Acronym tet
TOCID symbol T, (2)Q
Name tetrahedron,
3D simplex3),
pyrochor(id),
regular trigonal pyramid,
digonal antiprism,
regular (di)sphenoid,
hemicube,
smaller Delone cell of face-centered cubic (fcc) lattice,
regular line-scalene,
regular (point-)tettene,
vertex figure of pen,
Gosset polytope 02
|,>,O device line pyramid pyramid = |>>

` © ©`
Vertex figure [33] = x3o
Snub derivation
Vertex layers
 Layer Symmetry Subsymmetries o3o3o o3o . o . o . o3o 1 x3o3o x3o .{3} first x . oedge first . o3overtex first 2 o3o .opposite vertex o . xopposite edge . x3overtex figureopposite {3}
Lace city
in approx. ASCII-art
``` o
x o
```
Coordinates (1/sqrt(8), 1/sqrt(8), 1/sqrt(8))   & all permutations, all even changes of sign
Volume sqrt(2)/12 = 0.117851
Surface sqrt(3) = 1.732051
Rel. Roundness π sqrt(3)/18 = 30.229989 %
General of army (is itself convex)
Colonel of regiment (is itself locally convex – no other uniform polyhedral members)
Dual (selfdual, in different orientation)
Dihedral angles
• between {3} and {3}:   arccos(1/3) = 70.528779°
Confer
more general:
xPoPo   n/d-py   n/d-ap
variations:
xo ox&#q   xo oq&#q   ho oh&#q   xo ox&#h   xo ox&#k   xo3oo&#q   qo3oo&#x
Grünbaumian relatives:
2tet   3tet   4tet   6tet
blends:
tridpy
compounds:
so   ki   e   sis   snu   dis
general polytopal classes:
deltahedra   regular   noble polytopes   simplex   scalene   tettene   partial Stott expansions   segmentohedra   fundamental lace prisms   lace simplices   Coxeter-Elte-Gosset polytopes
analogs:
regular simplex Sn   birectified simplex brSn   demihypercube Dn
External

The number of ways to color the tetrahedron with different colors per face is 4!/12 = 2. – This is because the color group is the permutation group of 4 elements and has size 4!, while the order of the pure rotational tetrahedral group is 12. (The reflectional tetrahedral group would have twice as many, i.e. 24 elements.)

3D simplices with 3 alike faces are trigonal pyramids (which thus is describable by ox3oo&#y). Those with 2 alike faces are sphenoids. Those with 2 pairs of alike faces then are disphenoids. The (regular) tetrahedron hence is just a special case of all these. More specially some authors even want to distinguish the various types of those disphenoids by means of additional attributions: a tetragonal disphenoid will have four identical isosceles triangles (which thus is describable by xo ox&#y or as digonal antiprism of arbitrary height), a digonal disphenoid has two types of isosceles triangles (which thus is xo oy&#z), a rhombic disphenoid has four identical scalene triangles, and a phyllic disphenoid has two types of scalene triangles, i.e. the latter two just are chiral versions of the formers.

Incidence matrix according to Dynkin symbol

```x3o3o

. . . | 4 | 3 | 3
------+---+---+--
x . . | 2 | 6 | 2
------+---+---+--
x3o . | 3 | 3 | 4

snubbed forms: β3o3o
```

```x3o3/2o

. .   . | 4 | 3 | 3
--------+---+---+--
x .   . | 2 | 6 | 2
--------+---+---+--
x3o   . | 3 | 3 | 4

snubbed forms: β3o3/2o
```

```x3/2o3o

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

snubbed forms: β3/2o3o
```

```x3/2o3/2o

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

snubbed forms: β3/2o3/2o
```

```s4o3o

demi( . . . ) | 4 | 3 | 3
--------------+---+---+--
s4o .   ♦ 2 | 6 | 2
--------------+---+---+--
sefa( s4o3o ) | 3 | 3 | 4

starting figure: x4o3o
```

```s2s4o

demi( . . . ) | 4 | 2 1 | 3
--------------+---+-----+--
s2s .   ♦ 2 | 4 * | 2
. s4o   ♦ 2 | * 2 | 2
--------------+---+-----+--
sefa( s2s4o ) | 3 | 2 1 | 4

starting figure: x x4o
```

```s2s2s

demi( . . . ) | 4 | 1 1 1 | 3
--------------+---+-------+--
s2s .   ♦ 2 | 2 * * | 2
s 2 s   ♦ 2 | * 2 * | 2
. s2s   ♦ 2 | * * 2 | 2
--------------+---+-------+--
sefa( s2s2s ) | 3 | 1 1 1 | 4

starting figure: x x x
```

```xo3oo&#x   → height = sqrt(2/3) = 0.816497
({3} || pt)

o.3o.    | 3 * | 2 1 | 1 2
.o3.o    | * 1 | 0 3 | 0 3
---------+-----+-----+----
x. ..    | 2 0 | 3 * | 1 1
oo3oo&#x | 1 1 | * 3 | 0 2
---------+-----+-----+----
x.3o.    | 3 0 | 3 0 | 1 *
xo ..&#x | 2 1 | 1 2 | * 3
```

```xo ox&#x   → height = 1/sqrt(2) = 0.707107
(line || perp line)

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

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

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

```oooo&#x   → all pairwise heights = 1

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