Acronym	tet
TOCID symbol	T, (2)Q
Name	tetrahedron, 3D simplex (α3), 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, Waterman polyhedron number 1 wrt. face-centered cubic lattice A3 centered at a shallow hole
|,>,O device	line pyramid pyramid = |>>
VRML	⭳ ©
Circumradius	sqrt(3/8) = 0.612372
Edge radius	1/sqrt(8) = 0.353553
Inradius	1/sqrt(24) = 0.204124
Vertex figure	[33] = x3o
Snub derivation / VRML	⭳
Vertex layers	Layer Symmetry Subsymmetries   o3o3o o3o . o . o . o3o 1 x3o3o x3o . {3} first x . o edge first . o3o vertex first 2 o3o . opposite vertex o . x opposite edge . x3o vertex figure opposite {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°
Face vector	4, 6, 4
Confer	more general: xPoPo   n/d-py   n/d-ap   variations: xo ox&#q   xo oq&#q   ho oh&#q   xo ox&#h   qo oq&#h   qo ou&#h   uo ou&#h   xo ox&#k   xo3oo&#q   qo3oo&#x   qo3oo&#h   Grünbaumian relatives: 2tet   3tet   4tet   6tet   blends: tridpy   compounds: so   ki   e   sis   snu   dis   general polytopal classes: Wythoffian polyhedra   Catalan polyhedra   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   rectified simplex pyramid rSn-py
External links

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.

Somehow off-topic there are some neet number relations between the tet and the oct:

• face count ratio: F_oct / F_tet = 8/4 = 2
• face type: n_oct = n_tet = 3 (triangles each)
• radius ratio: (circumradius_oct / inradius_oct)² = circumradius_tet/inradius_tet = 3
• volume ratio: V_oct / V_tet = 2

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