Acronym | ike (alt.: snit) | |||||||||||||||||||||||||||||||
TOCID symbol | I, sO, sTT | |||||||||||||||||||||||||||||||
Name |
icosahedron, snub tetrahedron, snub tetratetrahedron, snub triangle antiprism, hydrohedron, gyroelongated pentagonal bipyramid, vertex figure of ex | |||||||||||||||||||||||||||||||
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Circumradius | sqrt[(5+sqrt(5))/8] = 0.951057 | |||||||||||||||||||||||||||||||
Edge radius | (1+sqrt(5))/4 = 0.809017 | |||||||||||||||||||||||||||||||
Inradius | sqrt[(7+3 sqrt(5))/24] = 0.755761 | |||||||||||||||||||||||||||||||
Vertex figure | [35] = x5o | |||||||||||||||||||||||||||||||
Snub derivation |
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Vertex layers |
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Lace city in approx. ASCII-art |
o o f x x f o o | |||||||||||||||||||||||||||||||
Coordinates |
(f/2, 1/2, 0) & even permutations, all changes of sign where f = (1+sqrt(5))/2 | |||||||||||||||||||||||||||||||
General of army | (is itself convex) | |||||||||||||||||||||||||||||||
Colonel of regiment | (is itself locally convex – other uniform polyhedral member: gad – other edge facetings) | |||||||||||||||||||||||||||||||
Dual | doe | |||||||||||||||||||||||||||||||
Dihedral angles |
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Face vector | 12, 30, 20 | |||||||||||||||||||||||||||||||
Confer |
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External links |
As abstract polytope ike is isomorphic to gike, thereby replacing vertex figure pentagons by corresponding pentagrams.
A q-ike can be face-inscribed into an F-oct. Its edges then are to be dissected in the ratio f:x, cf. the pic at the right.
The number of ways to color the icosahedron with different colors per face is 20!/60 = 40 548 366 802 944 000. – This is because the color group is the permutation group of 20 elements and has size 20!, while the order of the pure rotational icosahedral group is 60. (The reflectional icosahedral group would have twice as many, i.e. 120 elements.)
The (direct) snub derivational process features 2 operations, first the mere alternated faceting, here wrt. a to be alternated vertices of toe as its starting figure, which then would result in qQo oqQ Qoq&#zh (where Q = 2q), a pyritohedrally gyrated ike variant, and a secondary edge rescaling to all unit sizes only.
→ s before s' |
The quartersnub derivation of s3s4s':
The first stellation of ike happens to be the dual of sidtid, i.e. stai. | ||
x3x4x |
s3s4x | ||
↓ s' before s | ↓ s' after s | ||
→ s after s' | |||
x3x4s' |
s3s4s' | © |
Incidence matrix according to Dynkin symbol
x3o5o . . . | 12 | 5 | 5 ------+----+----+--- x . . | 2 | 30 | 2 ------+----+----+--- x3o . | 3 | 3 | 20 snubbed forms: β3o5o
x3/2o5o . . . | 12 | 5 | 5 --------+----+----+--- x . . | 2 | 30 | 2 --------+----+----+--- x3/2o . | 3 | 3 | 20
o5/4o3x . . . | 12 | 5 | 5 --------+----+----+--- . . x | 2 | 30 | 2 --------+----+----+--- . o3x | 3 | 3 | 20
o5/4o3/2x . . . | 12 | 5 | 5 ----------+----+----+--- . . x | 2 | 30 | 2 ----------+----+----+--- . o3/2x | 3 | 3 | 20
in pyritohedral symmetry: 12 | 4 1 | 2 3 ---+------+----- 2 | 24 * | 1 1 per cube-vertex 2 | * 6 | 0 2 ---+------+----- 3 | 3 0 | 8 * 3 | 2 1 | * 12
s3s3s demi( . . . ) | 12 | 1 2 2 | 1 1 3 --------------+----+---------+------- s 2 s | 2 | 6 * * | 0 0 2 sefa( s3s . ) | 2 | * 12 * | 1 0 1 sefa( . s3s ) | 2 | * * 12 | 0 1 1 --------------+----+---------+------- s3s . ♦ 3 | 0 3 0 | 4 * * . s3s ♦ 3 | 0 0 3 | * 4 * sefa( s3s3s ) | 3 | 1 1 1 | * * 12
or demi( . . . ) | 12 | 1 4 | 2 3 --------------------------------+----+------+----- s 2 s | 2 | 6 * | 0 2 sefa( s3s . ) & sefa( . s3s ) | 2 | * 24 | 1 1 --------------------------------+----+------+----- s3s . & . s3s ♦ 3 | 0 3 | 8 * sefa( s3s3s ) | 3 | 1 2 | * 12 starting figure: x3x3x
s3s4o demi( . . . ) | 12 | 1 4 | 2 3 --------------+----+------+----- . s4o | 2 | 6 * | 0 2 sefa( s3s . ) | 2 | * 24 | 1 1 --------------+----+------+----- s3s . ♦ 3 | 0 3 | 8 * sefa( s3s4o ) | 3 | 1 2 | * 12 starting figure: x3x4o
s3s4/3o demi( . . . ) | 12 | 1 4 | 2 3 ----------------+----+------+----- . s4/3o | 2 | 6 * | 0 2 sefa( s3s . ) | 2 | * 24 | 1 1 ----------------+----+------+----- s3s . ♦ 3 | 0 3 | 8 * sefa( s3s4/3o ) | 3 | 1 2 | * 12 starting figure: x3x4/3o
oxoo5ooxo&#xt → outer heights = sqrt((5-sqrt(5))/10) = 0.525731, inner height = sqrt((5+sqrt(5))/10) = 0.850651 (pt || pseudo {5} || dual pseudo {5} || pt) o...5o... | 1 * * * | 5 0 0 0 0 | 5 0 0 0 .o..5.o.. | * 5 * * | 1 2 2 0 0 | 2 2 1 0 ..o.5..o. | * * 5 * | 0 0 2 2 1 | 0 1 2 2 ...o5...o | * * * 1 | 0 0 0 0 5 | 0 0 0 5 -------------+---------+------------+-------- oo..5oo..&#x | 1 1 0 0 | 5 * * * * | 2 0 0 0 .x.. .... | 0 2 0 0 | * 5 * * * | 1 1 0 0 .oo. .oo.&#x | 0 1 1 0 | * * 10 * * | 0 1 1 0 .... ..x. | 0 0 2 0 | * * * 5 * | 0 0 1 1 ..oo5..oo&#x | 0 0 1 1 | * * * * 5 | 0 0 0 2 -------------+---------+------------+-------- ox.. ....&#x | 1 2 0 0 | 2 1 0 0 0 | 5 * * * .xo. ....&#x | 0 2 1 0 | 0 1 2 0 0 | * 5 * * .... .ox.&#x | 0 1 2 0 | 0 0 2 1 0 | * * 5 * .... ..xo&#x | 0 0 2 1 | 0 0 0 1 2 | * * * 5
or o...5o... & | 2 * | 5 0 0 | 5 0 .o..5.o.. & | * 10 | 1 2 2 | 2 3 ---------------+------+----------+------ oo..5oo..&#x & | 1 1 | 10 * * | 2 0 .x.. .... & | 0 2 | * 10 * | 1 1 .oo. .oo.&#x | 0 2 | * * 10 | 0 2 ---------------+------+----------+------ ox.. ....&#x & | 1 2 | 2 1 0 | 10 * .xo. ....&#x & | 0 3 | 0 1 2 | * 10
xofo3ofox&#xt → outer heights = 1/sqrt(3) = 0.577350 inner height = sqrt[(3-sqrt(5))/6] = 0.356822 ({3} || pseudo dual f-{3} || pseudo f-{3} || dual {3}) o...3o... | 3 * * * | 2 2 1 0 0 0 0 | 1 2 2 0 0 0 .o..3.o.. | * 3 * * | 0 2 0 2 1 0 0 | 0 1 2 2 0 0 ..o.3..o. | * * 3 * | 0 0 1 2 0 2 0 | 0 0 2 2 1 0 ...o3...o | * * * 3 | 0 0 0 0 1 2 2 | 0 0 0 2 2 1 --------------+---------+---------------+------------ x... .... | 2 0 0 0 | 3 * * * * * * | 1 1 0 0 0 0 oo..3oo..&#x | 1 1 0 0 | * 6 * * * * * | 0 1 1 0 0 0 o.o.3o.o.&#x | 1 0 1 0 | * * 3 * * * * | 0 0 2 0 0 0 .oo.3.oo.&#x | 0 1 1 0 | * * * 6 * * * | 0 0 1 1 0 0 .o.o3.o.o&#x | 0 1 0 1 | * * * * 3 * * | 0 0 0 2 0 0 ..oo3..oo&#x | 0 0 1 1 | * * * * * 6 * | 0 0 0 1 1 0 .... ...x | 0 0 0 2 | * * * * * * 3 | 0 0 0 0 1 1 --------------+---------+---------------+------------ x...3o... | 3 0 0 0 | 3 0 0 0 0 0 0 | 1 * * * * * xo.. ....&#x | 2 1 0 0 | 1 2 0 0 0 0 0 | * 3 * * * * ooo.3ooo.&#xt | 1 1 1 0 | 0 1 1 1 0 0 0 | * * 6 * * * .ooo3.ooo&#xt | 0 1 1 1 | 0 0 0 1 1 1 0 | * * * 6 * * .... ..ox&#x | 0 0 1 2 | 0 0 0 0 0 2 1 | * * * * 3 * ...o3...x | 0 0 0 3 | 0 0 0 0 0 0 3 | * * * * * 1
or o...3o... & | 6 * | 2 2 1 0 | 1 2 2 .o..3.o.. & | * 6 | 0 2 1 2 | 0 1 4 -----------------+-----+----------+------- x... .... & | 2 0 | 6 * * * | 1 1 0 oo..3oo..&#x & | 1 1 | * 12 * * | 0 1 1 o.o.3o.o.&#x & | 1 1 | * * 6 * | 0 0 2 .oo.3.oo.&#x | 0 2 | * * * 6 | 0 0 2 -----------------+-----+----------+------- x...3o... & | 3 0 | 3 0 0 0 | 2 * * xo.. ....&#x & | 2 1 | 1 2 0 0 | * 6 * ooo.3ooo.&#xt & | 1 2 | 0 1 1 1 | * * 12
xofox ofxfo&#xt → outer heights = (sqrt(5)-1)/4 = 0.309017 inner heights = 1/2 (line || pseudo ortho f-line || pseudo (f,x)-{4} || pseudo ortho f-line || line) o.... o.... | 2 * * * * | 1 2 2 0 0 0 0 0 0 0 | 2 2 1 0 0 0 0 .o... .o... | * 2 * * * | 0 2 0 2 1 0 0 0 0 0 | 1 2 0 2 0 0 0 ..o.. ..o.. | * * 4 * * | 0 0 1 1 0 1 1 1 0 0 | 0 1 1 1 1 1 0 ...o. ...o. | * * * 2 * | 0 0 0 0 1 0 2 0 2 0 | 0 0 0 2 2 0 1 ....o ....o | * * * * 2 | 0 0 0 0 0 0 0 2 2 1 | 0 0 0 0 2 1 2 ----------------+-----------+---------------------+-------------- x.... ..... | 2 0 0 0 0 | 1 * * * * * * * * * | 2 0 0 0 0 0 0 oo... oo...&#x | 1 1 0 0 0 | * 4 * * * * * * * * | 1 1 0 0 0 0 0 o.o.. o.o..&#x | 1 0 1 0 0 | * * 4 * * * * * * * | 0 1 1 0 0 0 0 .oo.. .oo..&#x | 0 1 1 0 0 | * * * 4 * * * * * * | 0 1 0 1 0 0 0 .o.o. .o.o.&#x | 0 1 0 1 0 | * * * * 2 * * * * * | 0 0 0 2 0 0 0 ..... ..x.. | 0 0 2 0 0 | * * * * * 2 * * * * | 0 0 1 0 0 1 0 ..oo. ..oo.&#x | 0 0 1 1 0 | * * * * * * 4 * * * | 0 0 0 1 1 0 0 ..o.o ..o.o&#x | 0 0 1 0 1 | * * * * * * * 4 * * | 0 0 0 0 1 1 0 ...oo ...oo&#x | 0 0 0 1 1 | * * * * * * * * 4 * | 0 0 0 0 1 0 1 ....x ..... | 0 0 0 0 2 | * * * * * * * * * 1 | 0 0 0 0 0 0 2 ----------------+-----------+---------------------+-------------- xo... .....&#x | 2 1 0 0 0 | 1 2 0 0 0 0 0 0 0 0 | 2 * * * * * * ooo.. ooo..&#xt | 1 1 1 0 0 | 0 1 1 1 0 0 0 0 0 0 | * 4 * * * * * ..... o.x..&#x | 1 0 2 0 0 | 0 0 2 0 0 1 0 0 0 0 | * * 2 * * * * .ooo. .ooo.&#xt | 0 1 1 1 0 | 0 0 0 1 1 0 1 0 0 0 | * * * 4 * * * ..ooo ..ooo&#xt | 0 0 1 1 1 | 0 0 0 0 0 0 1 1 1 0 | * * * * 4 * * ..... ..x.o&#x | 0 0 2 0 1 | 0 0 0 0 0 1 0 2 0 0 | * * * * * 2 * ...ox .....&#x | 0 0 0 1 2 | 0 0 0 0 0 0 0 0 2 1 | * * * * * * 2
or o.... o.... & | 4 * * | 1 2 2 0 0 0 | 2 2 1 0 .o... .o... & | * 4 * | 0 2 0 2 1 0 | 1 2 0 2 ..o.. ..o.. | * * 4 | 0 0 2 2 0 1 | 0 2 2 1 -------------------+-------+-------------+-------- x.... ..... & | 2 0 0 | 2 * * * * * | 2 0 0 0 oo... oo...&#x & | 1 1 0 | * 8 * * * * | 1 1 0 0 o.o.. o.o..&#x & | 1 0 1 | * * 8 * * * | 0 1 1 0 .oo.. .oo..&#x & | 0 1 1 | * * * 8 * * | 0 1 0 1 .o.o. .o.o.&#x | 0 2 0 | * * * * 2 * | 0 0 0 2 ..... ..x.. | 0 0 2 | * * * * * 2 | 0 0 2 0 -------------------+-------+-------------+-------- xo... .....&#x & | 2 1 0 | 1 2 0 0 0 0 | 4 * * * ooo.. ooo..&#xt & | 1 1 1 | 0 1 1 1 0 0 | * 8 * * ..... o.x..&#x & | 1 0 2 | 0 0 2 0 0 1 | * * 4 * .ooo. .ooo.&#xt | 0 2 1 | 0 0 0 2 1 0 | * * * 4
fxo ofx xof&#zx → all heights = 0 (tegum sum of 3 pairwise perpendicular (x,f)-{4}'s) o.. o.. o.. | 4 * * | 1 2 2 0 0 0 | 2 1 2 0 .o. .o. .o. | * 4 * | 0 2 0 1 2 0 | 1 0 2 2 ..o ..o ..o | * * 4 | 0 0 2 0 2 1 | 0 2 2 1 ---------------+-------+-------------+-------- ... ... x.. | 2 0 0 | 2 * * * * * | 2 0 0 0 oo. oo. oo.&#x | 1 1 0 | * 8 * * * * | 1 0 1 0 o.o o.o o.o&#x | 1 0 1 | * * 8 * * * | 0 1 1 0 .x. ... ... | 0 2 0 | * * * 2 * * | 0 0 0 2 .oo .oo .oo&#x | 0 1 1 | * * * * 8 * | 0 0 1 1 ... ..x ... | 0 0 2 | * * * * * 2 | 0 2 0 0 ---------------+-------+-------------+-------- ... ... xo.&#x | 2 1 0 | 1 2 0 0 0 0 | 4 * * * ... o.x ...&#x | 1 0 2 | 0 0 2 0 0 1 | * 4 * * ooo ooo ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * 8 * .xo ... ...&#x | 0 2 1 | 0 0 0 1 2 0 | * * * 4
or o.. o.. o.. & | 12 | 1 4 | 3 2 ------------------+----+------+----- ... ... x.. & | 2 | 6 * | 2 0 oo. oo. oo.&#x & | 2 | * 24 | 1 1 ------------------+----+------+----- ... ... xo.&#x & | 3 | 1 2 | 12 * ooo ooo ooo&#x | 3 | 0 3 | * 8
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