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,
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
 Layer Symmetry Subsymmetries o3o3o3o o3o3o . o3o . o o . o3o . o3o3o 1 x3o3o3o x3o3o .tet first x3o . o{3} first x . o3oedge first . o3o3overtex first 2 o3o3o .opposite vertex o3o . xopposite edge o . x3oopposite {3} . x3o3overtex figureopposite 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
analogs:
regular simplex Sn   Gossetic 2n,1   Gossetic 1n,2
External

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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