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 = |>>
 
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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
Vertex layers
LayerSymmetrySubsymmetries
 o3o3oo3o .o . o. o3o
1x3o3ox3o .
{3} first
x . o
edge first
. o3o
vertex first
2o3o .
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°
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
links
hedrondude   wikipedia   polytopewiki   WikiChoron   mathworld   quickfur

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

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