5D simplex (α5),
pyroteron,
pentachoral pyramid,
regular tet-scalene,
regular triangle-tettene,
regular line-pennene,
regular (point-)hixene
©
- general polytopal classes:
- simplex
links
| Acronym | hix |
| Name |
hexateron, 5D simplex (α5), pyroteron, pentachoral pyramid, regular tet-scalene, regular triangle-tettene, regular line-pennene, regular (point-)hixene |
| Circumradius | sqrt(5/12) = 0.645497 |
| Inradius | 1/sqrt(60) = 0.129099 |
| Vertex figure |
|
| Dual | (selfdual, in different orientation) |
| Confer |
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External links |
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Incidence matrix according to Dynkin symbol
x3o3o3o3o . . . . . | 6 ♦ 5 | 10 | 10 | 5 ----------+---+----+----+----+-- x . . . . | 2 | 15 ♦ 4 | 6 | 4 ----------+---+----+----+----+-- x3o . . . | 3 | 3 | 20 | 3 | 3 ----------+---+----+----+----+-- x3o3o . . ♦ 4 | 6 | 4 | 15 | 2 ----------+---+----+----+----+-- x3o3o3o . ♦ 5 | 10 | 10 | 5 | 6
ox3oo3oo3oo&#x → height = sqrt(3/5) = 0.774597
(pt || pen)
o.3o.3o.3o. | 1 * ♦ 5 0 | 10 0 | 10 0 | 5 0
.o3.o3.o3.o | * 5 ♦ 1 4 | 4 6 | 6 4 | 4 1
---------------+-----+------+-------+------+----
oo3oo3oo3oo&#x | 1 1 | 5 * ♦ 4 0 | 6 0 | 4 0
.x .. .. .. | 0 2 | * 10 ♦ 1 3 | 3 3 | 3 1
---------------+-----+------+-------+------+----
ox .. .. ..&#x | 1 2 | 2 1 | 10 * | 3 0 | 3 0
.x3.o .. .. | 0 3 | 0 3 | * 10 | 1 2 | 2 1
---------------+-----+------+-------+------+----
ox3oo .. ..&#x ♦ 1 3 | 3 3 | 3 1 | 10 * | 2 0
.x3.o3.o .. ♦ 0 4 | 0 6 | 0 4 | * 5 | 1 1
---------------+-----+------+-------+------+----
ox3oo3oo ..&#x ♦ 1 4 | 4 6 | 6 4 | 4 1 | 5 *
.x3.o3.o3.o ♦ 0 5 | 0 10 | 0 10 | 0 5 | * 1
xo ox3oo3oo&#x → height = sqrt(3/8) = 0.612372
(line || ortho tet)
o. o.3o.3o. | 2 * ♦ 1 4 0 | 4 6 0 | 6 4 0 | 4 1
.o .o3.o3.o | * 4 ♦ 0 2 3 | 1 6 3 | 3 6 1 | 3 2
---------------+-----+-------+--------+-------+----
x. .. .. .. | 2 0 | 1 * * ♦ 4 0 0 | 6 0 0 | 4 0
oo oo3oo3oo&#x | 1 1 | * 8 * ♦ 1 3 0 | 3 3 0 | 3 1
.. .x .. .. | 0 2 | * * 6 ♦ 0 2 2 | 1 4 1 | 2 2
---------------+-----+-------+--------+-------+----
xo .. .. ..&#x | 2 1 | 1 2 0 | 4 * * | 3 0 0 | 3 0
.. ox .. ..&#x | 1 2 | 0 2 1 | * 12 * | 1 2 0 | 2 1
.. .x3.o .. | 0 3 | 0 0 3 | * * 4 | 0 2 1 | 1 2
---------------+-----+-------+--------+-------+----
xo ox .. ..&#x ♦ 2 2 | 1 4 1 | 2 2 0 | 6 * * | 2 0
.. ox3oo ..&#x ♦ 1 3 | 0 3 3 | 0 3 1 | * 8 * | 1 1
.. .x3.o3.o ♦ 0 4 | 0 0 6 | 0 0 4 | * * 1 | 0 2
---------------+-----+-------+--------+-------+----
xo ox3oo ..&#x ♦ 2 3 | 1 6 3 | 3 6 1 | 3 2 0 | 4 *
.. ox3oo3oo&#x ♦ 1 4 | 0 4 6 | 0 6 4 | 0 4 1 | * 2
xo3oo ox3oo&#x → height = 1/sqrt(3) = 0.577350
({3} || ortho {3})
o.3o. o.3o. | 3 * ♦ 2 3 0 | 1 6 3 0 | 3 6 1 | 3 2
.o3.o .o3.o | * 3 ♦ 0 3 2 | 0 3 6 1 | 1 6 3 | 2 3
---------------+-----+-------+---------+-------+----
x. .. .. .. | 2 0 | 3 * * ♦ 1 3 0 0 | 3 3 0 | 3 1
oo3oo oo3oo&#x | 1 1 | * 9 * ♦ 0 2 2 0 | 1 4 1 | 2 2
.. .. .x .. | 0 2 | * * 3 ♦ 0 0 3 1 | 0 3 3 | 1 3
---------------+-----+-------+---------+-------+----
x.3o. .. .. | 3 0 | 3 0 0 | 1 * * * | 3 0 0 | 3 0
xo .. .. ..&#x | 2 1 | 1 2 0 | * 9 * * | 1 2 0 | 2 1
.. .. ox ..&#x | 1 2 | 0 2 1 | * * 9 * | 0 2 1 | 1 2
.. .. .x3.o | 0 3 | 0 0 3 | * * * 1 | 0 0 3 | 0 3
---------------+-----+-------+---------+-------+----
xo3oo .. ..&#x ♦ 3 1 | 3 3 0 | 1 3 0 0 | 3 * * | 2 0
xo .. ox ..&#x ♦ 2 2 | 1 4 1 | 0 2 2 0 | * 9 * | 1 1
.. .. ox3oo&#x ♦ 1 3 | 0 3 3 | 0 0 3 1 | * * 3 | 0 2
---------------+-----+-------+---------+-------+----
xo3oo ox ..&#x ♦ 3 2 | 3 6 1 | 1 6 3 0 | 2 3 0 | 3 *
xo .. ox3oo&#x ♦ 2 3 | 1 6 3 | 0 3 6 1 | 0 3 2 | * 3
or
o.3o. o.3o. & | 6 ♦ 2 3 | 1 9 | 4 3 | 5
-----------------+---+-----+------+-----+--
x. .. .. .. & | 2 | 6 * ♦ 1 3 | 3 3 | 4
oo3oo oo3oo&#x | 2 | * 9 ♦ 0 4 | 2 4 | 4
-----------------+---+-----+------+-----+--
x.3o. .. .. & | 3 | 3 0 | 2 * | 3 0 | 3
xo .. .. ..&#x & | 3 | 1 2 | * 18 | 1 2 | 3
-----------------+---+-----+------+-----+--
xo3oo .. ..&#x & ♦ 4 | 3 3 | 1 3 | 6 * | 2
xo .. ox ..&#x ♦ 4 | 2 4 | 0 4 | * 9 | 2
-----------------+---+-----+------+-----+--
xo3oo ox ..&#x & ♦ 5 | 4 6 | 1 9 | 2 3 | 6
oxo3ooo3ooo&#x → height(1,2) = height(2,3) = sqrt(5/8) = 0.790569
height(1,3) = 1
( (pt || tet) || pt )
o..3o..3o.. | 1 * * ♦ 4 1 0 0 | 6 4 0 0 | 4 6 0 0 | 1 4 0
.o.3.o.3.o. | * 4 * ♦ 1 0 3 1 | 3 1 3 3 | 3 3 1 3 | 1 3 1
..o3..o3..o | * * 1 ♦ 0 1 0 4 | 0 4 0 6 | 0 6 0 4 | 0 4 1
---------------+-------+---------+---------+---------+------
oo.3oo.3oo.&#x | 1 1 0 | 4 * * * ♦ 3 1 0 0 | 3 3 0 0 | 1 3 0
o.o3o.o3o.o&#x | 1 0 1 | * 1 * * ♦ 0 4 0 0 | 0 6 0 0 | 0 4 0
.x. ... ... | 0 2 0 | * * 6 * ♦ 1 0 2 1 | 2 1 1 2 | 1 2 1
.oo3.oo3.oo&#x | 0 1 1 | * * * 4 ♦ 0 1 0 3 | 0 3 0 3 | 0 3 1
---------------+-------+---------+---------+---------+------
ox. ... ...&#x | 1 2 0 | 2 0 1 0 | 6 * * * | 2 1 0 0 | 1 2 0
ooo3ooo3ooo&#x | 1 1 1 | 1 1 0 1 | * 4 * * | 0 3 0 0 | 0 3 0
.x.3.o. ... | 0 3 0 | 0 0 3 0 | * * 4 * | 1 0 1 1 | 1 1 1
.xo ... ...&#x | 0 2 1 | 0 0 1 2 | * * * 6 | 0 1 0 2 | 0 2 1
---------------+-------+---------+---------+---------+------
ox.3oo. ...&#x ♦ 1 3 0 | 3 0 3 0 | 3 0 1 0 | 4 * * * | 1 1 0
oxo ... ...&#x ♦ 1 2 1 | 2 1 1 2 | 1 2 0 1 | * 6 * * | 0 2 0
.x.3.o.3.o. ♦ 0 4 0 | 0 0 6 0 | 0 0 4 0 | * * 1 * | 1 0 1
.xo3.oo ...&#x ♦ 0 3 1 | 0 0 3 3 | 0 0 1 3 | * * * 4 | 0 1 1
---------------+-------+---------+---------+---------+------
ox.3oo.3oo.&#x ♦ 1 4 0 | 4 0 6 0 | 6 0 4 0 | 4 0 1 0 | 1 * *
oxo3ooo ...&#x ♦ 1 3 1 | 3 1 3 3 | 3 3 1 3 | 1 3 0 1 | * 4 *
.xo3.oo3.oo&#x ♦ 0 4 1 | 0 0 6 4 | 0 0 4 6 | 0 0 1 4 | * * 1
oxo3ooo oox&#x → height(1,2) = sqrt(2/3) = 0.816497
height(1,3) = sqrt(3)/2 = 0.866025
height(2,3) = sqrt(5/12) = 0.645497
( (pt || {3}) || line )
o..3o.. o.. | 1 * * ♦ 3 2 0 0 0 | 3 1 6 0 0 0 | 1 6 3 0 0 | 2 3 0
.o.3.o. .o. | * 3 * ♦ 1 0 2 2 0 | 2 0 2 1 4 1 | 1 4 1 2 2 | 2 2 1
..o3..o ..o | * * 2 ♦ 0 1 0 3 1 | 0 1 3 0 3 3 | 0 3 3 1 3 | 1 3 1
---------------+-------+-----------+-------------+-----------+------
oo.3oo. oo.&#x | 1 1 0 | 3 * * * * ♦ 2 0 2 0 0 0 | 1 4 1 0 0 | 2 2 0
o.o3o.o o.o&#x | 1 0 1 | * 2 * * * ♦ 0 1 3 0 0 0 | 0 3 3 0 0 | 1 3 0
.x. ... ... | 0 2 0 | * * 3 * * ♦ 1 0 0 1 2 0 | 1 2 0 2 1 | 2 1 1
.oo3.oo .oo&#x | 0 1 1 | * * * 6 * ♦ 0 0 1 0 2 1 | 0 2 1 1 2 | 1 2 1
... ... ..x | 0 0 2 | * * * * 1 ♦ 0 1 0 0 0 3 | 0 0 3 0 3 | 0 3 1
---------------+-------+-----------+-------------+-----------+------
ox. ... ...&#x | 1 2 0 | 2 0 1 0 0 | 3 * * * * * | 1 2 0 0 0 | 2 1 0
... ... o.x&#x | 1 0 2 | 0 2 0 0 1 | * 1 * * * * | 0 0 3 0 0 | 0 3 0
ooo3ooo ooo&#x | 1 1 1 | 1 1 0 1 0 | * * 6 * * * | 0 2 1 0 0 | 1 2 0
.x.3.o. ... | 0 3 0 | 0 0 3 0 0 | * * * 1 * * | 1 0 0 2 0 | 2 0 1
.xo ... ...&#x | 0 2 1 | 0 0 1 2 0 | * * * * 6 * | 0 1 0 1 1 | 1 1 1
... ... .ox&#x | 0 1 2 | 0 0 0 2 1 | * * * * * 3 | 0 0 1 0 2 | 0 2 1
---------------+-------+-----------+-------------+-----------+------
ox.3oo. ...&#x ♦ 1 3 0 | 3 0 3 0 0 | 3 0 0 1 0 0 | 1 * * * * | 2 0 0
oxo ... ...&#x ♦ 1 2 1 | 2 1 1 2 0 | 1 0 2 0 1 0 | * 6 * * * | 1 1 0
... ... oox&#x ♦ 1 1 2 | 1 2 0 2 1 | 0 1 2 0 0 1 | * * 3 * * | 0 2 0
.xo3.oo ...&#x ♦ 0 3 1 | 0 0 3 3 0 | 0 0 0 1 3 0 | * * * 2 * | 1 0 1
.xo ... .ox&#x ♦ 0 2 2 | 0 0 1 4 1 | 0 0 0 0 2 2 | * * * * 3 | 0 1 1
---------------+-------+-----------+-------------+-----------+------
oxo3ooo ...&#x ♦ 1 3 1 | 3 1 3 3 0 | 3 0 3 1 3 0 | 1 3 0 1 0 | 2 * *
oxo ... oox&#x ♦ 1 2 2 | 2 2 1 4 1 | 1 1 4 0 2 2 | 0 2 2 0 1 | * 3 *
.xo3.oo .ox&#x ♦ 0 3 2 | 0 0 3 6 1 | 0 0 0 1 6 3 | 0 0 0 2 3 | * * 1
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