Acronym tet
TOCID symbol T, (2)Q
Name tetrahedron,
3D simplex3),
pyrochor(id),
triangle pyramid,
digonal antiprism,
hemicube,
smaller Delone cell of face-centered cubic (fcc) lattice,
regular line-scalene,
regular (point-)tettene,
vertex figure of pen
|,>,O device line pyramid pyramid = |>>
 
 © ©
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 even permutations, all even changes of sign
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   xo3oo&#q  
Grünbaumian relatives:
2tet   3tet   4tet   6tet  
blends:
tridpy  
compounds:
so   ki   e   sis   snu   dis  
general polytopal classes:
deltahedra   regular   noble polytopes   simplex   partial Stott expansions   segmentohedra   fundamental lace prisms  
External
links
hedrondude   wikipedia   WikiChoron   mathworld   quickfur

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

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