Acronym tut
TOCID symbol tT
Name truncated tetrahedron,
cantic cube,
Waterman polyhedron number 2 wrt. face-centered cubic lattice A3 centered at a shallow hole
 
 © ©
Circumradius sqrt(11/8) = 1.172604
Edge radius 3/sqrt(8) = 1.060660
Inradius
wrt. {3}
5/sqrt(24) = 1.020621
Inradius
wrt. {6}
sqrt(3/8) = 0.612372
Vertex figure [3,6,6] = xo&#h
Snub derivation
Vertex layers
LayerSymmetrySubsymmetries
 o3o3oo3o .o . o. o3o
1x3x3ox3x .
{6} first
x . o
edge first
. x3o
{3} first
2u3o .u . x. u3o
3x3o .
opposite {3}
x . u. x3x
opposite{6}
4 o . x
opposite edge
 
Lace city
in approx. ASCII-art
   o     o   
             
x           x
             
   u     u   
             
      x      
Coordinates (3/sqrt(8), 1/sqrt(8), 1/sqrt(8))   all permutations, even changes of sign
Volume 23 sqrt(2)/12 = 2.710576
Surface 7 sqrt(3) = 12.124356
General of army (is itself convex)
Colonel of regiment (is itself locally convex – no other uniform polyhedral members)
Dihedral angles
  • between {3} and {6}:   arccos(-1/3) = 109.471221°
  • between {6} and {6}:   arccos(1/3) = 70.528779°
Dual tikit
Face vector 12, 18, 8
Confer
Grünbaumian relatives:
2tut  
variations:
a3b3c   x3q3o   x3v3o   x3u3o   x3w3o   v3x3o   q3x3o   u3x3o   w3x3o   (-x)3x3o  
compounds:
tisso   taki   te  
blends:
tutut  
ambification:
retut  
general polytopal classes:
Wythoffian polyhedra   tutsatopes   partial Stott expansions   bistratic lace towers   lace simplices  
analogs:
truncated simplex tSn   bitruncated simplex btSn   truncated demihypercube tDn   maximal expanded demihypercube eDn  
External
links
hedrondude   wikipedia   polytopewiki   WikiChoron   mathworld   quickfur

Note that tut can be thought of as the external blend of 1 tet + 4 octs + 4 hippies. This decomposition is described as the degenerate segmentochoron ox3ox3xo&#xt.


Incidence matrix according to Dynkin symbol

x3x3o

. . . | 12 | 1  2 | 2 1
------+----+------+----
x . . |  2 | 6  * | 2 0
. x . |  2 | * 12 | 1 1
------+----+------+----
x3x . |  6 | 3  3 | 4 *
. x3o |  3 | 0  3 | * 4

snubbed forms: β3x3o (or as mere faceting β3x3o), x3β3o, β3β3o

o3/2x3x

.   . . | 12 |  2 1 | 1 2
--------+----+------+----
.   x . |  2 | 12 * | 1 1
.   . x |  2 |  * 6 | 0 2
--------+----+------+----
o3/2x . |  3 | 3  0 | 4 *
.   x3x |  6 | 3  3 | * 4

snubbed forms: o3/2x3β (or as mere faceting o3/2x3β), o3/2β3x, o3/2β3β

s4o3x

demi( . . . ) | 12 | 1  2 | 1 2
--------------+----+------+----
      s4o . )   2 | 6  * | 0 2
demi( . . x ) |  2 | * 12 | 1 1
--------------+----+------+----
demi( . o3x ) |  3 | 0  3 | 4 *
sefa( s4o3x ) |  6 | 3  3 | * 4

starting figure: x4o3x

xux3oox&#xt   → both heights = sqrt(2/3) = 0.816497
({3} || pseudo u-{3} || {6})

o..3o..     | 3 * * | 2 1 0 0 0 | 1 2 0 0
.o.3.o.     | * 3 * | 0 1 2 0 0 | 0 2 1 0
..o3..o     | * * 6 | 0 0 1 1 1 | 0 1 1 1
------------+-------+-----------+--------
x.. ...     | 2 0 0 | 3 * * * * | 1 1 0 0
oo.3oo.&#x  | 1 1 0 | * 3 * * * | 0 2 0 0
.oo3.oo&#x  | 0 1 1 | * * 6 * * | 0 1 1 0
..x ...     | 0 0 2 | * * * 3 * | 0 1 0 1
... ..x     | 0 0 2 | * * * * 3 | 0 0 1 1
------------+-------+-----------+--------
x..3o..     | 3 0 0 | 3 0 0 0 0 | 1 * * *
xux ...&#xt | 2 2 2 | 1 2 2 1 0 | * 3 * *
... .ox&#x  | 0 1 2 | 0 0 2 0 1 | * * 3 *
..x3..x     | 0 0 6 | 0 0 0 3 3 | * * * 1

xuxo oxux&#xt   → all heights = 1/sqrt(2) = 0.707107
(line || pseudo (u,x)-{4} || pseudo (x,u)-{4} || ortho line)

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

© 2004-2024
top of page