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 | |||||||||||||||||||||||||||
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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 |
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Vertex layers |
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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 |
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Dual | tikit | |||||||||||||||||||||||||||
Face vector | 12, 18, 8 | |||||||||||||||||||||||||||
Confer |
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External links |
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
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