Acronym | tic | |||||||||||||||||||||||||||||||||||||
TOCID symbol | tC | |||||||||||||||||||||||||||||||||||||
Name |
truncated cube, truncated hexahedron | |||||||||||||||||||||||||||||||||||||
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Circumradius | sqrt[7+4 sqrt(2)]/2 = 1.778824 | |||||||||||||||||||||||||||||||||||||
Inradius wrt. {3} | (3+2 sqrt(2))/sqrt(12) = 1.682522 | |||||||||||||||||||||||||||||||||||||
Inradius wrt. {8} | [1+sqrt(2)]/2 = 1.207107 | |||||||||||||||||||||||||||||||||||||
Vertex figure | [3,82] = xo&#x(8,2) | |||||||||||||||||||||||||||||||||||||
Vertex layers |
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Lace city in approx. ASCII-art |
x w w x w w w w x w w x | |||||||||||||||||||||||||||||||||||||
x w w x o U o (U=qw=u+q=x+w) o U o x w w x | ||||||||||||||||||||||||||||||||||||||
Coordinates | ((1+sqrt(2))/2, (1+sqrt(2))/2, 1/2) & all permutations, all changes of sign | |||||||||||||||||||||||||||||||||||||
Volume | (21+14 sqrt(2))/3 = 13.599663 | |||||||||||||||||||||||||||||||||||||
Surface | 12+12 sqrt(2)+2 sqrt(3) = 32.434664 | |||||||||||||||||||||||||||||||||||||
General of army | (is itself convex) | |||||||||||||||||||||||||||||||||||||
Colonel of regiment | (is itself locally convex – no other uniform polyhedral members) | |||||||||||||||||||||||||||||||||||||
Dihedral angles |
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Dual | tikko | |||||||||||||||||||||||||||||||||||||
Face vector | 24, 36, 14 | |||||||||||||||||||||||||||||||||||||
Confer |
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External links |
As abstract polytope tic is isomorphic to quith, thereby replacing octagons by octagrams.
Note that tic can be thought of as the external blend of 1 cube + 8 tets + 6 squacues, cf. the Steward toroid T4 \ Q4P4Q4. This decomposition is also described as the degenerate segmentochoron oo3ox4xx&#xt.
Incidence matrix according to Dynkin symbol
o3x4x . . . | 24 | 2 1 | 1 2 ------+----+-------+---- . x . | 2 | 24 * | 1 1 . . x | 2 | * 12 | 0 2 ------+----+-------+---- o3x . | 3 | 3 0 | 8 * . x4x | 8 | 4 4 | * 6 snubbed forms: o3β4x, o3x4s, o3β4β
o3/2x4x . . . | 24 | 2 1 | 1 2 --------+----+-------+---- . x . | 2 | 24 * | 1 1 . . x | 2 | * 12 | 0 2 --------+----+-------+---- o3/2x . | 3 | 3 0 | 8 * . x4x | 8 | 4 4 | * 6
xwwx4xoox&#xt → outer heights = 1/sqrt(2) = 0.707107 inner height = 1 ({8} || pseudo w-{4} || pseudo w-{4} || {8}) o...4o... | 8 * * * | 1 1 1 0 0 0 0 | 1 1 1 0 0 .o..4.o.. | * 4 * * | 0 0 2 1 0 0 0 | 0 1 2 0 0 ..o.4..o. | * * 4 * | 0 0 0 1 2 0 0 | 0 0 2 1 0 ...o4...o | * * * 8 | 0 0 0 0 1 1 1 | 0 0 1 1 1 --------------+---------+---------------+---------- x... .... | 2 0 0 0 | 4 * * * * * * | 1 0 1 0 0 .... x... | 2 0 0 0 | * 4 * * * * * | 1 1 0 0 0 oo..4oo..&#x | 1 1 0 0 | * * 8 * * * * | 0 1 1 0 0 .oo.4.oo.&#x | 0 1 1 0 | * * * 4 * * * | 0 0 2 0 0 ..oo4..oo&#x | 0 0 1 1 | * * * * 8 * * | 0 0 1 1 0 ...x .... | 0 0 0 2 | * * * * * 4 * | 0 0 1 0 1 .... ...x | 0 0 0 2 | * * * * * * 4 | 0 0 0 1 1 --------------+---------+---------------+---------- x...4x... | 8 0 0 0 | 4 4 0 0 0 0 0 | 1 * * * * .... xo..&#x | 2 1 0 0 | 0 1 2 0 0 0 0 | * 4 * * * xwwx ....&#xt | 2 2 2 2 | 1 0 2 2 2 1 0 | * * 4 * * .... ..ox&#x | 0 0 1 2 | 0 0 0 0 2 0 1 | * * * 4 * ...x4...x | 0 0 0 8 | 0 0 0 0 0 4 4 | * * * * 1
or o...4o... & | 16 * | 1 1 1 0 | 1 1 1 .o..4.o.. & | * 8 | 0 0 2 1 | 0 1 2 -----------------+------+----------+------ x... .... & | 2 0 | 8 * * * | 1 0 1 .... x... & | 2 0 | * 8 * * | 1 1 0 oo..4oo..&#x & | 1 1 | * * 16 * | 0 1 1 .oo.4.oo.&#x | 0 2 | * * * 4 | 0 0 2 -----------------+------+----------+------ x...4x... & | 8 0 | 4 4 0 0 | 2 * * .... xo..&#x & | 2 1 | 0 1 2 0 | * 8 * xwwx ....&#xt | 4 4 | 2 0 4 2 | * * 4
xwwxoo3ooxwwx&#xt → height(1,2) = height(3,4) = height(5,6) = 1/sqrt(3) = 0.577350 height(2,3) = height(4,5) = sqrt(2/3) = 0.816497 ({3} || pseudo w-{3} || pseudo (w,x)-{6} || pseudo (x,w)-{6} || pseudo dual w-{3} || dual {3}) o.....3o..... | 3 * * * * * | 2 1 0 0 0 0 0 0 0 | 1 2 0 0 0 0 .o....3.o.... | * 3 * * * * | 0 1 2 0 0 0 0 0 0 | 0 2 1 0 0 0 ..o...3..o... | * * 6 * * * | 0 0 1 1 1 0 0 0 0 | 0 1 1 1 0 0 ...o..3...o.. | * * * 6 * * | 0 0 0 0 1 1 1 0 0 | 0 1 0 1 1 0 ....o.3....o. | * * * * 3 * | 0 0 0 0 0 0 2 1 0 | 0 0 0 2 1 0 .....o3.....o | * * * * * 3 | 0 0 0 0 0 0 0 1 2 | 0 0 0 2 0 1 ------------------+-------------+-------------------+------------ x..... ...... | 2 0 0 0 0 0 | 3 * * * * * * * * | 1 1 0 0 0 0 oo....3oo....&#x | 1 1 0 0 0 0 | * 3 * * * * * * * | 0 2 0 0 0 0 .oo...3.oo...&#x | 0 1 1 0 0 0 | * * 6 * * * * * * | 0 1 1 0 0 0 ...... ..x... | 0 0 2 0 0 0 | * * * 3 * * * * * | 0 0 1 1 0 0 ..oo..3..oo..&#x | 0 0 1 1 0 0 | * * * * 6 * * * * | 0 1 0 1 0 0 ...x.. ...... | 0 0 0 2 0 0 | * * * * * 3 * * * | 0 1 0 0 1 0 ...oo.3...oo.&#x | 0 0 0 1 1 0 | * * * * * * 6 * * | 0 0 0 1 1 0 ....oo3....oo&#x | 0 0 0 0 1 1 | * * * * * * * 3 * | 0 0 0 2 0 0 ...... .....x | 0 0 0 0 0 2 | * * * * * * * * 3 | 0 0 0 1 0 1 ------------------+-------------+-------------------+------------ x.....3o..... | 3 0 0 0 0 0 | 3 0 0 0 0 0 0 0 0 | 1 * * * * * xwwx.. ......&#xt | 2 2 2 2 0 0 | 1 2 2 0 2 1 0 0 0 | * 3 * * * * ...... .ox...&#x | 0 1 2 0 0 0 | 0 0 2 1 0 0 0 0 0 | * * 3 * * * ...... ..xwwx&#xt | 0 0 2 2 2 2 | 0 0 0 1 2 0 2 2 1 | * * * 3 * * ...xo. ......&#x | 0 0 0 2 1 0 | 0 0 0 0 0 1 2 0 0 | * * * * 3 * .....o3.....x | 0 0 0 0 0 3 | 0 0 0 0 0 0 0 0 3 | * * * * * 1
or o.....3o..... & | 6 * * | 2 1 0 0 0 | 1 2 0 .o....3.o.... & | * 6 * | 0 1 2 0 0 | 0 2 1 ..o...3..o... & | * * 12 | 0 0 1 1 1 | 0 2 1 ---------------------+--------+------------+------ x..... ...... & | 2 0 0 | 6 * * * * | 1 1 0 oo....3oo....&#x & | 1 1 0 | * 6 * * * | 0 2 0 .oo...3.oo...&#x & | 0 1 1 | * * 12 * * | 0 1 1 ...... ..x... & | 0 0 2 | * * * 6 * | 0 1 1 ..oo..3..oo..&#x | 0 0 2 | * * * * 6 | 0 2 0 ---------------------+--------+------------+------ x.....3o..... & | 3 0 0 | 3 0 0 0 0 | 2 * * xwwx.. ......&#xt & | 2 2 4 | 1 2 2 1 2 | * 6 * ...... .ox...&#x & | 0 1 2 | 0 0 2 1 0 | * * 6
wx3oo3xw&#zx → height = 0 (tegum sum of 2 mutually inverted (x,w)-coes) o.3o.3o. | 12 * | 2 1 0 | 1 2 0 .o3.o3.o | * 12 | 0 1 2 | 0 2 1 -------------+-------+----------+------ .. .. x. | 2 0 | 12 * * | 1 1 0 oo3oo3oo&#x | 1 1 | * 12 * | 0 2 0 .x .. .. | 0 2 | * * 12 | 0 1 1 -------------+-------+----------+------ .. o.3x. | 3 0 | 3 0 0 | 4 * * wx .. xw&#zx | 4 4 | 2 4 2 | * 6 * .x3.o .. | 0 3 | 0 0 3 | * * 4
or o.3o.3o. & | 24 | 2 1 | 1 2 ---------------+----+-------+---- .. .. x. & | 2 | 24 * | 1 1 oo3oo3oo&#x | 2 | * 12 | 0 2 ---------------+----+-------+---- .. o.3x. & | 3 | 3 0 | 8 * wx .. xw&#zx | 8 | 4 4 | * 6
wx xw4xo&#zx → height = 0 (tegum sum of (w,x,x)-op and (x,w,w)-cube) o. o.4o. | 16 * | 1 1 1 0 | 1 1 1 .o .o4.o | * 8 | 0 0 2 1 | 0 2 1 -------------+------+----------+------ .. x. .. | 2 0 | 8 * * * | 1 1 0 .. .. x. | 2 0 | * 8 * * | 1 0 1 oo oo4oo&#x | 1 1 | * * 16 * | 0 1 1 .x .. .. | 0 2 | * * * 4 | 0 2 0 -------------+------+----------+------ .. x.4x. | 8 0 | 4 4 0 0 | 2 * * wx xw ..&#zx | 4 4 | 2 0 4 2 | * 4 * .. .. xo&#x | 2 1 | 0 1 2 0 | * * 8
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