Acronym oct
TOCID symbol O, TT, (3)Q
Name octahedron,
rectified tetrahedron,
tricross3),
tetratetrahedron,
aerochor(id),
trigonal antiprism,
larger Delone cell of face-centered cubic (fcc) lattice,
equatorial cross-section of (vertex first) 1/q-tes,
vertex figure of hex
 
 © ©
Circumradius 1/sqrt(2) = 0.707107
Edge radius 1/2
Inradius 1/sqrt(6) = 0.408248
Vertex figure [34] = x4o
Snub derivation
Vertex layers
LayerSymmetrySubsymmetries
 o3o4oo3o .o . o. o4o
1x3o4ox3o .
{3} first
x . o
edge first
. o4o
vertex first
2o3x .
opposite {3}
o . q. x4o
vertex figure
3 x . o
opposite edge
. o4o
opposite vertex
 o3o3oo3o .o . o. o3o
1o3x3oo3x .
{3} first
o . o
vertex first
. x3o
{3} first
2x3o .
opposite {3}
x . x
vertex figure
. o3x
opposite {3}
3 o . o
opposite vertex
 
Lace city
in approx. ASCII-art
 x o
o x 
  o  
o q o
  o  
Coordinates (1/sqrt(2), 0, 0)   & all permutations, all changes of sign
General of army (is itself convex)
Colonel of regiment (is itself locally convex – other uniform polyhedral member: thah – other edge facetings)
Dual cube
Dihedral angles
  • between {3} and {3}:   arccos(-1/3) = 109.471221°
Confer
general antiprisms:
n-ap   n/2-ap   n/d-ap  
special bipyramids:
m mNo  
variations:
xo3ox&#q
Grünbaumian relatives:
oct+6{4}   2oct   2oct+6{4}   2oct+8{3}   2oct+12{4}   4oct  
related Johnson solids:
squippy  
compounds:
se   sno   doso   dissit   si   gissi   addasi   dasi  
general polytopal classes:
deltahedra   regular   noble polytopes   orthoplex   partial Stott expansions   segmentohedra   fundamental lace prisms   bistratic lace towers  
External
links
hedrondude   wikipedia   WikiChoron   mathworld   Polyedergarten   quickfur

Incidence matrix according to Dynkin symbol

x3o4o

. . . | 6 |  4 | 4
------+---+----+--
x . . | 2 | 12 | 2
------+---+----+--
x3o . | 3 |  3 | 8

snubbed forms: β3o4o

x3/2o4o

.   . . | 6 |  4 | 4
--------+---+----+--
x   . . | 2 | 12 | 2
--------+---+----+--
x3/2o . | 3 |  3 | 8

snubbed forms: β3/2o4o

o4/3o3x

.   . . | 6 |  4 | 4
--------+---+----+--
.   . x | 2 | 12 | 2
--------+---+----+--
.   o3x | 3 |  3 | 8

snubbed forms: o4/3o3β

o4/3o3/2x

.   .   . | 6 |  4 | 4
----------+---+----+--
.   .   x | 2 | 12 | 2
----------+---+----+--
.   o3/2x | 3 |  3 | 8

snubbed forms: o4/3o3/2β

o3x3o

. . . | 6 |  4 | 2 2
------+---+----+----
. x . | 2 | 12 | 1 1
------+---+----+----
o3x . | 3 |  3 | 4 *
. x3o | 3 |  3 | * 4

snubbed forms: o3β3o

o3/2x3o

.   . . | 6 |  4 | 2 2
--------+---+----+----
.   x . | 2 | 12 | 1 1
--------+---+----+----
o3/2x . | 3 |  3 | 4 *
.   x3o | 3 |  3 | * 4

snubbed forms: o3/2β3o

o3/2x3/2o

.   .   . | 6 |  4 | 2 2
----------+---+----+----
.   x   . | 2 | 12 | 1 1
----------+---+----+----
o3/2x   . | 3 |  3 | 4 *
.   x3/2o | 3 |  3 | * 4

snubbed forms: o3/2β3/2o

s2s3s

demi( . . .  ) | 6 | 1 1 2 | 1 3
---------------+---+-------+----
      s2s .    | 2 | 3 * * | 0 2
      s . s2*a | 2 | * 3 * | 0 2
sefa( . s3s  ) | 2 | * * 6 | 1 1
---------------+---+-------+----
      . s3s     3 | 0 0 3 | 2 *
sefa( s2s3s  ) | 3 | 1 1 1 | * 6
or
demi( . . . )            | 6 | 2 2 | 1 3
-------------------------+---+-----+----
      s2s .  &  s . s2*a | 2 | 6 * | 0 2
sefa( . s3s )            | 2 | * 6 | 1 1
-------------------------+---+-----+----
      . s3s               3 | 0 3 | 2 *
sefa( s2s3s )            | 3 | 2 1 | * 6

starting figure: x x3x

s2s6o

demi( . . . ) | 6 | 2 2 | 1 3
--------------+---+-----+----
      s2s .   | 2 | 6 * | 0 2
sefa( . s6o ) | 2 | * 6 | 1 1
--------------+---+-----+----
      . s6o )  3 | 0 3 | 2 *
sefa( s2s6o ) | 3 | 2 1 | * 6

starting figure: x x6o

xo3ox&#x   → height = sqrt(2/3) = 0.816497
({3} || dual {3})

o.3o.    | 3 * | 2 2 0 | 1 2 1 0
.o3.o    | * 3 | 0 2 2 | 0 1 2 1
---------+-----+-------+--------
x. ..    | 2 0 | 3 * * | 1 1 0 0
oo3oo&#x | 1 1 | * 6 * | 0 1 1 0
.. .x    | 0 2 | * * 3 | 0 0 1 1
---------+-----+-------+--------
x.3o.    | 3 0 | 3 0 0 | 1 * * *
xo ..&#x | 2 1 | 1 2 0 | * 3 * *
.. ox&#x | 1 2 | 0 2 1 | * * 3 *
.o3.x    | 0 3 | 0 0 3 | * * * 1

oxo4ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo {4} || pt)

o..4o..    | 1 * * | 4 0 0 | 4 0
.o.4.o.    | * 4 * | 1 2 1 | 2 2
..o4..o    | * * 1 | 0 0 4 | 0 4
-----------+-------+-------+----
oo.4oo.&#x | 1 1 0 | 4 * * | 2 0
.x. ...    | 0 2 0 | * 4 * | 1 1
.oo4.oo&#x | 0 1 1 | * * 4 | 0 2
-----------+-------+-------+----
ox. ...&#x | 1 2 0 | 2 1 0 | 4 *
.xo ...&#x | 0 2 1 | 0 1 2 | * 4
or
o..4o..    & | 2 * | 4 0 | 4
.o.4.o.      | * 4 | 2 2 | 4
-------------+-----+-----+--
oo.4oo.&#x & | 1 1 | 8 * | 2
.x. ...      | 0 2 | * 4 | 2
-------------+-----+-----+--
ox. ...&#x & | 1 2 | 2 1 | 8

oxo oxo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo {4} || pt)

o.. o..    | 1 * * | 4 0 0 0 | 2 2 0 0
.o. .o.    | * 4 * | 1 1 1 1 | 1 1 1 1
..o ..o    | * * 1 | 0 0 0 4 | 0 0 2 2
-----------+-------+---------+--------
oo. oo.&#x | 1 1 0 | 4 * * * | 1 1 0 0
.x. ...    | 0 2 0 | * 2 * * | 1 0 1 0
... .x.    | 0 2 0 | * * 2 * | 0 1 0 1
.oo .oo&#x | 0 1 1 | * * * 4 | 0 0 1 1
-----------+-------+---------+--------
ox. ...&#x | 1 2 0 | 2 1 0 0 | 2 * * *
... ox.&#x | 1 2 0 | 2 0 1 0 | * 2 * *
.xo ...&#x | 0 2 1 | 0 1 0 2 | * * 2 *
... .xo&#x | 0 2 1 | 0 0 1 2 | * * * 2
or
o.. o..    & | 2 * | 4 0 0 | 2 2
.o. .o.      | * 4 | 2 1 1 | 2 2
-------------+-----+-------+----
oo. oo.&#x & | 1 1 | 8 * * | 1 1
.x. ...      | 0 2 | * 2 * | 2 0
... .x.      | 0 2 | * * 2 | 0 2
-------------+-----+-------+----
ox. ...&#x & | 1 2 | 2 1 0 | 4 *
... ox.&#x & | 1 2 | 2 0 1 | * 4

xox oqo&#xt   → both heights = 1/2
(line || perp pseudo q-line || line)

o.. o..     | 2 * * | 1 2 1 0 0 | 2 2 0
.o. .o.     | * 2 * | 0 2 0 2 0 | 1 2 1
..o ..o     | * * 2 | 0 0 1 2 1 | 0 2 2
------------+-------+-----------+------
x.. ...     | 2 0 0 | 1 * * * * | 2 0 0
oo. oo.&#x  | 1 1 0 | * 4 * * * | 1 1 0
o.o o.o&#x  | 1 0 1 | * * 2 * * | 0 2 0
.oo .oo&#x  | 0 1 1 | * * * 4 * | 0 1 1
..x ...     | 0 0 2 | * * * * 1 | 0 0 2
------------+-------+-----------+------
xo. ...&#x  | 2 1 0 | 1 2 0 0 0 | 2 * *
ooo ooo&#xt | 1 1 1 | 0 1 1 1 0 | * 4 *
.ox ...&#x  | 0 1 2 | 0 0 0 2 1 | * * 2
or
o.. o..     & | 4 * | 1 2 1 | 2 2
.o. .o.       | * 2 | 0 4 0 | 2 2
--------------+-----+-------+----
x.. ...     & | 2 0 | 2 * * | 2 0
oo. oo.&#x  & | 1 1 | * 8 * | 1 1
o.o o.o&#x    | 2 0 | * * 2 | 0 2
--------------+-----+-------+----
xo. ...&#x  & | 2 1 | 1 2 0 | 4 *
ooo ooo&#xt   | 2 1 | 0 2 1 | * 4

oxox&#xr   → all cyclical heights = sqrt(3)/2 = 0.866025
             in fact this lace simplex degenerates into a rhomb with diagonals:
             height(1,3) = sqrt(2) = 1.414214
             height(2,4) = 1

o...    | 1 * * * | 2 2 0 0 0 0 0 | 1 2 1 0 0 0
.o..    | * 2 * * | 1 0 1 1 1 0 0 | 1 1 0 1 1 0
..o.    | * * 1 * | 0 0 0 2 0 2 0 | 0 0 0 1 2 1
...o    | * * * 2 | 0 1 0 0 1 1 1 | 0 1 1 0 1 1
--------+---------+---------------+------------
oo..&#x | 1 1 0 0 | 2 * * * * * * | 1 1 0 0 0 0
o..o&#x | 1 0 0 1 | * 2 * * * * * | 0 1 1 0 0 0
.x..    | 0 2 0 0 | * * 1 * * * * | 1 0 0 1 0 0
.oo.&#x | 0 1 1 0 | * * * 2 * * * | 0 0 0 1 1 0
.o.o&#x | 0 1 0 1 | * * * * 2 * * | 0 1 0 0 1 0
..oo&#x | 0 0 1 1 | * * * * * 2 * | 0 0 0 0 1 1
...x    | 0 0 0 2 | * * * * * * 1 | 0 0 1 0 0 1
--------+---------+---------------+------------
ox..&#x | 1 2 0 0 | 2 0 1 0 0 0 0 | 1 * * * * *
oo.o&#x | 1 1 0 1 | 1 1 0 0 1 0 0 | * 2 * * * *
o..x&#x | 1 0 0 2 | 0 2 0 0 0 0 1 | * * 1 * * *
.xo.&#x | 0 2 1 0 | 0 0 1 2 0 0 0 | * * * 1 * *
.ooo&#x | 0 1 1 1 | 0 0 0 1 1 1 0 | * * * * 2 *
..ox&#x | 0 0 1 2 | 0 0 0 0 0 2 1 | * * * * * 1

qo ox4oo&#zx   → height = 0
(tegum sum of q-line and perp {4})
(tegum product of q-line with {4})

o. o.4o.    | 2 * | 4 0 | 4
.o .o4.o    | * 4 | 2 2 | 4
------------+-----+-----+--
oo oo4oo&#x | 1 1 | 8 * | 2
.. .x ..    | 0 2 | * 4 | 2
------------+-----+-----+--
.. ox ..&#x | 1 2 | 2 1 | 8

qo ox ox&#zx   → height = 0
(tegum sum of q-line and perp {4})
(tegum product of q-line with {4})

... 

qoo oqo ooq&#zx   → all heights = 0
(tegum sum of 3 perp q-lines)
(tegum product of 3 q-lines)

o.. o.. o..    | 2 * * | 2 2 0 | 4
.o. .o. .o.    | * 2 * | 2 0 2 | 4
..o ..o ..o    | * * 2 | 0 2 2 | 4
---------------+-------+-------+--
oo. oo. oo.&#x | 1 1 0 | 4 * * | 2
o.o o.o o.o&#x | 1 0 1 | * 4 * | 2
.oo .oo .oo&#x | 0 1 1 | * * 4 | 2
---------------+-------+-------+--
ooo ooo ooo&#x | 1 1 1 | 1 1 1 | 8

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