Acronym hix
Name hexateron,
5D simplex5),
pyropet(id),
pentachoral pyramid,
regular tet-scalene,
regular triangle-tettene,
regular line-pennene,
regular (point-)hixene,
vertex figure of hop
|,>,O device line pyramid pyramid pyramid pyramid = |>>>>
Field of sections
 ©
Circumradius sqrt(5/12) = 0.645497
Inradius 1/sqrt(60) = 0.129099
Vertex figure
 ©
Lace city
in approx. ASCII-art
o3o3o   o3o3o
             
             
    x3o3o    
Dual (selfdual, in different orientation)
Dihedral angles
  • at tet between pen and pen:   arccos(1/5) = 78.463041°
Confer
related segmentotera:
pexhix   pabex hix  
general polytopal classes:
simplex   tettene   partial Stott expansions   segmentotera   fundamental lace prisms  
External
links
hedrondude   wikipedia   mathworld

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

x3/2o3o3o3o

.   . . . . | 6   5 | 10 | 10 | 5
------------+---+----+----+----+--
x   . . . . | 2 | 15   4 |  6 | 4
------------+---+----+----+----+--
x3/2o . . . | 3 |  3 | 20 |  3 | 3
------------+---+----+----+----+--
x3/2o3o . .  4 |  6 |  4 | 15 | 2
------------+---+----+----+----+--
x3/2o3o3o .  5 | 10 | 10 |  5 | 6

x3o3o3o3/2o

. . . .   . | 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 || perp 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} || perp {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}) || perp 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

xoo oxo oox&#x   → all heights = 1/sqrt(2) = 0.707107
( (line || perp line) || perp line)

o.. o.. o..    | 2 * *  1 2 2 0 0 0 | 2 2 1 1 4 0 0 | 4 1 1 2 2 0 | 2 2 1
.o. .o. .o.    | * 2 *  0 2 0 1 2 0 | 1 0 2 0 4 2 1 | 2 1 0 4 2 1 | 2 1 2
..o ..o ..o    | * * 2  0 0 2 0 2 1 | 0 1 0 2 4 1 2 | 2 0 1 2 4 1 | 1 2 2
---------------+-------+-------------+---------------+-------------+------
x.. ... ...    | 2 0 0 | 1 * * * * *  2 2 0 0 0 0 0 | 4 1 1 0 0 0 | 2 2 0
oo. oo. oo.&#x | 1 1 0 | * 4 * * * *  1 0 1 0 2 0 0 | 2 1 0 2 1 0 | 2 1 1
o.o o.o o.o&#x | 1 0 1 | * * 4 * * *  0 1 0 1 2 0 0 | 2 0 1 1 2 0 | 1 2 1
... .x. ...    | 0 2 0 | * * * 1 * *  0 0 2 0 0 2 0 | 0 1 0 4 0 1 | 2 0 2
.oo .oo .oo&#x | 0 1 1 | * * * * 4 *  0 0 0 0 2 1 1 | 1 0 0 2 2 1 | 1 1 2
... ... ..x    | 0 0 2 | * * * * * 1  0 0 0 2 0 0 2 | 0 0 1 0 4 1 | 0 2 2
---------------+-------+-------------+---------------+-------------+------
xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 0 | 2 * * * * * * | 2 1 0 0 0 0 | 2 1 0
x.o ... ...&#x | 2 0 1 | 1 0 2 0 0 0 | * 2 * * * * * | 2 0 1 0 0 0 | 1 2 0
... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 | * * 2 * * * * | 0 1 0 2 0 0 | 2 0 1
... ... o.x&#x | 1 0 2 | 0 0 2 0 0 1 | * * * 2 * * * | 0 0 1 0 2 0 | 0 2 1
ooo ooo ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * * * 8 * * | 1 0 0 1 1 0 | 1 1 1
... .xo ...&#x | 0 2 1 | 0 0 0 1 2 0 | * * * * * 2 * | 0 0 0 2 0 1 | 1 0 2
... ... .ox&#x | 0 1 2 | 0 0 0 0 2 1 | * * * * * * 2 | 0 0 0 0 2 1 | 0 1 2
---------------+-------+-------------+---------------+-------------+------
xoo ... ...&#x  2 1 1 | 1 2 2 0 1 0 | 1 1 0 0 2 0 0 | 4 * * * * * | 1 1 0
xo. ox. ...&#x  2 2 0 | 1 4 0 1 0 0 | 2 0 2 0 0 0 0 | * 1 * * * * | 2 0 0
x.o ... o.x&#x  2 0 2 | 1 0 4 0 0 1 | 0 2 0 2 0 0 0 | * * 1 * * * | 0 2 0
... oxo ...&#x  1 2 1 | 0 2 1 1 2 0 | 0 0 1 0 2 1 0 | * * * 4 * * | 1 0 1
... ... oox&#x  1 1 2 | 0 1 2 0 2 1 | 0 0 0 1 2 0 1 | * * * * 4 * | 0 1 1
... .xo .ox&#x  0 2 2 | 0 0 0 1 4 1 | 0 0 0 0 0 2 2 | * * * * * 1 | 0 0 2
---------------+-------+-------------+---------------+-------------+------
xoo oxo ...&#x  2 2 1 | 1 4 2 1 2 0 | 2 1 2 0 4 1 0 | 2 1 0 2 0 0 | 2 * *
xoo ... oox&#x  2 1 2 | 1 2 4 0 2 1 | 1 2 0 2 4 0 1 | 2 0 1 0 2 0 | * 2 *
... oxo oox&#x  1 2 2 | 0 2 2 1 4 1 | 0 0 1 1 4 2 2 | 0 0 0 2 2 1 | * * 2

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