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Acronym	tet
TOCID symbol	T, (2)Q
Name	tetrahedron, 3Dsimplex (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 ofpen, Gosset polytope 02, Waterman polyhedron number 1 wrt. face-centered cubiclattice 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 aretrigonal pyramids (which thus is describable byox3oo&#y). Those with 2 alike faces aresphenoids. Those with 2 pairs of alike faces then aredisphenoids.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:atetragonal disphenoid will have four identicalisosceles triangles (which thus is describable byxo ox&#y or as digonalantiprism of arbitrary height), adigonal disphenoid has two types of isosceles triangles (which thus isxo oy&#z), arhombic disphenoid has four identicalscalene triangles, and aphyllic 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 theoct:

• 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 toDynkin symbol

x3o3o. . . | 4 | 3 | 3------+---+---+--x . . | 2 | 6 | 2------+---+---+--x3o . | 3 | 3 | 4snubbed forms:β3o3o

x3o3/2o. .   . | 4 | 3 | 3--------+---+---+--x .   . | 2 | 6 | 2--------+---+---+--x3o   . | 3 | 3 | 4snubbed forms:β3o3/2o

x3/2o3o.   . . | 4 | 3 | 3--------+---+---+--x   . . | 2 | 6 | 2--------+---+---+--x3/2o . | 3 | 3 | 4snubbed forms:β3/2o3o

x3/2o3/2o.   .   . | 4 | 3 | 3----------+---+---+--x   .   . | 2 | 6 | 2----------+---+---+--x3/2o   . | 3 | 3 | 4snubbed forms:β3/2o3/2o

s4o3odemi( . . . ) |4 | 3 | 3--------------+---+---+--      s4o .♦ 2 | 6 | 2--------------+---+---+--sefa( s4o3o ) | 3 | 3 | 4starting figure:x4o3o

s2s4odemi( . . . ) |4 | 2 1 | 3--------------+---+-----+--      s2s .♦ 2 | 4 * | 2      . s4o♦ 2 | * 2 | 2--------------+---+-----+--sefa( s2s4o ) | 3 | 2 1 | 4starting figure:x x4o

s2s2sdemi( . . . ) |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 | 4starting 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 1oo3oo&#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 0oo 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 0o.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 = 1o...    |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 0o.o.&#x | 1 0 1 0 | * 1 * * * * | 1 0 1 0o..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