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 = |>> | |||||||||||||||||||
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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 |
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Vertex layers |
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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 |
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Face vector | 4, 6, 4 | |||||||||||||||||||
Confer |
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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:
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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