Acronym pen, K-4.1
Name pentachoron,
4D simplex4),
5-cell,
pyroter(id),
tetrahedral pyramid,
triangle-pyramidal pyramid,
regular triangle-scalene,
regular line-tettene,
regular (point-)pennene,
3 tet rosette,
vertex figure of hix,
Gosset polytope 03
|,>,O device line pyramid pyramid pyramid = |>>>
  ©  
Segmentochoron display
Cross sections
 ©
Circumradius sqrt(2/5) = 0.632456
Edge radius sqrt(3/20) = 0.387298
Face radius 1/sqrt(15) = 0.258199
Inradius 1/sqrt(40) = 0.158114
Vertex figure
 ©
Vertex layers
LayerSymmetrySubsymmetries
 o3o3o3o o3o3o . o3o . o o . o3o . o3o3o
1x3o3o3o x3o3o .
tet first
x3o . o
{3} first
x . o3o
edge first
. o3o3o
vertex first
2 o3o3o .
opposite vertex
o3o . x
opposite edge
o . x3o
opposite {3}
. x3o3o
vertex figure
opposite tet
Lace city
in approx. ASCII-art
o3o   o3o
         
   x3o   
   o o   
         
x o   o x
Lace hyper city
in approx. ASCII-art
 ©  
         
         
    x    
         
         
edge
  o      
         
        o
         
  o      
perp {3}
Volume sqrt(5)/96 = 0.023292
Surface 5 sqrt(2)/12 = 0.589256
Rel. Roundness 3 π2 sqrt(5)/500 = 13.241464 %
General of army (is itself convex)
Colonel of regiment (is itself locally convex – uniform polychoral members:
by cells: tet
pen 5
)
Dual (selfdual, in different orientation)
Dihedral angles
  • at {3} between tet and tet:   arccos(1/4) = 75.522488°
Confer
general segmentochora:
n-appy   line || perp {n}  
compounds:
sted   mix  
variations:
qo3oo3oo&#x  
Grünbaumian relatives:
2pen (?)   3pen (?)   4pen+160{3} (?)  
general polytopal classes:
tetrahedrochora   regular   noble polytopes   simplex   scalene   tettene   pennene   segmentochora   fundamental lace prisms   lace simplices   Coxeter-Elte-Gosset polytopes  
External
links
hedrondude   wikipedia   polytopewiki   WikiChoron   mathworld   quickfur

A selfdual polychoron.

The number of ways to color the pentachoron with different colors per cell is 5!/60 = 2. – This is because the color group is the permutation group of 5 elements and has size 5!, while the order of the pure rotational pentachoral group is 60. (The reflectional pentachoral group would have twice as many, i.e. 120 elements.)


Incidence matrix according to Dynkin symbol

x3o3o3o

. . . . | 5   4 |  6 | 4
--------+---+----+----+--
x . . . | 2 | 10 |  3 | 3
--------+---+----+----+--
x3o . . | 3 |  3 | 10 | 2
--------+---+----+----+--
x3o3o .  4 |  6 |  4 | 5

snubbed forms: β3o3o3o

x3o3o3/2o

. . .   . | 5   4 |  6 | 4
----------+---+----+----+--
x . .   . | 2 | 10 |  3 | 3
----------+---+----+----+--
x3o .   . | 3 |  3 | 10 | 2
----------+---+----+----+--
x3o3o   .  4 |  6 |  4 | 5

x3o3/2o3o

. .   . . | 5   4 |  6 | 4
----------+---+----+----+--
x .   . . | 2 | 10 |  3 | 3
----------+---+----+----+--
x3o   . . | 3 |  3 | 10 | 2
----------+---+----+----+--
x3o3/2o .  4 |  6 |  4 | 5

x3o3/2o3/2o

. .   .   . | 5   4 |  6 | 4
------------+---+----+----+--
x .   .   . | 2 | 10 |  3 | 3
------------+---+----+----+--
x3o   .   . | 3 |  3 | 10 | 2
------------+---+----+----+--
x3o3/2o   .  4 |  6 |  4 | 5

x3/2o3o3o

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

x3/2o3o3/2o

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

x3/2o3/2o3o

.   .   . . | 5   4 |  6 | 4
------------+---+----+----+--
x   .   . . | 2 | 10 |  3 | 3
------------+---+----+----+--
x3/2o   . . | 3 |  3 | 10 | 2
------------+---+----+----+--
x3/2o3/2o .  4 |  6 |  4 | 5

x3/2o3/2o3/2o

.   .   .   . | 5   4 |  6 | 4
--------------+---+----+----+--
x   .   .   . | 2 | 10 |  3 | 3
--------------+---+----+----+--
x3/2o   .   . | 3 |  3 | 10 | 2
--------------+---+----+----+--
x3/2o3/2o   .  4 |  6 |  4 | 5

ox3oo3oo&#x   → height = sqrt(5/8) = 0.790569
(pt || tet)

o.3o.3o.    | 1 *  4 0 | 6 0 | 4 0
.o3.o3.o    | * 4  1 3 | 3 3 | 3 1
------------+-----+-----+-----+----
oo3oo3oo&#x | 1 1 | 4 * | 3 0 | 3 0
.x .. ..    | 0 2 | * 6 | 1 2 | 2 1
------------+-----+-----+-----+----
ox .. ..&#x | 1 2 | 2 1 | 6 * | 2 0
.x3.o ..    | 0 3 | 0 3 | * 4 | 1 1
------------+-----+-----+-----+----
ox3oo ..&#x  1 3 | 3 3 | 3 1 | 4 *
.x3.o3.o     0 4 | 0 6 | 0 4 | * 1

xo ox3oo&#x   → height = sqrt(5/12) = 0.645497
(line || perp {3})

o. o.3o.    | 2 *  1 3 0 | 3 3 0 | 3 1
.o .o3.o    | * 3  0 2 2 | 1 4 1 | 2 2
------------+-----+-------+-------+----
x. .. ..    | 2 0 | 1 * * | 3 0 0 | 3 0
oo oo3oo&#x | 1 1 | * 6 * | 1 2 0 | 2 1
.. .x ..    | 0 2 | * * 3 | 0 2 1 | 1 2
------------+-----+-------+-------+----
xo .. ..&#x | 2 1 | 1 2 0 | 3 * * | 2 0
.. ox ..&#x | 1 2 | 0 2 1 | * 6 * | 1 1
.. .x3.o    | 0 3 | 0 0 3 | * * 1 | 0 2
------------+-----+-------+-------+----
xo ox ..&#x  2 2 | 1 4 1 | 2 2 0 | 3 *
.. ox3oo&#x  1 3 | 0 3 3 | 0 3 1 | * 2

oxo3ooo&#x   → height(1,2) = height(2,3) = sqrt(2/3) = 0.816497
               height(1,3) = 1
( (pt || {3}) || pt)

o..3o..    | 1 * *  3 1 0 0 | 3 3 0 0 | 1 3 0
.o.3.o.    | * 3 *  1 0 2 1 | 2 1 1 2 | 1 2 1
..o3..o    | * * 1  0 1 0 3 | 0 3 0 3 | 0 3 1
-----------+-------+---------+---------+------
oo.3oo.&#x | 1 1 0 | 3 * * * | 2 1 0 0 | 1 2 0
o.o3o.o&#x | 1 0 1 | * 1 * * | 0 3 0 0 | 0 3 0
.x. ...    | 0 2 0 | * * 3 * | 1 0 1 1 | 1 1 1
.oo3.oo&#x | 0 1 1 | * * * 3 | 0 1 0 2 | 0 2 1
-----------+-------+---------+---------+------
ox. ...&#x | 1 2 0 | 2 0 1 0 | 3 * * * | 1 1 0
ooo ...&#x | 1 1 1 | 1 1 0 1 | * 3 * * | 0 2 0
.x.3.o.    | 0 3 0 | 0 0 3 0 | * * 1 * | 1 0 1
.xo ...&#x | 0 2 1 | 0 0 1 2 | * * * 3 | 0 1 1
-----------+-------+---------+---------+------
ox.3oo.&#x  1 3 0 | 3 0 3 0 | 3 0 1 0 | 1 * *
oxo ...&#x  1 2 1 | 2 1 1 2 | 1 2 0 1 | * 3 *
.xo3.oo&#x  0 3 1 | 0 0 3 3 | 0 0 1 3 | * * 1

oxo oox&#x   → height(1,2) = height(1,3) = sqrt(3)/2 = 0.866025
               height(2,3) = 1/sqrt(2) = 0.707107
( (pt || line) || perp line)

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

ooox&#x   → height(1,2) = height(1,3) = height(2,3) = 1
            height(1,4) = height(2,4) = height(3,4) = sqrt(3)/2 = 0.866025

o...    | 1 * * *  1 1 2 0 0 0 0 | 1 2 2 1 0 0 0 | 2 1 1 0
.o..    | * 1 * *  1 0 0 1 2 0 0 | 1 2 0 0 2 1 0 | 2 1 0 1
..o.    | * * 1 *  0 1 0 1 0 2 0 | 1 0 2 0 2 0 1 | 2 0 1 1
...o    | * * * 2  0 0 1 0 1 1 1 | 0 1 1 1 1 1 1 | 1 1 1 1
--------+---------+---------------+---------------+--------
oo..&#x | 1 1 0 0 | 1 * * * * * * | 1 2 0 0 0 0 0 | 2 1 0 0
o.o.&#x | 1 0 1 0 | * 1 * * * * * | 1 0 2 0 0 0 0 | 2 0 1 0
o..o&#x | 1 0 0 1 | * * 2 * * * * | 0 1 1 1 0 0 0 | 1 1 1 0
.oo.&#x | 0 1 1 0 | * * * 1 * * * | 1 0 0 0 2 0 0 | 2 0 0 1
.o.o&#x | 0 1 0 1 | * * * * 2 * * | 0 1 0 0 1 1 0 | 1 1 0 1
..oo&#x | 0 0 1 1 | * * * * * 2 * | 0 0 1 0 1 0 1 | 1 0 1 1
...x    | 0 0 0 2 | * * * * * * 1 | 0 0 0 1 0 1 1 | 0 1 1 1
--------+---------+---------------+---------------+--------
ooo.&#x | 1 1 1 0 | 1 1 0 1 0 0 0 | 1 * * * * * * | 2 0 0 0
oo.o&#x | 1 1 0 1 | 1 0 1 0 1 0 0 | * 2 * * * * * | 1 1 0 0
o.oo&#x | 1 0 1 1 | 0 1 1 0 0 1 0 | * * 2 * * * * | 1 0 1 0
o..x&#x | 1 0 0 2 | 0 0 2 0 0 0 1 | * * * 1 * * * | 0 1 1 0
.ooo&#x | 0 1 1 1 | 0 0 0 1 1 1 0 | * * * * 2 * * | 1 0 0 1
.o.x&#x | 0 1 0 2 | 0 0 0 0 2 0 1 | * * * * * 1 * | 0 1 0 1
..ox&#x | 0 0 1 2 | 0 0 0 0 0 2 1 | * * * * * * 1 | 0 0 1 1
--------+---------+---------------+---------------+--------
oooo&#x  1 1 1 1 | 1 1 1 1 1 1 0 | 1 1 1 0 1 0 0 | 2 * * *
oo.x&#x  1 1 0 2 | 1 0 2 0 2 0 1 | 0 2 0 1 0 1 0 | * 1 * *
o.ox&#x  1 0 1 2 | 0 1 2 0 0 2 1 | 0 0 2 1 0 0 1 | * * 1 *
.oox&#x  0 1 1 2 | 0 0 0 1 2 2 1 | 0 0 0 0 2 1 1 | * * * 1

ooooo&#x   → all pairwise heights = 1

o....    | 1 * * * *  1 1 1 1 0 0 0 0 0 0 | 1 1 1 1 1 1 0 0 0 0 | 1 1 1 1 0
.o...    | * 1 * * *  1 0 0 0 1 1 1 0 0 0 | 1 1 1 0 0 0 1 1 1 0 | 1 1 1 0 1
..o..    | * * 1 * *  0 1 0 0 1 0 0 1 1 0 | 1 0 0 1 1 0 1 1 0 1 | 1 1 0 1 1
...o.    | * * * 1 *  0 0 1 0 0 1 0 1 0 1 | 0 1 0 1 0 1 1 0 1 1 | 1 0 1 1 1
....o    | * * * * 1  0 0 0 1 0 0 1 0 1 1 | 0 0 1 0 1 1 0 1 1 1 | 0 1 1 1 1
---------+-----------+---------------------+---------------------+----------
oo...&#x | 1 1 0 0 0 | 1 * * * * * * * * * | 1 1 1 0 0 0 0 0 0 0 | 1 1 1 0 0
o.o..&#x | 1 0 1 0 0 | * 1 * * * * * * * * | 1 0 0 1 1 0 0 0 0 0 | 1 1 0 1 0
o..o.&#x | 1 0 0 1 0 | * * 1 * * * * * * * | 0 1 0 1 0 1 0 0 0 0 | 1 0 1 1 0
o...o&#x | 1 0 0 0 1 | * * * 1 * * * * * * | 0 0 1 0 1 1 0 0 0 0 | 0 1 1 1 0
.oo..&#x | 0 1 1 0 0 | * * * * 1 * * * * * | 1 0 0 0 0 0 1 1 0 0 | 1 1 0 0 1
.o.o.&#x | 0 1 0 1 0 | * * * * * 1 * * * * | 0 1 0 0 0 0 1 0 1 0 | 1 0 1 0 1
.o..o&#x | 0 1 0 0 1 | * * * * * * 1 * * * | 0 0 1 0 0 0 0 1 1 0 | 0 1 1 0 1
..oo.&#x | 0 0 1 1 0 | * * * * * * * 1 * * | 0 0 0 1 0 0 1 0 0 1 | 1 0 0 1 1
..o.o&#x | 0 0 1 0 1 | * * * * * * * * 1 * | 0 0 0 0 1 0 0 1 0 1 | 0 1 0 1 1
...oo&#x | 0 0 0 1 1 | * * * * * * * * * 1 | 0 0 0 0 0 1 0 0 1 1 | 0 0 1 1 1
---------+-----------+---------------------+---------------------+----------
ooo..&#x | 1 1 1 0 0 | 1 1 0 0 1 0 0 0 0 0 | 1 * * * * * * * * * | 1 1 0 0 0
oo.o.&#x | 1 1 0 1 0 | 1 0 1 0 0 1 0 0 0 0 | * 1 * * * * * * * * | 1 0 1 0 0
oo..o&#x | 1 1 0 0 1 | 1 0 0 1 0 0 1 0 0 0 | * * 1 * * * * * * * | 0 1 1 0 0
o.oo.&#x | 1 0 1 1 0 | 0 1 1 0 0 0 0 1 0 0 | * * * 1 * * * * * * | 1 0 0 1 0
o.o.o&#x | 1 0 1 0 1 | 0 1 0 1 0 0 0 0 1 0 | * * * * 1 * * * * * | 0 1 0 1 0
o..oo&#x | 1 0 0 1 1 | 0 0 1 1 0 0 0 0 0 1 | * * * * * 1 * * * * | 0 0 1 1 0
.ooo.&#x | 0 1 1 1 0 | 0 0 0 0 1 1 0 1 0 0 | * * * * * * 1 * * * | 1 0 0 0 1
.oo.o&#x | 0 1 1 0 1 | 0 0 0 0 1 0 1 0 1 0 | * * * * * * * 1 * * | 0 1 0 0 1
.o.oo&#x | 0 1 0 1 1 | 0 0 0 0 0 1 1 0 0 1 | * * * * * * * * 1 * | 0 0 1 0 1
..ooo&#x | 0 0 1 1 1 | 0 0 0 0 0 0 0 1 1 1 | * * * * * * * * * 1 | 0 0 0 1 1
---------+-----------+---------------------+---------------------+----------
oooo.&#x  1 1 1 1 0 | 1 1 1 0 1 1 0 1 0 0 | 1 1 0 1 0 0 1 0 0 0 | 1 * * * *
ooo.o&#x  1 1 1 0 1 | 1 1 0 1 1 0 1 0 1 0 | 1 0 1 0 1 0 0 1 0 0 | * 1 * * *
oo.oo&#x  1 1 0 1 1 | 1 0 1 1 0 1 1 0 0 1 | 0 1 1 0 0 1 0 0 1 0 | * * 1 * *
o.ooo&#x  1 0 1 1 1 | 0 1 1 1 0 0 0 1 1 1 | 0 0 0 1 1 1 0 0 0 1 | * * * 1 *
.oooo&#x  0 1 1 1 1 | 0 0 0 0 1 1 1 1 1 1 | 0 0 0 0 0 0 1 1 1 1 | * * * * 1

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