+-------- s2s6o ♦ 6 0 | 6 6 0 0 0 | 2 6 0 0 0 0 | 1 * * * . s6o || . s6o ♦ 3 3 | 0 3 3 0 3 | 1 0 0 3 1 0 | * 2 * * sefa( s2s6o ) || sefa( s2s6o ) ♦ 3 3 | 2 1 3 2 1 | 0 1 2 1 0 1 | * * 6 * s2s6o ♦ 0 6 | 0 0 0 6 6 | 0 0 0 0 2 6 | * * * 1
xxx oxo4ooo&#xt → both heights = 1/sqrt(2) = 0.707107 (line || pseudo cube || line: dyadic tepuf) o.. o..4o.. | 2 * * ♦ 1 4 0 0 0 0 | 4 4 0 0 0 | 4 1 0 .o. .o.4.o. | * 8 * ♦ 0 1 1 2 1 0 | 1 2 2 1 2 | 2 1 2 ..o ..o4..o | * * 2 ♦ 0 0 0 0 4 1 | 0 0 0 4 4 | 0 1 4 ----------------+-------+-------------+-----------+------ x.. ... ... | 2 0 0 | 1 * * * * * | 4 0 0 0 0 | 4 0 0 oo. oo.4oo.&#x | 1 1 0 | * 8 * * * * | 1 2 0 0 0 | 2 1 0 .x. ... ... | 0 2 0 | * * 4 * * * | 1 0 2 1 0 | 2 0 2 ... .x. ... | 0 2 0 | * * * 8 * * | 0 1 1 0 1 | 1 1 1 .oo .oo4.oo&#x | 0 1 1 | * * * * 8 * | 0 0 0 1 2 | 0 1 2 ..x ... ... | 0 0 2 | * * * * * 1 | 0 0 0 4 0 | 0 0 4 ----------------+-------+-------------+-----------+------ xx. ... ...&#x | 2 2 0 | 1 2 1 0 0 0 | 4 * * * * | 2 0 0 ... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 | * 8 * * * | 1 1 0 .x. .x. ... | 0 4 0 | 0 0 2 2 0 0 | * * 4 * * | 1 0 1 .xx ... ...&#x | 0 2 2 | 0 0 1 0 2 1 | * * * 4 * | 0 0 2 ... .xo ...&#x | 0 2 1 | 0 0 0 1 2 0 | * * * * 8 | 0 1 1 ----------------+-------+-------------+-----------+------ xx. ox. ...&#x ♦ 2 4 0 | 1 4 2 2 0 0 | 2 2 1 0 0 | 4 * * ... oxo4ooo&#xt ♦ 1 4 1 | 0 4 0 4 4 0 | 0 4 0 0 4 | * 2 * .xx .xo ...&#x ♦ 0 4 2 | 0 0 2 2 4 1 | 0 0 1 2 2 | * * 4
or o.. o..4o.. & | 4 * ♦ 1 4 0 0 | 4 4 0 | 4 1 .o. .o.4.o. | * 8 ♦ 0 2 1 2 | 2 4 2 | 4 1 ------------------+-----+----------+--------+---- x.. ... ... & | 2 0 | 2 * * * | 4 0 0 | 4 0 oo. oo.4oo.&#x & | 1 1 | * 16 * * | 1 2 0 | 2 1 .x. ... ... | 0 2 | * * 4 * | 2 0 2 | 4 0 ... .x. ... | 0 2 | * * * 8 | 0 2 1 | 2 1 ------------------+-----+----------+--------+---- xx. ... ...&#x & | 2 2 | 1 2 1 0 | 8 * * | 2 0 ... ox. ...&#x & | 1 2 | 0 2 0 1 | * 16 * | 1 1 .x. .x. ... | 0 4 | 0 0 2 2 | * * 4 | 2 0 ------------------+-----+----------+--------+---- xx. ox. ...&#x & ♦ 2 4 | 1 4 2 2 | 2 2 1 | 8 * ... oxo4ooo&#xt ♦ 2 4 | 0 8 0 4 | 0 8 0 | * 2
xxx oxo oxo&#xt → both heights = 1/sqrt(2) = 0.707107 (line || pseudo cube || line: dyadic tepuf) o.. o.. o.. | 2 * * ♦ 1 4 0 0 0 0 0 | 4 2 2 0 0 0 0 0 | 2 2 1 0 0 .o. .o. .o. | * 8 * ♦ 0 1 1 1 1 1 0 | 1 1 1 1 1 1 1 1 | 1 1 1 1 1 ..o ..o ..o | * * 2 ♦ 0 0 0 0 0 4 1 | 0 0 0 0 0 4 2 2 | 0 0 1 2 2 ----------------+-------+---------------+-----------------+---------- x.. ... ... | 2 0 0 | 1 * * * * * * | 4 0 0 0 0 0 0 0 | 2 2 0 0 0 oo. oo. oo.&#x | 1 1 0 | * 8 * * * * * | 1 1 1 0 0 0 0 0 | 1 1 1 0 0 .x. ... ... | 0 2 0 | * * 4 * * * * | 1 0 0 1 1 1 0 0 | 1 1 0 1 1 ... .x. ... | 0 2 0 | * * * 4 * * * | 0 1 0 1 0 0 1 0 | 1 0 1 1 0 ... ... .x. | 0 2 0 | * * * * 4 * * | 0 0 1 0 1 0 0 1 | 0 1 1 0 1 .oo .oo .oo&#x | 0 1 1 | * * * * * 8 * | 0 0 0 0 0 1 1 1 | 0 0 1 1 1 ..x ... ... | 0 0 2 | * * * * * * 1 | 0 0 0 0 0 4 0 0 | 0 0 0 2 2 ----------------+-------+---------------+-----------------+---------- xx. ... ...&#x | 2 2 0 | 1 2 1 0 0 0 0 | 4 * * * * * * * | 1 1 0 0 0 ... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 0 | * 4 * * * * * * | 1 0 1 0 0 ... ... ox.&#x | 1 2 0 | 0 2 0 0 1 0 0 | * * 4 * * * * * | 0 1 1 0 0 .x. .x. ... | 0 4 0 | 0 0 2 2 0 0 0 | * * * 2 * * * * | 1 0 0 1 0 .x. ... .x. | 0 4 0 | 0 0 2 0 2 0 0 | * * * * 2 * * * | 0 1 0 0 1 .xx ... ...&#x | 0 2 2 | 0 0 1 0 0 2 1 | * * * * * 4 * * | 0 0 0 1 1 ... .xo ...&#x | 0 2 1 | 0 0 0 1 0 2 0 | * * * * * * 4 * | 0 0 1 1 0 ... ... .xo&#x | 0 2 1 | 0 0 0 0 1 2 0 | * * * * * * * 4 | 0 0 1 0 1 ----------------+-------+---------------+-----------------+---------- xx. ox. ...&#x ♦ 2 4 0 | 1 4 2 2 0 0 0 | 2 2 0 1 0 0 0 0 | 2 * * * * xx. ... ox.&#x ♦ 2 4 0 | 1 4 2 0 2 0 0 | 2 0 2 0 1 0 0 0 | * 2 * * * ... oxo oxo&#xt ♦ 1 4 1 | 0 4 0 2 2 4 0 | 0 2 2 0 0 0 2 2 | * * 2 * * .xx .xo ...&#x ♦ 0 4 2 | 0 0 2 2 0 4 1 | 0 0 0 1 0 2 2 0 | * * * 2 * .xx ... .xo&#x ♦ 0 4 2 | 0 0 2 0 2 4 1 | 0 0 0 0 1 2 0 2 | * * * * 2
or o.. o.. o.. & | 4 * ♦ 1 4 0 0 0 | 4 2 2 0 0 | 2 2 1 .o. .o. .o. | * 8 ♦ 0 2 1 1 1 | 2 2 2 1 1 | 2 2 1 ------------------+-----+------------+-----------+------ x.. ... ... & | 2 0 | 2 * * * * | 4 0 0 0 0 | 2 2 0 oo. oo. oo.&#x & | 1 1 | * 16 * * * | 1 1 1 0 0 | 1 1 1 .x. ... ... | 0 2 | * * 4 * * | 2 0 0 1 1 | 2 2 0 ... .x. ... | 0 2 | * * * 4 * | 0 2 0 1 0 | 2 0 1 ... ... .x. | 0 2 | * * * * 4 | 0 0 2 0 1 | 0 2 1 ------------------+-----+------------+-----------+------ xx. ... ...&#x & | 2 2 | 1 2 1 0 0 | 8 * * * * | 1 1 0 ... ox. ...&#x & | 1 2 | 0 2 0 1 0 | * 8 * * * | 1 0 1 ... ... ox.&#x & | 1 2 | 0 2 0 0 1 | * * 8 * * | 0 1 1 .x. .x. ... | 0 4 | 0 0 2 2 0 | * * * 2 * | 2 0 0 .x. ... .x. | 0 4 | 0 0 2 0 2 | * * * * 2 | 0 2 0 ------------------+-----+------------+-----------+------ xx. ox. ...&#x & ♦ 2 4 | 1 4 2 2 0 | 2 2 0 1 0 | 4 * * xx. ... ox.&#x & ♦ 2 4 | 1 4 2 0 2 | 2 0 2 0 1 | * 4 * ... oxo oxo&#xt ♦ 2 4 | 0 8 0 2 2 | 0 4 4 0 0 | * * 2
oqo xox xxx&#xt → both heights = 1/2 ({4} || ortho pseudo (q,x)-{4} || {4}) o.. o.. o.. & | 8 * ♦ 1 1 2 1 0 | 1 2 2 2 1 | 1 2 2 .o. .o. .o. | * 4 ♦ 0 0 4 0 1 | 0 2 2 4 0 | 1 2 2 ------------------+-----+------------+-----------+------ ... x.. ... & | 2 0 | 4 * * * * | 1 0 2 0 0 | 1 2 0 ... ... x.. & | 2 0 | * 4 * * * | 1 0 0 2 1 | 0 2 2 oo. oo. oo.&#x & | 1 1 | * * 16 * * | 0 1 1 1 0 | 1 1 1 o.o o.o o.o&#x | 2 0 | * * * 4 * | 0 2 0 0 1 | 1 0 2 ... ... .x. | 0 2 | * * * * 2 | 0 0 0 4 0 | 0 2 2 ------------------+-----+------------+-----------+------ ... x.. x.. & | 4 0 | 2 2 0 0 0 | 2 * * * * | 0 2 0 ooo ooo ooo&#xt | 2 1 | 0 0 2 1 0 | * 8 * * * | 1 0 1 ... xo. ...&#x & | 2 1 | 1 0 2 0 0 | * * 8 * * | 1 1 0 ... ... xx.&#x & | 2 2 | 0 1 2 0 1 | * * * 8 * | 0 1 1 ... ... x.x&#x | 4 0 | 0 2 0 2 0 | * * * * 2 | 0 0 2 ------------------+-----+------------+-----------+------ oqo xox ...&#xt ♦ 4 2 | 2 0 8 2 0 | 0 4 4 0 0 | 2 * * ... xo. xx.&#x & ♦ 4 2 | 2 2 4 0 1 | 1 0 2 2 0 | * 4 * ... ... xxx&#x ♦ 4 2 | 0 2 4 2 1 | 0 2 0 2 1 | * * 4
xxx qoo oqo ooq&#zx → height = 0 (tegum sum of 3 ortho (q,x)-{4}) o.. o.. o.. o.. | 4 * * ♦ 1 2 2 0 0 0 | 2 2 4 0 | 4 1 .o. .o. .o. .o. | * 4 * ♦ 0 2 0 1 2 0 | 2 0 4 2 | 4 1 ..o ..o ..o ..o | * * 4 ♦ 0 0 2 0 2 1 | 0 2 4 2 | 4 1 --------------------+-------+-------------+----------+---- x.. ... ... ... | 2 0 0 | 2 * * * * * | 2 2 0 0 | 4 0 oo. oo. oo. oo.&#x | 1 1 0 | * 8 * * * * | 1 0 2 0 | 2 1 o.o o.o o.o o.o&#x | 1 0 1 | * * 8 * * * | 0 1 2 0 | 2 1 .x. ... ... ... | 0 2 0 | * * * 2 * * | 2 0 0 2 | 4 0 .oo .oo .oo .oo&#x | 0 1 1 | * * * * 8 * | 0 0 2 1 | 2 1 ..x ... ... ... | 0 0 2 | * * * * * 2 | 0 2 0 2 | 4 0 --------------------+-------+-------------+----------+---- xx. ... ... ...&#x | 2 2 0 | 1 2 0 1 0 0 | 4 * * * | 2 0 x.x ... ... ...&#x | 2 0 2 | 1 0 2 0 0 1 | * 4 * * | 2 0 ooo ooo ooo ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * 16 * | 1 1 .xx ... ... ...&#x | 0 2 2 | 0 0 0 1 2 1 | * * * 4 | 2 0 --------------------+-------+-------------+----------+---- xxx ... ... ...&#x ♦ 2 2 2 | 1 2 2 1 2 1 | 1 1 2 1 | 8 * ... qoo oqo ooq&#zx ♦ 2 2 2 | 0 4 4 0 4 0 | 0 0 8 0 | * 2
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