Acronym | id | |||||||||||||||||||||||||||||||||||||||||||||
TOCID symbol | ID | |||||||||||||||||||||||||||||||||||||||||||||
Name |
icosidodecahedron, rectified icosahedron, rectified dodecahedron, equatorial cross-section of vertex-first ex | |||||||||||||||||||||||||||||||||||||||||||||
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Circumradius | (1+sqrt(5))/2 = 1.618034 | |||||||||||||||||||||||||||||||||||||||||||||
Inradius wrt. {3} | sqrt[(7+3 sqrt(5))/6] = 1.511523 | |||||||||||||||||||||||||||||||||||||||||||||
Inradius wrt. {5} | sqrt[(5+2 sqrt(5))/5] = 1.376382 | |||||||||||||||||||||||||||||||||||||||||||||
Vertex figure | [(3,5)2] = x f | |||||||||||||||||||||||||||||||||||||||||||||
Vertex layers |
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Coordinates |
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General of army | (is itself convex) | |||||||||||||||||||||||||||||||||||||||||||||
Colonel of regiment | (is itself locally convex – other uniform polyhedral members: sidhid seihid – other edge facetings) | |||||||||||||||||||||||||||||||||||||||||||||
Dual | rhote | |||||||||||||||||||||||||||||||||||||||||||||
Dihedral angles |
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Face vector | 30, 60, 32 | |||||||||||||||||||||||||||||||||||||||||||||
Confer |
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External links |
As abstract polytope id is isomorphic to gid, thereby replacing pentagons by pentagrams.
Incidence matrix according to Dynkin symbol
o3x5o . . . | 30 | 4 | 2 2 ------+----+----+------ . x . | 2 | 60 | 1 1 ------+----+----+------ o3x . | 3 | 3 | 20 * . x5o | 5 | 5 | * 12 snubbed forms: o3β5o
o3/2x5o . . . | 30 | 4 | 2 2 --------+----+----+------ . x . | 2 | 60 | 1 1 --------+----+----+------ o3/2x . | 3 | 3 | 20 * . x5o | 5 | 5 | * 12
o5/4x3o . . . | 30 | 4 | 2 2 --------+----+----+------ . x . | 2 | 60 | 1 1 --------+----+----+------ o5/4x . | 5 | 5 | 12 * . x3o | 3 | 3 | * 20
o5/4x3/2o . . . | 30 | 4 | 2 2 ----------+----+----+------ . x . | 2 | 60 | 1 1 ----------+----+----+------ o5/4x . | 5 | 5 | 12 * . x3/2o | 3 | 3 | * 20
xoxfo5ofxox&#xt → outer heights = sqrt[(5-sqrt(5))/10] = 0.525731 inner heights = sqrt[(5+sqrt(5))/10] = 0.850651 ({5} || pseudo dual f-{5} || pseudo {10} || pseudo f-{5} || dual {5}) o....5o.... | 5 * * * * | 2 2 0 0 0 0 0 0 | 1 2 1 0 0 0 0 0 .o...5.o... | * 5 * * * | 0 2 2 0 0 0 0 0 | 0 1 2 1 0 0 0 0 ..o..5..o.. | * * 10 * * | 0 0 1 1 1 1 0 0 | 0 0 1 1 1 1 0 0 ...o.5...o. | * * * 5 * | 0 0 0 0 0 2 2 0 | 0 0 0 0 2 1 1 0 ....o5....o | * * * * 5 | 0 0 0 0 0 0 2 2 | 0 0 0 0 1 0 2 1 ----------------+------------+---------------------+---------------- x.... ..... | 2 0 0 0 0 | 5 * * * * * * * | 1 1 0 0 0 0 0 0 oo...5oo...&#x | 1 1 0 0 0 | * 10 * * * * * * | 0 1 1 0 0 0 0 0 .oo..5.oo..&#x | 0 1 1 0 0 | * * 10 * * * * * | 0 0 1 1 0 0 0 0 ..x.. ..... | 0 0 2 0 0 | * * * 5 * * * * | 0 0 0 1 1 0 0 0 ..... ..x.. | 0 0 2 0 0 | * * * * 5 * * * | 0 0 1 0 0 1 0 0 ..oo.5..oo.&#x | 0 0 1 1 0 | * * * * * 10 * * | 0 0 0 0 1 1 0 0 ...oo5...oo&#x | 0 0 0 1 1 | * * * * * * 10 * | 0 0 0 0 1 0 1 0 ..... ....x | 0 0 0 0 2 | * * * * * * * 5 | 0 0 0 0 0 0 1 1 ----------------+------------+---------------------+---------------- x....5o.... | 5 0 0 0 0 | 5 0 0 0 0 0 0 0 | 1 * * * * * * * xo... .....&#x | 2 1 0 0 0 | 1 2 0 0 0 0 0 0 | * 5 * * * * * * ..... ofx..&#xt | 1 2 2 0 0 | 0 2 2 0 1 0 0 0 | * * 5 * * * * * .ox.. .....&#x | 0 1 2 0 0 | 0 0 2 1 0 0 0 0 | * * * 5 * * * * ..xfo .....&#xt | 0 0 2 2 1 | 0 0 0 1 0 2 2 0 | * * * * 5 * * * ..... ..xo.&#x | 0 0 2 1 0 | 0 0 0 0 1 2 0 0 | * * * * * 5 * * ..... ...ox&#x | 0 0 0 1 2 | 0 0 0 0 0 0 2 1 | * * * * * * 5 * ....o5....x | 0 0 0 0 5 | 0 0 0 0 0 0 0 5 | * * * * * * * 1
or o....5o.... & | 10 * * | 2 2 0 0 | 1 2 1 0 .o...5.o... & | * 10 * | 0 2 2 0 | 0 1 2 1 ..o..5..o.. | * * 10 | 0 0 2 2 | 0 0 2 2 -------------------+----------+-------------+----------- x.... ..... & | 2 0 0 | 10 * * * | 1 1 0 0 oo...5oo...&#x & | 1 1 0 | * 20 * * | 0 1 1 0 .oo..5.oo..&#x & | 0 1 1 | * * 20 * | 0 0 1 1 ..x.. ..... & | 0 0 2 | * * * 10 | 0 0 1 1 -------------------+----------+-------------+----------- x....5o.... & | 5 0 0 | 5 0 0 0 | 2 * * * xo... .....&#x & | 2 1 0 | 1 2 0 0 | * 10 * * ..... ofx..&#xt & | 1 2 2 | 0 2 2 1 | * * 10 * .ox.. .....&#x & | 0 1 2 | 0 0 2 1 | * * * 10
oxFfofx3xfofFxo&#xt → height(1,2) = height(3,4) = height(4,5) = height(6,7) = 1/sqrt(3) = 0.577350 (F=ff=x+f) height(2,3) = height(5,6) = sqrt[(3-sqrt(5))/6] = 0.356822 ({3} || pseudo (x,f)-{6} || pseudo dual F-{3} || pseudo f-{6} || pseudo F-{3} || pseudo gyro {x,f)-{6} || dual {3}) o......3o...... & | 6 * * * | 2 2 0 0 0 0 | 1 1 2 0 0 .o.....3.o..... & | * 12 * * | 0 1 1 1 1 0 | 0 1 1 1 1 ..o....3..o.... & | * * 6 * | 0 0 0 2 0 2 | 0 0 1 1 2 ...o...3...o... | * * * 6 | 0 0 0 0 2 2 | 0 0 0 2 2 ----------------------+----------+-----------------+----------- ....... x...... & | 2 0 0 0 | 6 * * * * * | 1 0 1 0 0 oo.....3oo.....&#x & | 1 1 0 0 | * 12 * * * * | 0 1 1 0 0 .x..... ....... & | 0 2 0 0 | * * 6 * * * | 0 1 0 1 0 .oo....3.oo....&#x & | 0 1 1 0 | * * * 12 * * | 0 0 1 0 1 .o.o...3.o.o...&#x & | 0 1 0 1 | * * * * 12 * | 0 0 0 1 1 ..oo...3..oo...&#x & | 0 0 1 1 | * * * * * 12 | 0 0 0 1 1 ----------------------+----------+-----------------+----------- o......3x...... & | 3 0 0 0 | 3 0 0 0 0 0 | 2 * * * * ox..... .......&#x & | 1 2 0 0 | 0 2 1 0 0 0 | * 6 * * * ....... xfo....&#xt & | 2 2 1 0 | 1 2 0 2 0 0 | * * 6 * * .x.fo.. .......&#xt & | 0 2 1 2 | 0 0 1 0 2 2 | * * * 6 * .ooo...3.ooo...&#x & | 0 1 1 1 | 0 0 0 1 1 1 | * * * * 12
VooFxf oVofFx ooVxfF&#zx (F=ff=x+f, V=2f) → all heights = 0 (but not all pw. vertex combis exist as lacings) (tegum sum of 3 orthogonal V-edges and 3 each parallely oriented (x,f,F)-cubes.) o..... o..... o..... | 2 * * * * * | 4 0 0 0 0 0 0 0 0 | 2 2 0 0 0 0 0 .o.... .o.... .o.... | * 2 * * * * | 0 4 0 0 0 0 0 0 0 | 0 0 2 2 0 0 0 ..o... ..o... ..o... | * * 2 * * * | 0 0 4 0 0 0 0 0 0 | 0 0 0 0 2 2 0 ...o.. ...o.. ...o.. | * * * 8 * * | 1 0 0 1 1 1 0 0 0 | 1 1 0 1 0 0 1 ....o. ....o. ....o. | * * * * 8 * | 0 1 0 0 1 0 1 1 0 | 0 0 1 1 1 0 1 .....o .....o .....o | * * * * * 8 | 0 0 1 0 0 1 0 1 1 | 1 0 0 0 1 1 1 -------------------------+-------------+-------------------+-------------- o..o.. o..o.. o..o..&#x | 1 0 0 1 0 0 | 8 * * * * * * * * | 1 1 0 0 0 0 0 .o..o. .o..o. .o..o.&#x | 0 1 0 0 1 0 | * 8 * * * * * * * | 0 0 1 1 0 0 0 ..o..o ..o..o ..o..o&#x | 0 0 1 0 0 1 | * * 8 * * * * * * | 0 0 0 0 1 1 0 ...... ...... ...x.. | 0 0 0 2 0 0 | * * * 4 * * * * * | 0 1 0 1 0 0 0 ...oo. ...oo. ...oo.&#x | 0 0 0 1 1 0 | * * * * 8 * * * * | 0 0 0 1 0 0 1 ...o.o ...o.o ...o.o&#x | 0 0 0 1 0 1 | * * * * * 8 * * * | 1 0 0 0 0 0 1 ....x. ...... ...... | 0 0 0 0 2 0 | * * * * * * 4 * * | 0 0 1 0 1 0 0 ....oo ....oo ....oo&#x | 0 0 0 0 1 1 | * * * * * * * 8 * | 0 0 0 0 1 0 1 ...... .....x ...... | 0 0 0 0 0 2 | * * * * * * * * 4 | 1 0 0 0 0 1 0 -------------------------+-------------+-------------------+-------------- ...... o..f.x ......&#xt | 1 0 0 2 0 2 | 2 0 0 0 0 2 0 0 1 | 4 * * * * * * ...... ...... o..x..&#x | 1 0 0 2 0 0 | 2 0 0 1 0 0 0 0 0 | * 4 * * * * * .o..x. ...... ......&#x | 0 1 0 0 2 0 | 0 2 0 0 0 0 1 0 0 | * * 4 * * * * ...... ...... .o.xf.&#xt | 0 1 0 2 2 0 | 0 2 0 1 2 0 0 0 0 | * * * 4 * * * tower: B-E-D ..o.xf ...... ......&#xt | 0 0 1 0 2 2 | 0 0 2 0 0 0 1 2 0 | * * * * 4 * * tower: C-F-E ...... ..o..x ......&#x | 0 0 1 0 0 2 | 0 0 2 0 0 0 0 0 1 | * * * * * 4 * ...ooo ...ooo ...ooo&#x | 0 0 0 1 1 1 | 0 0 0 0 1 1 0 1 0 | * * * * * * 8
or o..... o..... o..... & | 6 * | 4 0 0 | 2 2 0 ...o.. ...o.. ...o.. & | * 24 | 1 1 2 | 2 1 1 ---------------------------+------+----------+-------- o..o.. o..o.. o..o..&#x & | 1 1 | 24 * * | 1 1 0 ...... ...... ...x.. & | 0 2 | * 12 * | 1 1 0 ...oo. ...oo. ...oo.&#x & | 0 2 | * * 24 | 1 0 1 ---------------------------+------+----------+-------- ...... o..f.x ......&#xt & | 1 4 | 2 1 2 | 12 * * ...... ...... o..x..&#x & | 1 2 | 2 1 0 | * 12 * ...ooo ...ooo ...ooo&#x | 0 3 | 0 0 3 | * * 8
with tet subsym. (chiral choice) 12 * * | 2 1 1 0 0 | 1 2 1 0 tetrahedral {3}-vertices * 6 * | 0 2 0 2 0 | 0 2 2 0 inscribed octahedral vertices * * 12 | 0 0 1 1 2 | 0 2 1 1 other tetrahedral {3}-vertices --------+----------------+---------- 2 0 0 | 12 * * * * | 1 1 0 0 1 1 0 | * 12 * * * | 0 1 1 0 1 0 1 | * * 12 * * | 0 1 1 0 0 1 1 | * * * 12 * | 0 1 1 0 0 0 2 | * * * * 12 | 0 1 0 1 --------+----------------+---------- 3 0 0 | 3 0 0 0 0 | 4 * * * 2 1 2 | 1 1 1 1 1 | * 12 * * {5} 1 1 1 | 0 1 1 1 0 | * * 12 * 0 0 3 | 0 0 0 0 3 | * * * 4
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