Acronym | traf, K-4.6 | ||
Name |
triangular antifastegium, triangle oct wedge, {3} || gyro trip, {3} || oct, (vertex-) diminished rap | ||
Segmentochoron display | |||
Circumradius | sqrt(3/5) = 0.774597 | ||
Lace city in approx. ASCII-art |
x3o x3o o3x | ||
o x x o x x o o | |||
Lace hyper city in approx. ASCII-art |
| ||
Dihedral angles |
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Face vector | 9, 24, 24, 9 | ||
Confer |
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External links |
Incidence matrix according to Dynkin symbol
xoo3oxx&#x → height(1,2) = height(1,3) = sqrt(2/3) = 0.816497 height(2,3) = 1 ( {3} || (dual {3} || dual {3}) ) o..3o.. | 3 * * ♦ 2 2 2 0 0 0 | 1 2 1 2 1 2 0 0 0 | 1 1 2 1 0 .o.3.o. | * 3 * | 0 2 0 2 1 0 | 0 1 2 0 0 2 1 2 0 | 1 0 1 2 1 ..o3..o | * * 3 | 0 0 2 0 1 2 | 0 0 0 1 2 2 0 2 1 | 0 1 1 2 1 -----------+-------+-------------+-------------------+---------- x.. ... | 2 0 0 | 3 * * * * * | 1 1 0 1 0 0 0 0 0 | 1 1 1 0 0 oo.3oo.&#x | 1 1 0 | * 6 * * * * | 0 1 1 0 0 1 0 0 0 | 1 0 1 1 0 o.o3o.o&#x | 1 0 1 | * * 6 * * * | 0 0 0 1 1 1 0 0 0 | 0 1 1 1 0 ... .x. | 0 2 0 | * * * 3 * * | 0 0 1 0 0 0 1 1 0 | 1 0 0 1 1 .oo3.oo&#x | 0 1 1 | * * * * 3 * | 0 0 0 0 0 2 0 2 0 | 0 0 1 2 1 ... ..x | 0 0 2 | * * * * * 3 | 0 0 0 0 1 0 0 1 1 | 0 1 0 1 1 -----------+-------+-------------+-------------------+---------- x..3o.. | 3 0 0 | 3 0 0 0 0 0 | 1 * * * * * * * * | 1 1 0 0 0 xo. ...&#x | 2 1 0 | 1 2 0 0 0 0 | * 3 * * * * * * * | 1 0 1 0 0 ... ox.&#x | 1 2 0 | 0 2 0 1 0 0 | * * 3 * * * * * * | 1 0 0 1 0 x.o ...&#x | 2 0 1 | 1 0 2 0 0 0 | * * * 3 * * * * * | 0 1 1 0 0 ... o.x&#x | 1 0 2 | 0 0 2 0 0 1 | * * * * 3 * * * * | 0 1 0 1 0 ooo3ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * * * * 6 * * * | 0 0 1 1 0 .o.3.x. | 0 3 0 | 0 0 0 3 0 0 | * * * * * * 1 * * | 1 0 0 0 1 ... .xx&#x | 0 2 2 | 0 0 0 1 2 1 | * * * * * * * 3 * | 0 0 0 1 1 ..o3..x | 0 0 3 | 0 0 0 0 0 3 | * * * * * * * * 1 | 0 1 0 0 1 -----------+-------+-------------+-------------------+---------- xo.3ox.&#x ♦ 3 3 0 | 3 6 0 3 0 0 | 1 3 3 0 0 0 1 0 0 | 1 * * * * x.o3o.x&#x ♦ 3 0 3 | 3 0 6 0 0 3 | 1 0 0 3 3 0 0 0 1 | * 1 * * * xoo ...&#x ♦ 2 1 1 | 1 2 2 0 1 0 | 0 1 0 1 0 2 0 0 0 | * * 3 * * ... oxx&#x ♦ 1 2 2 | 0 2 2 1 2 1 | 0 0 1 0 1 2 0 1 0 | * * * 3 * .oo3.xx&#x ♦ 0 3 3 | 0 0 0 3 3 3 | 0 0 0 0 0 0 1 3 1 | * * * * 1
xo3ox ox&#x → height = sqrt(5/12) = 0.645497
({3} || gyro trip)
o.3o. o. | 3 * ♦ 2 4 0 0 | 1 4 2 2 0 0 | 2 2 1 0
.o3.o .o | * 6 | 0 2 2 1 | 0 1 2 2 1 2 | 1 1 2 1
------------+-----+----------+-------------+--------
x. .. .. | 2 0 | 3 * * * | 1 2 0 0 0 0 | 2 1 0 0
oo3oo oo&#x | 1 1 | * 12 * * | 0 1 1 1 0 0 | 1 1 1 0
.. .x .. | 0 2 | * * 6 * | 0 0 1 0 1 1 | 1 0 1 1
.. .. .x | 0 2 | * * * 3 | 0 0 0 2 0 2 | 0 1 2 1
------------+-----+----------+-------------+--------
x.3o. .. | 3 0 | 3 0 0 0 | 1 * * * * * | 2 0 0 0
xo .. ..&#x | 2 1 | 1 2 0 0 | * 6 * * * * | 1 1 0 0
.. ox ..&#x | 1 2 | 0 2 1 0 | * * 6 * * * | 1 0 1 0
.. .. ox&#x | 1 2 | 0 2 0 1 | * * * 6 * * | 0 1 1 0
.o3.x .. | 0 3 | 0 0 3 0 | * * * * 2 * | 1 0 0 1
.. .x .x | 0 4 | 0 0 2 2 | * * * * * 3 | 0 0 1 1
------------+-----+----------+-------------+--------
xo3ox ..&#x ♦ 3 3 | 3 6 3 0 | 1 3 3 0 1 0 | 2 * * *
xo .. ox&#x ♦ 2 2 | 1 4 0 1 | 0 2 0 2 0 0 | * 3 * *
.. ox ox&#x ♦ 1 4 | 0 4 2 2 | 0 0 2 2 0 1 | * * 3 *
.o3.x .x ♦ 0 6 | 0 0 6 3 | 0 0 0 0 2 3 | * * * 1
(xx)(oo) (xo)(ox)&#x → height = sqrt(5/12) = 0.645497
(trip || dual {3})
(o.)(..) (o.)(..) | 4 * * * | 1 1 1 1 1 0 0 0 0 | 1 1 1 1 1 1 1 1 0 0 0 | 1 1 1 1 1 0
(.o)(..) (.o)(..) | * 2 * * | 0 0 2 0 0 1 2 0 0 | 0 2 1 2 0 0 0 0 2 1 0 | 1 0 2 1 0 1
(..)(o.) (..)(o.) | * * 1 * ♦ 0 0 0 4 0 0 0 2 0 | 0 0 0 0 2 2 4 0 0 0 1 | 0 1 0 2 2 0
(..)(.o) (..)(.o) | * * * 2 ♦ 0 0 0 0 2 0 2 1 1 | 0 0 0 2 0 0 2 1 1 2 1 | 0 0 1 2 1 1
---------------------+---------+-------------------+-----------------------+------------
(x.)(..) (..)(..) | 2 0 0 0 | 2 * * * * * * * * | 1 1 0 0 1 0 0 1 0 0 0 | 1 1 1 0 1 0
(..)(..) (x.)(..) | 2 0 0 0 | * 2 * * * * * * * | 1 0 1 0 0 1 0 0 0 0 0 | 1 1 0 1 0 0
(oo)(..) (oo)(..)&#x | 1 1 0 0 | * * 4 * * * * * * | 0 1 1 1 0 0 0 0 0 0 0 | 1 0 1 1 0 0
(o.)(o.) (o.)(o.)&#x | 1 0 1 0 | * * * 4 * * * * * | 0 0 0 0 1 1 1 0 0 0 0 | 0 1 0 1 1 0
(o.)(.o) (o.)(.o)&#x | 1 0 0 1 | * * * * 4 * * * * | 0 0 0 1 0 0 1 1 0 0 0 | 0 0 1 1 1 0
(.x)(..) (..)(..) | 0 2 0 0 | * * * * * 1 * * * | 0 2 0 0 0 0 0 0 2 0 0 | 1 0 2 0 0 1
(.o)(.o) (.o)(.o)&#x | 0 1 0 1 | * * * * * * 4 * * | 0 0 0 1 0 0 0 0 1 1 0 | 0 0 1 1 0 1
(..)(oo) (..)(oo)&#x | 0 0 1 1 | * * * * * * * 2 * | 0 0 0 0 0 0 2 0 0 0 1 | 0 0 0 2 1 0
(..)(..) (..)(.x) | 0 0 0 2 | * * * * * * * * 1 | 0 0 0 0 0 0 0 0 0 2 1 | 0 0 0 2 0 1
---------------------+---------+-------------------+-----------------------+------------
(x.)(..) (x.)(..) | 4 0 0 0 | 2 2 0 0 0 0 0 0 0 | 1 * * * * * * * * * * | 1 1 0 0 0 0
(xx)(..) (..)(..)&#x | 2 2 0 0 | 1 0 2 0 0 1 0 0 0 | * 2 * * * * * * * * * | 1 0 1 0 0 0
(..)(..) (xo)(..)&#x | 2 1 0 0 | 0 1 2 0 0 0 0 0 0 | * * 2 * * * * * * * * | 1 0 0 1 0 0
(oo)(.o) (oo)(.o)&#x | 1 1 0 1 | 0 0 1 0 1 0 1 0 0 | * * * 4 * * * * * * * | 0 0 1 1 0 0
(x.)(o.) (..)(..)&#x | 2 0 1 0 | 1 0 0 2 0 0 0 0 0 | * * * * 2 * * * * * * | 0 1 0 0 1 0
(..)(..) (x.)(o.)&#x | 2 0 1 0 | 0 1 0 2 0 0 0 0 0 | * * * * * 2 * * * * * | 0 1 0 1 0 0
(o.)(oo) (o.)(oo)&#x | 1 0 1 1 | 0 0 0 1 1 0 0 1 0 | * * * * * * 4 * * * * | 0 0 0 1 1 0
(x.)(.o) (..)(..)&#x | 2 0 0 1 | 1 0 0 0 2 0 0 0 0 | * * * * * * * 2 * * * | 0 0 1 0 1 0
(.x)(.o) (..)(..)&#x | 0 2 0 1 | 0 0 0 0 0 1 2 0 0 | * * * * * * * * 2 * * | 0 0 1 0 0 1
(..)(..) (.o)(.x)&#x | 0 1 0 2 | 0 0 0 0 0 0 2 0 1 | * * * * * * * * * 2 * | 0 0 0 1 0 1
(..)(..) (..)(ox)&#x | 0 0 1 2 | 0 0 0 0 0 0 0 2 1 | * * * * * * * * * * 1 | 0 0 0 2 0 0
---------------------+---------+-------------------+-----------------------+------------
(xx)(..) (xo)(..)&#x ♦ 4 2 0 0 | 2 2 4 0 0 1 0 0 0 | 1 2 2 0 0 0 0 0 0 0 0 | 1 * * * * *
(x.)(o.) (x.)(o.)&#x ♦ 4 0 1 0 | 2 2 0 4 0 0 0 0 0 | 1 0 0 0 2 2 0 0 0 0 0 | * 1 * * * *
(xx)(.o) (..)(..)&#x ♦ 2 2 0 1 | 1 0 2 0 2 1 2 0 0 | 0 1 0 2 0 0 0 1 1 0 0 | * * 2 * * *
(..)(..) (xo)(ox)&#x ♦ 2 1 1 2 | 0 1 2 2 2 0 2 2 1 | 0 0 1 2 0 1 2 0 0 1 1 | * * * 2 * *
(x.)(oo) (..)(..)&#x ♦ 2 0 1 1 | 1 0 0 2 2 0 0 1 0 | 0 0 0 0 1 0 2 1 0 0 0 | * * * * 2 *
(.x)(.o) (.o)(.x)&#x ♦ 0 2 0 2 | 0 0 0 0 0 1 4 0 1 | 0 0 0 0 0 0 0 0 2 2 0 | * * * * * 1
{3} || oct → height = sqrt(5/8) = 0.790569
o..3o.. | 3 * * ♦ 2 2 2 0 0 0 | 1 2 1 2 1 2 0 0 0 | 1 1 2 1 0
.o.3.o. | * 3 * | 0 2 0 2 1 0 | 0 1 2 0 0 2 1 2 0 | 1 0 1 2 1
..o3..o | * * 3 | 0 0 2 0 1 2 | 0 0 0 1 2 2 0 2 1 | 0 1 1 2 1
-----------+-------+-------------+-------------------+----------
x.. ... | 2 0 0 | 3 * * * * * | 1 1 0 1 0 0 0 0 0 | 1 1 1 0 0
oo.3oo.&#x | 1 1 0 | * 6 * * * * | 0 1 1 0 0 1 0 0 0 | 1 0 1 1 0
o.o3o.o&#x | 1 0 1 | * * 6 * * * | 0 0 0 1 1 1 0 0 0 | 0 1 1 1 0
... .x. | 0 2 0 | * * * 3 * * | 0 0 1 0 0 0 1 1 0 | 1 0 0 1 1
.oo3.oo&#x | 0 1 1 | * * * * 3 * | 0 0 0 0 0 2 0 2 0 | 0 0 1 2 1
... ..x | 0 0 2 | * * * * * 3 | 0 0 0 0 1 0 0 1 1 | 0 1 0 1 1
-----------+-------+-------------+-------------------+----------
x..3o.. | 3 0 0 | 3 0 0 0 0 0 | 1 * * * * * * * * | 1 1 0 0 0
xo. ...&#x | 2 1 0 | 1 2 0 0 0 0 | * 3 * * * * * * * | 1 0 1 0 0
... ox.&#x | 1 2 0 | 0 2 0 1 0 0 | * * 3 * * * * * * | 1 0 0 1 0
x.o ...&#x | 2 0 1 | 1 0 2 0 0 0 | * * * 3 * * * * * | 0 1 1 0 0
... o.x&#x | 1 0 2 | 0 0 2 0 0 1 | * * * * 3 * * * * | 0 1 0 1 0
ooo3ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * * * * 6 * * * | 0 0 1 1 0
.o.3.x. | 0 3 0 | 0 0 0 3 0 0 | * * * * * * 1 * * | 1 0 0 0 1
... .xx&#x | 0 2 2 | 0 0 0 1 2 1 | * * * * * * * 3 * | 0 0 0 1 1
..o3..x | 0 0 3 | 0 0 0 0 0 3 | * * * * * * * * 1 | 0 1 0 0 1
-----------+-------+-------------+-------------------+----------
xo.3ox.&#x ♦ 3 3 0 | 3 6 0 3 0 0 | 1 3 3 0 0 0 1 0 0 | 1 * * * *
x.o3o.x&#x ♦ 3 0 3 | 3 0 6 0 0 3 | 1 0 0 3 3 0 0 0 1 | * 1 * * *
xoo ...&#x ♦ 2 1 1 | 1 2 2 0 1 0 | 0 1 0 1 0 2 0 0 0 | * * 3 * *
... oxx&#x ♦ 1 2 2 | 0 2 2 1 2 1 | 0 0 1 0 1 2 0 1 0 | * * * 3 *
.oo3.xx&#x ♦ 0 3 3 | 0 0 0 3 3 3 | 0 0 0 0 0 0 1 3 1 | * * * * 1
(ox)(xo) (xo)(xo)&#x → height = sqrt(5/8) = 0.790569
(tet || squippy)
(o.)(..) (o.)(..) | 2 * * * | 1 2 2 0 0 0 0 0 0 | 1 2 2 1 2 0 0 0 0 0 0 | 1 1 2 1 0 0
(.o)(..) (.o)(..) | * 2 * * ♦ 0 2 0 1 2 1 0 0 0 | 2 1 2 0 0 1 2 1 0 0 0 | 1 2 1 0 1 0
(..)(o.) (..)(o.) | * * 4 * | 0 0 1 0 1 0 1 1 1 | 0 0 1 1 1 1 1 0 1 1 1 | 0 1 1 1 1 1
(..)(.o) (..)(.o) | * * * 1 ♦ 0 0 0 0 0 2 0 0 4 | 0 0 0 0 0 0 4 1 0 2 2 | 0 2 0 0 2 1
----------------------+---------+-------------------+-----------------------+------------
(..)(..) (x.)(..) | 2 0 0 0 | 1 * * * * * * * * | 0 2 0 0 2 0 0 0 0 0 0 | 1 0 2 1 0 0
(oo)(..) (oo)(..)&#x | 1 1 0 0 | * 4 * * * * * * * | 1 1 1 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0
(o.)(o.) (o.)(o.)&#x | 1 0 1 0 | * * 4 * * * * * * | 0 0 1 1 1 0 0 0 0 0 0 | 0 1 1 1 0 0
(.x)(..) (..)(..) | 0 2 0 0 | * * * 1 * * * * * | 2 0 0 0 0 0 0 1 0 0 0 | 1 2 0 0 0 0
(.o)(o.) (.o)(o.)&#x | 0 1 1 0 | * * * * 4 * * * * | 0 0 1 0 0 1 1 0 0 0 0 | 0 1 1 0 1 0
(.o)(.o) (.o)(.o)&#x | 0 1 0 1 | * * * * * 2 * * * | 0 0 0 0 0 0 2 1 0 0 0 | 0 2 0 0 1 0
(..)(x.) (..)(..) | 0 0 2 0 | * * * * * * 2 * * | 0 0 0 1 0 0 0 0 1 1 0 | 0 1 0 1 0 1
(..)(..) (..)(x.) | 0 0 2 0 | * * * * * * * 2 * | 0 0 0 0 1 1 0 0 1 0 1 | 0 0 1 1 1 1
(..)(oo) (..)(oo)&#x | 0 0 1 1 | * * * * * * * * 4 | 0 0 0 0 0 0 1 0 0 1 1 | 0 1 0 0 1 1
----------------------+---------+-------------------+-----------------------+------------
(ox)(..) (..)(..)&#x | 1 2 0 0 | 0 2 0 1 0 0 0 0 0 | 2 * * * * * * * * * * | 1 1 0 0 0 0
(..)(..) (xo)(..)&#x | 2 1 0 0 | 1 2 0 0 0 0 0 0 0 | * 2 * * * * * * * * * | 1 0 1 0 0 0
(oo)(o.) (oo)(o.)&#x | 1 1 1 0 | 0 1 1 0 1 0 0 0 0 | * * 4 * * * * * * * * | 0 1 1 0 0 0
(o.)(x.) (..)(..)&#x | 1 0 2 0 | 0 0 2 0 0 0 1 0 0 | * * * 2 * * * * * * * | 0 1 0 1 0 0
(..)(..) (x.)(x.)&#x | 2 0 2 0 | 1 0 2 0 0 0 0 1 0 | * * * * 2 * * * * * * | 0 0 1 1 0 0
(..)(..) (.o)(x.)&#x | 0 1 2 0 | 0 0 0 0 2 0 0 1 0 | * * * * * 2 * * * * * | 0 0 1 0 1 0
(.o)(oo) (.o)(oo)&#x | 0 1 1 1 | 0 0 0 0 1 1 0 0 1 | * * * * * * 4 * * * * | 0 1 0 0 1 0
(.x)(.o) (..)(..)&#x | 0 2 0 1 | 0 0 0 1 0 2 0 0 0 | * * * * * * * 1 * * * | 0 2 0 0 0 0
(..)(x.) (..)(x.) | 0 0 4 0 | 0 0 0 0 0 0 2 2 0 | * * * * * * * * 1 * * | 0 0 0 1 0 1
(..)(xo) (..)(..)&#x | 0 0 2 1 | 0 0 0 0 0 0 1 0 2 | * * * * * * * * * 2 * | 0 1 0 0 0 1
(..)(..) (..)(xo)&#x | 0 0 2 1 | 0 0 0 0 0 0 0 1 2 | * * * * * * * * * * 2 | 0 0 0 0 1 1
----------------------+---------+-------------------+-----------------------+------------
(ox)(..) (xo)(..)&#x ♦ 2 2 0 0 | 1 4 0 1 0 0 0 0 0 | 2 2 0 0 0 0 0 0 0 0 0 | 1 * * * * *
(ox)(xo) (..)(..)&#xr ♦ 1 2 2 1 | 0 2 2 1 2 2 1 0 2 | 1 0 2 1 0 0 2 1 0 1 0 | * 2 * * * * cycle (ABDC)
(..)(..) (xo)(x.)&#x ♦ 2 1 2 0 | 1 2 2 0 2 0 0 1 0 | 0 1 2 0 1 1 0 0 0 0 0 | * * 2 * * *
(o.)(x.) (x.)(x.)&#x ♦ 2 0 4 0 | 1 0 4 0 0 0 2 2 0 | 0 0 0 2 2 0 0 0 1 0 0 | * * * 1 * *
(..)(..) (.o)(xo)&#x ♦ 0 1 2 1 | 0 0 0 0 2 1 0 1 2 | 0 0 0 0 0 1 2 0 0 0 1 | * * * * 2 *
(..)(xo) (..)(xo)&#x ♦ 0 0 4 1 | 0 0 0 0 0 0 2 2 4 | 0 0 0 0 0 0 0 0 1 2 2 | * * * * * 1
xoo&#x || oxx&#x → height = sqrt(5/8) = 0.790569 (tet || squippy) o.. ... | 2 * * * * * | 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 | 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 | 1 1 1 1 1 0 0 0 .o. ... | * 1 * * * * ♦ 0 2 0 0 0 1 1 2 0 0 0 0 0 0 0 | 1 2 2 0 0 0 0 0 1 2 1 0 0 0 0 0 0 | 1 2 1 0 0 1 0 0 ..o ... | * * 1 * * * ♦ 0 0 2 0 0 1 0 0 1 2 0 0 0 0 0 | 0 2 0 1 2 0 0 0 1 0 0 2 1 0 0 0 0 | 1 2 0 1 0 0 1 0 ... o.. | * * * 1 * * ♦ 0 0 0 0 0 0 1 0 1 0 2 2 0 0 0 | 0 0 0 0 0 0 0 0 1 2 0 2 0 1 2 1 0 | 0 2 0 0 0 1 1 1 ... .o. | * * * * 2 * | 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 | 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 1 | 0 1 0 1 1 0 1 1 ... ..o | * * * * * 2 | 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 | 0 0 1 0 0 0 1 1 0 1 1 0 0 0 1 1 1 | 0 1 1 0 1 1 0 1 ----------------+-------------+-------------------------------+-----------------------------------+---------------- x.. ... | 2 0 0 0 0 0 | 1 * * * * * * * * * * * * * * | 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 | 1 0 1 1 1 0 0 0 oo.&#x ... | 1 1 0 0 0 0 | * 2 * * * * * * * * * * * * * | 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0 0 0 o.o&#x ... | 1 0 1 0 0 0 | * * 2 * * * * * * * * * * * * | 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 0 1 0 0 0 0 o.. || .o. | 1 0 0 0 1 0 | * * * 2 * * * * * * * * * * * | 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 | 0 1 0 1 1 0 0 0 o.. || ..o | 1 0 0 0 0 1 | * * * * 2 * * * * * * * * * * | 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 | 0 1 1 0 1 0 0 0 .oo&#x ... | 0 1 1 0 0 0 | * * * * * 1 * * * * * * * * * | 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 | 1 2 0 0 0 0 0 0 .o. || o.. | 0 1 0 1 0 0 | * * * * * * 1 * * * * * * * * | 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 | 0 2 0 0 0 1 0 0 .o. || ..o | 0 1 0 0 0 1 | * * * * * * * 2 * * * * * * * | 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 | 0 1 1 0 0 1 0 0 ..o || o.. | 0 0 1 1 0 0 | * * * * * * * * 1 * * * * * * | 0 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 | 0 2 0 0 0 0 1 0 ..o || .o. | 0 0 1 0 1 0 | * * * * * * * * * 2 * * * * * | 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 | 0 1 0 1 0 0 1 0 ... oo.&#x | 0 0 0 1 1 0 | * * * * * * * * * * 2 * * * * | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 | 0 1 0 0 0 0 1 1 ... o.o&#x | 0 0 0 1 0 1 | * * * * * * * * * * * 2 * * * | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 | 0 1 0 0 0 1 0 1 ... .x. | 0 0 0 0 2 0 | * * * * * * * * * * * * 1 * * | 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 | 0 0 0 1 1 0 1 1 ... .oo&#x | 0 0 0 0 1 1 | * * * * * * * * * * * * * 2 * | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 | 0 1 0 0 1 0 0 1 ... ..x | 0 0 0 0 0 2 | * * * * * * * * * * * * * * 1 | 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 | 0 0 1 0 1 1 0 1 ----------------+-------------+-------------------------------+-----------------------------------+---------------- xo.&#x ... | 2 1 0 0 0 0 | 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 * * * * * * * * * * * * * * * * | 1 0 1 0 0 0 0 0 ooo&#x ... | 1 1 1 0 0 0 | 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 | * 2 * * * * * * * * * * * * * * * | 1 1 0 0 0 0 0 0 oo.&#x || ..o | 1 1 0 0 0 1 | 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 | * * 2 * * * * * * * * * * * * * * | 0 1 1 0 0 0 0 0 x.o&#x ... | 2 0 1 0 0 0 | 1 0 2 0 0 0 0 0 0 0 0 0 0 0 0 | * * * 1 * * * * * * * * * * * * * | 1 0 0 1 0 0 0 0 o.o&#x || .o. | 1 0 1 0 1 0 | 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 | * * * * 2 * * * * * * * * * * * * | 0 1 0 1 0 0 0 0 x.. || .x. | 2 0 0 0 2 0 | 1 0 0 2 0 0 0 0 0 0 0 0 1 0 0 | * * * * * 1 * * * * * * * * * * * | 0 0 0 1 1 0 0 0 o.. || .oo&#x | 1 0 0 0 1 1 | 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 | * * * * * * 2 * * * * * * * * * * | 0 1 0 0 1 0 0 0 x.. || ..x | 2 0 0 0 0 2 | 1 0 0 0 2 0 0 0 0 0 0 0 0 0 1 | * * * * * * * 1 * * * * * * * * * | 0 0 1 0 1 0 0 0 .oo&#x || o.. | 0 1 1 1 0 0 | 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 | * * * * * * * * 1 * * * * * * * * | 0 2 0 0 0 0 0 0 .o. || o.o&#x | 0 1 0 1 0 1 | 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 | * * * * * * * * * 2 * * * * * * * | 0 1 0 0 0 1 0 0 .o. || ..x | 0 1 0 0 0 2 | 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 | * * * * * * * * * * 1 * * * * * * | 0 0 1 0 0 1 0 0 ..o || oo.&#x | 0 0 1 1 1 0 | 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 | * * * * * * * * * * * 2 * * * * * | 0 1 0 0 0 0 1 0 ..o || .x. | 0 0 1 0 2 0 | 0 0 0 0 0 0 0 0 0 2 0 0 1 0 0 | * * * * * * * * * * * * 1 * * * * | 0 0 0 1 0 0 1 0 ... ox.&#x | 0 0 0 1 2 0 | 0 0 0 0 0 0 0 0 0 0 2 0 1 0 0 | * * * * * * * * * * * * * 1 * * * | 0 0 0 0 0 0 1 1 ... ooo&#x | 0 0 0 1 1 1 | 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 | * * * * * * * * * * * * * * 2 * * | 0 1 0 0 0 0 0 1 ... o.x&#x | 0 0 0 1 0 2 | 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 | * * * * * * * * * * * * * * * 1 * | 0 0 0 0 0 1 0 1 ... .xx&#x | 0 0 0 0 2 2 | 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 | * * * * * * * * * * * * * * * * 1 | 0 0 0 0 1 0 0 1 ----------------+-------------+-------------------------------+-----------------------------------+---------------- xoo&#x ... ♦ 2 1 1 0 0 0 | 1 2 2 0 0 1 0 0 0 0 0 0 0 0 0 | 1 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 * * * * * * * ooo&#x || ooo&#x ♦ 1 1 1 1 1 1 | 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 | 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 | * 2 * * * * * * xo.&#x || ..x ♦ 2 1 0 0 0 2 | 1 2 0 0 2 0 0 2 0 0 0 0 0 0 1 | 1 0 2 0 0 0 0 1 0 0 1 0 0 0 0 0 0 | * * 1 * * * * * x.o&#x || .x. ♦ 2 0 1 0 2 0 | 1 0 2 2 0 0 0 0 0 2 0 0 1 0 0 | 0 0 0 1 2 1 0 0 0 0 0 0 1 0 0 0 0 | * * * 1 * * * * x.. || .xx&#x ♦ 2 0 0 0 2 2 | 1 0 0 2 2 0 0 0 0 0 0 0 1 2 1 | 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 0 1 | * * * * 1 * * * .o. || o.x&#x ♦ 0 1 0 1 0 2 | 0 0 0 0 0 0 1 2 0 0 0 2 0 0 1 | 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 1 0 | * * * * * 1 * * ..o || ox.&#x ♦ 0 0 1 1 2 0 | 0 0 0 0 0 0 0 0 1 2 2 0 1 0 0 | 0 0 0 0 0 0 0 0 0 0 0 2 1 1 0 0 0 | * * * * * * 1 * ... oxx&#x ♦ 0 0 0 1 2 2 | 0 0 0 0 0 0 0 0 0 0 2 2 1 2 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 1 | * * * * * * * 1
ooxx oxox&#xr → height(1,2) = height(2,4) = height(3,4) = sqrt(3)/2 = 0.866025 height(1,4) = height(2,3) = 1/sqrt(2) = 0.707107 o... o... | 1 * * * ♦ 2 4 0 0 0 0 0 0 0 | 1 4 2 2 0 0 0 0 0 0 0 | 2 2 1 0 0 0 .o.. .o.. | * 2 * * ♦ 1 0 1 2 2 0 0 0 0 | 1 2 0 0 1 2 2 1 0 0 0 | 2 1 0 1 1 0 ..o. ..o. | * * 2 * | 0 0 0 2 0 1 2 0 0 | 0 0 0 0 2 1 2 0 2 1 0 | 1 0 0 1 2 1 ...o ...o | * * * 4 | 0 1 0 0 1 0 1 1 1 | 0 1 1 1 0 0 1 1 1 1 1 | 1 1 1 0 1 1 --------------+---------+-------------------+-----------------------+------------ oo.. oo..&#x | 1 1 0 0 | 2 * * * * * * * * | 1 2 0 0 0 0 0 0 0 0 0 | 2 1 0 0 0 0 o..o o..o&#x | 1 0 0 1 | * 4 * * * * * * * | 0 1 1 1 0 0 0 0 0 0 0 | 1 1 1 0 0 0 .... .x.. | 0 2 0 0 | * * 1 * * * * * * | 1 0 0 0 0 2 0 0 0 0 0 | 2 0 0 1 0 0 .oo. .oo.&#x | 0 1 1 0 | * * * 4 * * * * * | 0 0 0 0 1 1 1 0 0 0 0 | 1 0 0 1 1 0 .o.o .o.o&#x | 0 1 0 1 | * * * * 4 * * * * | 0 1 0 0 0 0 1 1 0 0 0 | 1 1 0 0 1 0 ..x. .... | 0 0 2 0 | * * * * * 1 * * * | 0 0 0 0 2 0 0 0 2 0 0 | 0 0 0 1 2 1 ..oo ..oo&#x | 0 0 1 1 | * * * * * * 4 * * | 0 0 0 0 0 0 1 0 1 1 0 | 1 0 0 0 1 1 ...x .... | 0 0 0 2 | * * * * * * * 2 * | 0 0 1 0 0 0 0 1 1 0 1 | 0 1 1 0 1 1 .... ...x | 0 0 0 2 | * * * * * * * * 2 | 0 0 0 1 0 0 0 0 0 1 1 | 1 0 1 0 0 1 --------------+---------+-------------------+-----------------------+------------ .... ox..&#x | 1 2 0 0 | 2 0 1 0 0 0 0 0 0 | 1 * * * * * * * * * * | 2 0 0 0 0 0 oo.o oo.o&#x | 1 1 0 1 | 1 1 0 0 1 0 0 0 0 | * 4 * * * * * * * * * | 1 1 0 0 0 0 o..x ....&#x | 1 0 0 2 | 0 2 0 0 0 0 0 1 0 | * * 2 * * * * * * * * | 0 1 1 0 0 0 .... o..x&#x | 1 0 0 2 | 0 2 0 0 0 0 0 0 1 | * * * 2 * * * * * * * | 1 0 1 0 0 0 .ox. ....&#x | 0 1 2 0 | 0 0 0 2 0 1 0 0 0 | * * * * 2 * * * * * * | 0 0 0 1 1 0 .... .xo.&#x | 0 2 1 0 | 0 0 1 2 0 0 0 0 0 | * * * * * 2 * * * * * | 1 0 0 1 0 0 .ooo .ooo&#x | 0 1 1 1 | 0 0 0 1 1 0 1 0 0 | * * * * * * 4 * * * * | 1 0 0 0 1 0 .o.x ....&#x | 0 1 0 2 | 0 0 0 0 2 0 0 1 0 | * * * * * * * 2 * * * | 0 1 0 0 1 0 ..xx ....&#x | 0 0 2 2 | 0 0 0 0 0 1 2 1 0 | * * * * * * * * 2 * * | 0 0 0 0 1 1 .... ..ox&#x | 0 0 1 2 | 0 0 0 0 0 0 2 0 1 | * * * * * * * * * 2 * | 1 0 0 0 0 1 ...x ...x | 0 0 0 4 | 0 0 0 0 0 0 0 2 2 | * * * * * * * * * * 1 | 0 0 1 0 0 1 --------------+---------+-------------------+-----------------------+------------ .... oxox&#xr ♦ 1 2 1 2 | 2 2 1 2 2 0 2 0 1 | 1 2 0 1 0 1 2 0 0 1 0 | 2 * * * * * oo.x ....&#x ♦ 1 1 0 2 | 1 2 0 0 2 0 0 1 0 | 0 2 1 0 0 0 0 1 0 0 0 | * 2 * * * * o..x o..x&#x ♦ 1 0 0 4 | 0 4 0 0 0 0 0 2 2 | 0 0 2 2 0 0 0 0 0 0 1 | * * 1 * * * .ox. .xo.&#x ♦ 0 2 2 0 | 0 0 1 4 0 1 0 0 0 | 0 0 0 0 2 2 0 0 0 0 0 | * * * 1 * * .oxx ....&#x ♦ 0 1 2 2 | 0 0 0 2 2 1 2 1 0 | 0 0 0 0 1 0 2 1 1 0 0 | * * * * 2 * ..xx ..ox&#x ♦ 0 0 2 4 | 0 0 0 0 0 1 4 2 2 | 0 0 0 0 0 0 0 0 2 2 1 | * * * * * 1
xoxoxo&#xr → all consecutive pairwise heights = sqrt(3)/2 = 0.866025 all alternating pairwise heights = 1 Note: these lengths show that this cycle is not flat, rather it is wobbling up and down! o..... & | 6 * | 1 2 2 0 | 2 2 2 1 1 0 | 2 1 1 1 .o.... & | * 3 ♦ 0 4 0 2 | 2 2 0 0 4 1 | 1 2 0 2 -------------+-----+----------+-------------+-------- x..... & | 2 0 | 3 * * * | 2 0 2 0 0 0 | 2 0 1 1 oo....&#x & | 1 1 | * 12 * * | 1 1 0 0 1 0 | 1 1 0 1 o.o...&#x & | 2 0 | * * 6 * | 0 1 1 1 0 0 | 1 1 1 0 .o.o..&#x & | 0 2 | * * * 3 | 0 0 0 0 2 1 | 0 2 0 1 -------------+-----+----------+-------------+-------- xo....&#x & | 2 1 | 1 2 0 0 | 6 * * * * * | 1 0 0 1 ooo...&#x & | 2 1 | 0 2 1 0 | * 6 * * * * | 1 1 0 0 x.x...&#x & | 4 0 | 2 0 2 0 | * * 3 * * * | 1 0 1 0 o.o.o.&#x | 3 0 | 0 0 3 0 | * * * 2 * * | 0 1 1 0 .ooo..&#x & | 1 2 | 0 2 0 1 | * * * * 6 * | 0 1 0 1 .o.o.o&#x | 0 3 | 0 0 0 3 | * * * * * 1 | 0 2 0 0 -------------+-----+----------+-------------+-------- xox...&#x & ♦ 4 1 | 2 4 2 0 | 2 2 1 0 0 0 | 3 * * * oooooo&#x ♦ 3 3 | 0 6 3 3 | 0 3 0 1 3 1 | * 2 * * x.x.x.&#x ♦ 6 0 | 3 0 6 0 | 0 0 3 2 0 0 | * * 1 * .oxo..&#x & ♦ 2 2 | 1 4 0 1 | 2 0 0 0 2 0 | * * * 3
o(xx)o x(ox)o&#xt → height(1,2) = height(3,4) = 1/sqrt(12) = 0.288675 height(1,3) = height(2,3) = height(2,4) = 1/sqrt(3) = 0.577350 height(1,23) = height(23,4) = sqrt(5/12) = 0.645497 o(..). o(..). | 2 * * * | 1 2 2 0 0 0 0 0 0 | 1 2 2 1 2 0 0 0 0 0 0 | 1 1 2 1 0 0 .(o.). .(o.). | * 2 * * ♦ 0 2 0 1 2 1 0 0 0 | 2 1 2 0 0 1 2 1 0 0 0 | 1 2 1 0 1 0 .(.o). .(.o). | * * 4 * | 0 0 1 0 1 0 1 1 1 | 0 0 1 1 1 1 1 0 1 1 1 | 0 1 1 1 1 1 .(..)o .(..)o | * * * 1 ♦ 0 0 0 0 0 2 0 0 4 | 0 0 0 0 0 0 4 1 0 2 2 | 0 2 0 0 2 1 ------------------+---------+-------------------+-----------------------+------------ .(..). x(..). | 2 0 0 0 | 1 * * * * * * * * | 0 2 0 0 2 0 0 0 0 0 0 | 1 0 2 1 0 0 o(o.). o(o.).&#x | 1 1 0 0 | * 4 * * * * * * * | 1 1 1 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0 o(.o). o(.o).&#x | 1 0 1 0 | * * 4 * * * * * * | 0 0 1 1 1 0 0 0 0 0 0 | 0 1 1 1 0 0 .(x.). .(..). | 0 2 0 0 | * * * 1 * * * * * | 2 0 0 0 0 0 0 1 0 0 0 | 1 2 0 0 0 0 .(oo). .(oo).&#x | 0 1 1 0 | * * * * 4 * * * * | 0 0 1 0 0 1 1 0 0 0 0 | 0 1 1 0 1 0 .(o.)o .(o.)o&#x | 0 1 0 1 | * * * * * 2 * * * | 0 0 0 0 0 0 2 1 0 0 0 | 0 2 0 0 1 0 .(.x). .(..). | 0 0 2 0 | * * * * * * 2 * * | 0 0 0 1 0 0 0 0 1 1 0 | 0 1 0 1 0 1 .(..). .(.x). | 0 0 2 0 | * * * * * * * 2 * | 0 0 0 0 1 1 0 0 1 0 1 | 0 0 1 1 1 1 .(.o)o .(.o)o&#x | 0 0 1 1 | * * * * * * * * 4 | 0 0 0 0 0 0 1 0 0 1 1 | 0 1 0 0 1 1 ------------------+---------+-------------------+-----------------------+------------ o(x.). .(..).&#x | 1 2 0 0 | 0 2 0 1 0 0 0 0 0 | 2 * * * * * * * * * * | 1 1 0 0 0 0 .(..). x(o.).&#x | 2 1 0 0 | 1 2 0 0 0 0 0 0 0 | * 2 * * * * * * * * * | 1 0 1 0 0 0 o(oo). o(oo).&#x | 1 1 1 0 | 0 1 1 0 1 0 0 0 0 | * * 4 * * * * * * * * | 0 1 1 0 0 0 o(.x). .(..).&#x | 1 0 2 0 | 0 0 2 0 0 0 1 0 0 | * * * 2 * * * * * * * | 0 1 0 1 0 0 .(..). x(.x).&#x | 2 0 2 0 | 1 0 2 0 0 0 0 1 0 | * * * * 2 * * * * * * | 0 0 1 1 0 0 .(..). .(ox).&#x | 0 1 2 0 | 0 0 0 0 2 0 0 1 0 | * * * * * 2 * * * * * | 0 0 1 0 1 0 .(oo)o .(oo)o&#x | 0 1 1 1 | 0 0 0 0 1 1 0 0 1 | * * * * * * 4 * * * * | 0 1 0 0 1 0 .(x.)o .(..).&#x | 0 2 0 1 | 0 0 0 1 0 2 0 0 0 | * * * * * * * 1 * * * | 0 2 0 0 0 0 .(.x). .(.x). | 0 0 4 0 | 0 0 0 0 0 0 2 2 0 | * * * * * * * * 1 * * | 0 0 0 1 0 1 .(.x)o .(..).&#x | 0 0 2 1 | 0 0 0 0 0 0 1 0 2 | * * * * * * * * * 2 * | 0 1 0 0 0 1 .(..). .(.x)o&#x | 0 0 2 1 | 0 0 0 0 0 0 0 1 2 | * * * * * * * * * * 2 | 0 0 0 0 1 1 ------------------+---------+-------------------+-----------------------+------------ o(x.). x(o.).&#x ♦ 2 2 0 0 | 1 4 0 1 0 0 0 0 0 | 2 2 0 0 0 0 0 0 0 0 0 | 1 * * * * * o(xx)o .(..).&#xt ♦ 1 2 2 1 | 0 2 2 1 2 2 1 0 2 | 1 0 2 1 0 0 2 1 0 1 0 | * 2 * * * * .(..). x(ox).&#x ♦ 2 1 2 0 | 1 2 2 0 2 0 0 1 0 | 0 1 2 0 1 1 0 0 0 0 0 | * * 2 * * * o(.x). x(.x).&#x ♦ 2 0 4 0 | 1 0 4 0 0 0 2 2 0 | 0 0 0 2 2 0 0 0 1 0 0 | * * * 1 * * .(..). .(ox)o&#x ♦ 0 1 2 1 | 0 0 0 0 2 1 0 1 2 | 0 0 0 0 0 1 2 0 0 0 1 | * * * * 2 * .(.x)o .(.x)o&#x ♦ 0 0 4 1 | 0 0 0 0 0 0 2 2 4 | 0 0 0 0 0 0 0 0 1 2 2 | * * * * * 1
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