Acronym pen, K-4.1
Name pentachoron,
4D simplex (α4),
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,
5-2-stepprism
|,>,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°
Face vector 5, 10, 10, 5
Confer
general segmentochora:
n-appy   line || perp {n}  
compounds:
sted   mix  
variations:
qo3oo3oo&#x   xo3oo3oo&#q   xo3oo ox&#q  
Grünbaumian relatives:
2pen   3pen   4pen   4pen+160{3}   6pen  
ambification:
rap  
general polytopal classes:
Wythoffian polychora   Catalan polychora   tetrahedrochora   regular   noble polytopes   simplex   scalene   tettene   pennene   segmentochora   fundamental lace prisms   lace simplices   Coxeter-Elte-Gosset polytopes  
analogs:
regular simplex Sn   Gossetic 2n,1   Gossetic 1n,2  
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.)

©

The pentachoron allows for a projection into 2D as the complete graph of 5 vertices K5, i.e. as the overlay of a convex pentagon and a vertex-inscribed pentagram, cf. to the right. In fact this corresponds to the folding of A4 into H2. 

folding A4 into H2

o   o
 \ / 
  X  
 / \ 
o---o

=>

o---o
  5

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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