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 |
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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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Confer |
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External links |
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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. Those with 2 alike faces are sphenoids. Those with 2 pairs of alike faces are disphenoids. Thus a tetrahedron is a special case of all these.
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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