tet

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

|,>,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

Layer	Symmetry	Subsymmetries
	`o3o3o`	`o3o .`	`o . o`	`. o3o`
1	`x3o3o`	`x3o .` {3} first	`x . o` edge first	`. o3o` vertex first
2	`x3o3o`	`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°

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 S_n birectified simplex brS_n demihypercube D_n

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.

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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