Acronym | bidhin (alt.: octahex, tetalrap, tetaltet octtric) | ||
Name |
bidiminished hemipenteract, oct || hex, tet || inverted rap, (tet,dual tet,oct)-trigonic | ||
Circumradius | sqrt(5/8) = 0.790569 | ||
Lace city in approx. ASCII-art |
O t T = x3o3o (tet) O = o3x3o (oct) T t = o3o3x (dual tet) | ||
Lace hyper city in approx. ASCII-art |
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Coordinates | |||
Face vector | 14, 60, 100, 69, 17 | ||
Confer |
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The hemipenteract (hin) could be given as hex || gyro hex. Here the top-hex (itself being pt || pseudo oct || pt) has to be bidiminished at opposite vertices as hex - 2x octpy. Simultanuously the bottom-hex only gets marginally rasped at opposite tet (as hex equally is: tet || dual tet) according to hex - 2x tet. The sefa clearly is the vertex figure of hin, i.e. rap. In other words, one chops off 2 rappies.
Aside: There is also a polyteron which likewise uses O, T, and t (cf. Lace city), but has a not as sharp angle at O: the dot. (That relation kind of is like cube to trip.) As dot had edges connecting the opposite pair of octs, these clearly are missing in here. Conversely, the structure of hex, except of the pair of opposite bases (tets), were missing in dot but occurs here for sure.
Incidence matrix according to Dynkin symbol
xoo3oxo3oox&#x → heights(1,2) = height(2,3) = sqrt(5/8) = 0.790569 height(1,3) = 1/sqrt(2) = 0.707107 (oct || (tet || dual tet) o..3o..3o.. | 4 * * ♦ 3 3 3 0 0 0 | 3 3 3 6 3 6 0 0 0 0 0 | 1 3 1 3 6 1 6 3 3 0 0 0 0 | 1 1 3 3 1 0 .o.3.o.3.o. | * 6 * ♦ 0 2 0 4 2 0 | 0 1 4 0 0 4 2 2 4 1 0 | 0 2 2 0 0 0 2 4 2 1 2 2 0 | 1 0 2 1 2 1 ..o3..o3..o | * * 4 ♦ 0 0 3 0 3 3 | 0 0 0 3 6 6 0 0 3 3 3 | 0 0 0 1 6 3 3 3 6 0 1 3 1 | 0 1 1 3 3 1 ---------------+-------+-----------------+----------------------------+------------------------------+------------ x.. ... ... | 2 0 0 | 6 * * * * * | 2 1 0 2 0 0 0 0 0 0 0 | 1 2 0 2 1 0 2 0 0 0 0 0 0 | 1 1 2 1 0 0 oo.3oo.3oo.&#x | 1 1 0 | * 12 * * * * | 0 1 2 0 0 2 0 0 0 0 0 | 0 2 1 0 0 0 2 2 1 0 0 0 0 | 1 0 2 1 1 0 o.o3o.o3o.o&#x | 1 0 1 | * * 12 * * * ♦ 0 0 0 2 2 2 0 0 0 0 0 | 0 0 0 1 2 1 2 1 2 0 0 0 0 | 0 1 1 2 1 0 ... .x. ... | 0 2 0 | * * * 12 * * | 0 0 1 0 0 0 1 1 1 0 0 | 0 1 1 0 0 0 0 1 0 1 1 1 0 | 1 0 1 0 1 1 .oo3.oo3.oo&#x | 0 1 1 | * * * * 12 * | 0 0 0 0 0 2 0 0 2 1 0 | 0 0 0 0 0 0 1 2 2 0 1 2 0 | 0 0 1 1 2 1 ... ... ..x | 0 0 2 | * * * * * 6 | 0 0 0 0 2 0 0 0 0 1 2 | 0 0 0 0 1 2 0 0 2 0 0 2 1 | 0 1 0 1 2 1 ---------------+-------+-----------------+----------------------------+------------------------------+------------ x..3o.. ... | 3 0 0 | 3 0 0 0 0 0 | 4 * * * * * * * * * * | 1 1 0 1 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0 xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 0 | * 6 * * * * * * * * * | 0 2 0 0 0 0 2 0 0 0 0 0 0 | 1 0 2 1 0 0 ... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 | * * 12 * * * * * * * * | 0 1 1 0 0 0 0 1 0 0 0 0 0 | 1 0 1 0 1 0 x.o ... ...&#x | 2 0 1 | 1 0 2 0 0 0 | * * * 12 * * * * * * * | 0 0 0 1 1 0 1 0 0 0 0 0 0 | 0 1 1 1 0 0 ... ... o.x&#x | 1 0 2 | 0 0 2 0 0 1 | * * * * 12 * * * * * * | 0 0 0 0 1 1 0 0 1 0 0 0 0 | 0 1 0 1 1 0 ooo3ooo3ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * * * * 24 * * * * * | 0 0 0 0 0 0 1 1 1 0 0 0 0 | 0 0 1 1 1 0 .o.3.x. ... | 0 3 0 | 0 0 0 3 0 0 | * * * * * * 4 * * * * | 0 1 0 0 0 0 0 0 0 1 1 0 0 | 1 0 1 0 0 1 ... .x.3.o. | 0 3 0 | 0 0 0 3 0 0 | * * * * * * * 4 * * * | 0 0 1 0 0 0 0 0 0 1 0 1 0 | 1 0 0 0 1 1 ... .xo ...&#x | 0 2 1 | 0 0 0 1 2 0 | * * * * * * * * 12 * * | 0 0 0 0 0 0 0 1 0 0 1 1 0 | 0 0 1 0 1 1 ... ... .ox&#x | 0 1 2 | 0 0 0 0 2 1 | * * * * * * * * * 6 * | 0 0 0 0 0 0 0 0 2 0 0 2 0 | 0 0 0 1 2 1 ... ..o3..x | 0 0 3 | 0 0 0 0 0 3 | * * * * * * * * * * 4 | 0 0 0 0 0 1 0 0 0 0 0 1 1 | 0 1 0 0 1 1 ---------------+-------+-----------------+----------------------------+------------------------------+------------ x..3o..3o.. ♦ 4 0 0 | 6 0 0 0 0 0 | 4 0 0 0 0 0 0 0 0 0 0 | 1 * * * * * * * * * * * * | 1 1 0 0 0 0 xo.3ox. ...&#x ♦ 3 3 0 | 3 6 0 3 0 0 | 1 3 3 0 0 0 1 0 0 0 0 | * 4 * * * * * * * * * * * | 1 0 1 0 0 0 ... ox.3oo.&#x ♦ 1 3 0 | 0 3 0 3 0 0 | 0 0 3 0 0 0 0 1 0 0 0 | * * 4 * * * * * * * * * * | 1 0 0 0 1 0 x.o3o.o ...&#x ♦ 3 0 1 | 3 0 3 0 0 0 | 1 0 0 3 0 0 0 0 0 0 0 | * * * 4 * * * * * * * * * | 0 1 1 0 0 0 x.o ... o.x&#x ♦ 2 0 2 | 1 0 4 0 0 1 | 0 0 0 2 2 0 0 0 0 0 0 | * * * * 6 * * * * * * * * | 0 1 0 1 0 0 ... o.o3o.x&#x ♦ 1 0 3 | 0 0 3 0 0 3 | 0 0 0 0 3 0 0 0 0 0 1 | * * * * * 4 * * * * * * * | 0 1 0 0 1 0 xoo ... ...&#x ♦ 2 1 1 | 1 2 2 0 1 0 | 0 1 0 1 0 2 0 0 0 0 0 | * * * * * * 12 * * * * * * | 0 0 1 1 0 0 ... oxo ...&#x ♦ 1 2 1 | 0 2 1 1 2 0 | 0 0 1 0 0 2 0 0 1 0 0 | * * * * * * * 12 * * * * * | 0 0 1 0 1 0 ... ... oox&#x ♦ 1 1 2 | 0 1 2 0 2 1 | 0 0 0 0 1 2 0 0 0 1 0 | * * * * * * * * 12 * * * * | 0 0 0 1 1 0 .o.3.x.3.o. ♦ 0 6 0 | 0 0 0 12 0 0 | 0 0 0 0 0 0 4 4 0 0 0 | * * * * * * * * * 1 * * * | 1 0 0 0 0 1 .oo3.xo ...&#x ♦ 0 3 1 | 0 0 0 3 3 0 | 0 0 0 0 0 0 1 0 3 0 0 | * * * * * * * * * * 4 * * | 0 0 1 0 0 1 ... .xo3.ox&#x ♦ 0 3 3 | 0 0 0 3 6 3 | 0 0 0 0 0 0 0 1 3 3 1 | * * * * * * * * * * * 4 * | 0 0 0 0 1 1 ..o3..o3..x ♦ 0 0 4 | 0 0 0 0 0 6 | 0 0 0 0 0 0 0 0 0 0 4 | * * * * * * * * * * * * 1 | 0 1 0 0 0 1 ---------------+-------+-----------------+----------------------------+------------------------------+------------ xo.3ox.3oo.&#x ♦ 4 6 0 | 6 12 0 12 0 0 | 4 6 12 0 0 0 4 4 0 0 0 | 1 4 4 0 0 0 0 0 0 1 0 0 0 | 1 * * * * * x.o3o.o3o.x&#x ♦ 4 0 4 | 6 0 12 0 0 6 | 4 0 0 12 12 0 0 0 0 0 4 | 1 0 0 4 6 4 0 0 0 0 0 0 1 | * 1 * * * * xoo3oxo ...&#x ♦ 3 3 1 | 3 6 3 3 3 0 | 1 3 3 3 0 6 1 0 3 0 0 | 0 1 0 1 0 0 3 3 0 0 1 0 0 | * * 4 * * * xoo ... oox&#x ♦ 2 1 2 | 1 2 4 0 2 1 | 0 1 0 2 2 4 0 0 0 1 0 | 0 0 0 0 1 0 2 0 2 0 0 0 0 | * * * 6 * * ... oxo3oox&#x ♦ 1 3 3 | 0 3 3 3 6 3 | 0 0 3 0 3 6 0 1 3 3 1 | 0 0 1 0 0 1 0 3 3 0 0 1 0 | * * * * 4 * .oo3.xo3.ox&#x ♦ 0 6 4 | 0 0 0 12 12 6 | 0 0 0 0 0 0 4 4 12 6 4 | 0 0 0 0 0 0 0 0 0 1 4 4 1 | * * * * * 1
oct || hex → height = 1/sqrt(2) = 0.707107
o..3o..3o.. & | 8 * ♦ 3 3 3 0 | 3 9 3 6 3 0 | 1 4 3 3 9 3 1 0 | 1 1 4 3
.o.3.o.3.o. | * 6 ♦ 0 0 4 4 | 0 0 2 4 8 4 | 0 0 0 4 4 4 4 1 | 0 2 4 1
------------------+-----+-------------+-----------------+-------------------+--------
x.. ... ... & | 2 0 | 12 * * * | 2 2 1 0 0 0 | 1 2 1 2 2 0 0 0 | 1 1 2 1
o.o3o.o3o.o&#x | 2 0 | * 12 * * ♦ 0 4 0 2 0 0 | 0 2 2 0 4 1 0 0 | 1 0 2 2
oo.3oo.3oo.&#x & | 1 1 | * * 24 * | 0 0 1 2 2 0 | 0 0 0 2 3 2 1 0 | 0 1 3 1
... .x. ... | 0 2 | * * * 12 | 0 0 0 0 2 2 | 0 0 0 2 0 1 2 1 | 0 2 2 0
------------------+-----+-------------+-----------------+-------------------+--------
x..3o.. ... & | 3 0 | 3 0 0 0 | 8 * * * * * | 1 1 0 1 0 0 0 0 | 1 1 1 0
x.o ... ...&#x & | 3 0 | 1 2 0 0 | * 24 * * * * | 0 1 1 0 1 0 0 0 | 1 0 1 1
xo. ... ...&#x & | 2 1 | 1 0 2 0 | * * 12 * * * | 0 0 0 2 2 0 0 0 | 0 1 2 1
ooo3ooo3ooo&#x | 2 1 | 0 1 2 0 | * * * 24 * * | 0 0 0 0 2 1 0 0 | 0 0 2 1
... ox. ...&#x & | 1 2 | 0 0 2 1 | * * * * 24 * | 0 0 0 1 0 1 1 0 | 0 1 2 0
.o.3.x. ... & | 0 3 | 0 0 0 3 | * * * * * 8 | 0 0 0 1 0 0 1 1 | 0 2 1 0
------------------+-----+-------------+-----------------+-------------------+--------
x..3o..3o.. & ♦ 4 0 | 6 0 0 0 | 4 0 0 0 0 0 | 2 * * * * * * * | 1 1 0 0
x.o3o.o ...&#x & ♦ 4 0 | 3 3 0 0 | 1 3 0 0 0 0 | * 8 * * * * * * | 1 0 1 0
x.o ... o.x&#x ♦ 4 0 | 2 4 0 0 | 0 4 0 0 0 0 | * * 6 * * * * * | 1 0 0 1
xo.3ox. ...&#x & ♦ 3 3 | 3 0 6 3 | 1 0 3 0 3 1 | * * * 8 * * * * | 0 1 1 0
xoo ... ...&#x & ♦ 3 1 | 1 2 3 0 | 0 1 1 2 0 0 | * * * * 24 * * * | 0 0 1 1
... oxo ...&#x ♦ 2 2 | 0 1 4 1 | 0 0 0 2 2 0 | * * * * * 12 * * | 0 0 2 0
... ox.3oo.&#x & ♦ 1 3 | 0 0 3 3 | 0 0 0 0 3 1 | * * * * * * 8 * | 0 1 1 0
.o.3.x.3.o. ♦ 0 6 | 0 0 0 12 | 0 0 0 0 0 8 | * * * * * * * 1 | 0 2 0 0
------------------+-----+-------------+-----------------+-------------------+--------
x.o3o.o3o.x&#x ♦ 8 0 | 12 12 0 0 | 8 24 0 0 0 0 | 2 8 6 0 0 0 0 0 | 1 * * *
xo.3ox.3oo.&#x & ♦ 4 6 | 6 0 12 12 | 4 0 6 0 12 8 | 1 0 0 4 0 0 4 1 | * 2 * *
xoo3oxo ...&#x & ♦ 4 3 | 3 3 9 3 | 1 3 3 6 6 1 | 0 1 0 1 3 3 1 0 | * * 8 *
xoo ... oox&#x ♦ 4 1 | 2 4 4 0 | 0 4 2 4 0 0 | 0 0 1 0 4 0 0 0 | * * * 6
tet || inv rap → height = sqrt(2/5) = 0.632456
o..3o..3o.. | 4 * * ♦ 3 3 3 0 0 0 | 3 3 3 6 3 6 0 0 0 0 0 | 1 3 1 3 6 1 6 3 3 0 0 0 0 | 1 1 3 3 1 0
.o.3.o.3.o. | * 6 * ♦ 0 2 0 4 2 0 | 0 1 4 0 0 4 2 2 4 1 0 | 0 2 2 0 0 0 2 4 2 1 2 2 0 | 1 0 2 1 2 1
..o3..o3..o | * * 4 ♦ 0 0 3 0 3 3 | 0 0 0 3 6 6 0 0 3 3 3 | 0 0 0 1 6 3 3 3 6 0 1 3 1 | 0 1 1 3 3 1
---------------+-------+-----------------+----------------------------+------------------------------+------------
x.. ... ... | 2 0 0 | 6 * * * * * | 2 1 0 2 0 0 0 0 0 0 0 | 1 2 0 2 1 0 2 0 0 0 0 0 0 | 1 1 2 1 0 0
oo.3oo.3oo.&#x | 1 1 0 | * 12 * * * * | 0 1 2 0 0 2 0 0 0 0 0 | 0 2 1 0 0 0 2 2 1 0 0 0 0 | 1 0 2 1 1 0
o.o3o.o3o.o&#x | 1 0 1 | * * 12 * * * ♦ 0 0 0 2 2 2 0 0 0 0 0 | 0 0 0 1 2 1 2 1 2 0 0 0 0 | 0 1 1 2 1 0
... .x. ... | 0 2 0 | * * * 12 * * | 0 0 1 0 0 0 1 1 1 0 0 | 0 1 1 0 0 0 0 1 0 1 1 1 0 | 1 0 1 0 1 1
.oo3.oo3.oo&#x | 0 1 1 | * * * * 12 * | 0 0 0 0 0 2 0 0 2 1 0 | 0 0 0 0 0 0 1 2 2 0 1 2 0 | 0 0 1 1 2 1
... ... ..x | 0 0 2 | * * * * * 6 | 0 0 0 0 2 0 0 0 0 1 2 | 0 0 0 0 1 2 0 0 2 0 0 2 1 | 0 1 0 1 2 1
---------------+-------+-----------------+----------------------------+------------------------------+------------
x..3o.. ... | 3 0 0 | 3 0 0 0 0 0 | 4 * * * * * * * * * * | 1 1 0 1 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0
xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 0 | * 6 * * * * * * * * * | 0 2 0 0 0 0 2 0 0 0 0 0 0 | 1 0 2 1 0 0
... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 | * * 12 * * * * * * * * | 0 1 1 0 0 0 0 1 0 0 0 0 0 | 1 0 1 0 1 0
x.o ... ...&#x | 2 0 1 | 1 0 2 0 0 0 | * * * 12 * * * * * * * | 0 0 0 1 1 0 1 0 0 0 0 0 0 | 0 1 1 1 0 0
... ... o.x&#x | 1 0 2 | 0 0 2 0 0 1 | * * * * 12 * * * * * * | 0 0 0 0 1 1 0 0 1 0 0 0 0 | 0 1 0 1 1 0
ooo3ooo3ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * * * * 24 * * * * * | 0 0 0 0 0 0 1 1 1 0 0 0 0 | 0 0 1 1 1 0
.o.3.x. ... | 0 3 0 | 0 0 0 3 0 0 | * * * * * * 4 * * * * | 0 1 0 0 0 0 0 0 0 1 1 0 0 | 1 0 1 0 0 1
... .x.3.o. | 0 3 0 | 0 0 0 3 0 0 | * * * * * * * 4 * * * | 0 0 1 0 0 0 0 0 0 1 0 1 0 | 1 0 0 0 1 1
... .xo ...&#x | 0 2 1 | 0 0 0 1 2 0 | * * * * * * * * 12 * * | 0 0 0 0 0 0 0 1 0 0 1 1 0 | 0 0 1 0 1 1
... ... .ox&#x | 0 1 2 | 0 0 0 0 2 1 | * * * * * * * * * 6 * | 0 0 0 0 0 0 0 0 2 0 0 2 0 | 0 0 0 1 2 1
... ..o3..x | 0 0 3 | 0 0 0 0 0 3 | * * * * * * * * * * 4 | 0 0 0 0 0 1 0 0 0 0 0 1 1 | 0 1 0 0 1 1
---------------+-------+-----------------+----------------------------+------------------------------+------------
x..3o..3o.. ♦ 4 0 0 | 6 0 0 0 0 0 | 4 0 0 0 0 0 0 0 0 0 0 | 1 * * * * * * * * * * * * | 1 1 0 0 0 0
xo.3ox. ...&#x ♦ 3 3 0 | 3 6 0 3 0 0 | 1 3 3 0 0 0 1 0 0 0 0 | * 4 * * * * * * * * * * * | 1 0 1 0 0 0
... ox.3oo.&#x ♦ 1 3 0 | 0 3 0 3 0 0 | 0 0 3 0 0 0 0 1 0 0 0 | * * 4 * * * * * * * * * * | 1 0 0 0 1 0
x.o3o.o ...&#x ♦ 3 0 1 | 3 0 3 0 0 0 | 1 0 0 3 0 0 0 0 0 0 0 | * * * 4 * * * * * * * * * | 0 1 1 0 0 0
x.o ... o.x&#x ♦ 2 0 2 | 1 0 4 0 0 1 | 0 0 0 2 2 0 0 0 0 0 0 | * * * * 6 * * * * * * * * | 0 1 0 1 0 0
... o.o3o.x&#x ♦ 1 0 3 | 0 0 3 0 0 3 | 0 0 0 0 3 0 0 0 0 0 1 | * * * * * 4 * * * * * * * | 0 1 0 0 1 0
xoo ... ...&#x ♦ 2 1 1 | 1 2 2 0 1 0 | 0 1 0 1 0 2 0 0 0 0 0 | * * * * * * 12 * * * * * * | 0 0 1 1 0 0
... oxo ...&#x ♦ 1 2 1 | 0 2 1 1 2 0 | 0 0 1 0 0 2 0 0 1 0 0 | * * * * * * * 12 * * * * * | 0 0 1 0 1 0
... ... oox&#x ♦ 1 1 2 | 0 1 2 0 2 1 | 0 0 0 0 1 2 0 0 0 1 0 | * * * * * * * * 12 * * * * | 0 0 0 1 1 0
.o.3.x.3.o. ♦ 0 6 0 | 0 0 0 12 0 0 | 0 0 0 0 0 0 4 4 0 0 0 | * * * * * * * * * 1 * * * | 1 0 0 0 0 1
.oo3.xo ...&#x ♦ 0 3 1 | 0 0 0 3 3 0 | 0 0 0 0 0 0 1 0 3 0 0 | * * * * * * * * * * 4 * * | 0 0 1 0 0 1
... .xo3.ox&#x ♦ 0 3 3 | 0 0 0 3 6 3 | 0 0 0 0 0 0 0 1 3 3 1 | * * * * * * * * * * * 4 * | 0 0 0 0 1 1
..o3..o3..x ♦ 0 0 4 | 0 0 0 0 0 6 | 0 0 0 0 0 0 0 0 0 0 4 | * * * * * * * * * * * * 1 | 0 1 0 0 0 1
---------------+-------+-----------------+----------------------------+------------------------------+------------
xo.3ox.3oo.&#x ♦ 4 6 0 | 6 12 0 12 0 0 | 4 6 12 0 0 0 4 4 0 0 0 | 1 4 4 0 0 0 0 0 0 1 0 0 0 | 1 * * * * *
x.o3o.o3o.x&#x ♦ 4 0 4 | 6 0 12 0 0 6 | 4 0 0 12 12 0 0 0 0 0 4 | 1 0 0 4 6 4 0 0 0 0 0 0 1 | * 1 * * * *
xoo3oxo ...&#x ♦ 3 3 1 | 3 6 3 3 3 0 | 1 3 3 3 0 6 1 0 3 0 0 | 0 1 0 1 0 0 3 3 0 0 1 0 0 | * * 4 * * *
xoo ... oox&#x ♦ 2 1 2 | 1 2 4 0 2 1 | 0 1 0 2 2 4 0 0 0 1 0 | 0 0 0 0 1 0 2 0 2 0 0 0 0 | * * * 6 * *
... oxo3oox&#x ♦ 1 3 3 | 0 3 3 3 6 3 | 0 0 3 0 3 6 0 1 3 3 1 | 0 0 1 0 0 1 0 3 3 0 0 1 0 | * * * * 4 *
.oo3.xo3.ox&#x ♦ 0 6 4 | 0 0 0 12 12 6 | 0 0 0 0 0 0 4 4 12 6 4 | 0 0 0 0 0 0 0 0 0 1 4 4 1 | * * * * * 1
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