Acronym | sirco (alt.: esquobcu) | |||||||||||||||||||||||||||||||||||||
TOCID symbol | rCO | |||||||||||||||||||||||||||||||||||||
Name |
small rhombicuboctahedron, expanded octahedron, expanded cube, elongated square-orthobicupola | |||||||||||||||||||||||||||||||||||||
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Circumradius | sqrt[5+2 sqrt(2)]/2 = 1.398966 | |||||||||||||||||||||||||||||||||||||
Inradius wrt. {3} | [3+sqrt(2)]/sqrt(12) = 1.274274 | |||||||||||||||||||||||||||||||||||||
Inradius wrt. {4} | (1+sqrt(2))/2 = 1.207107 | |||||||||||||||||||||||||||||||||||||
Vertex figure | [3,43] = xq&#q | |||||||||||||||||||||||||||||||||||||
Snub derivation |
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Vertex layers |
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Lace city in approx. ASCII-art |
x x x w w x x w w x x x | |||||||||||||||||||||||||||||||||||||
Coordinates | ((1+sqrt(2))/2, 1/2, 1/2) & all permutations, all changes of sign | |||||||||||||||||||||||||||||||||||||
Volume | [12+10 sqrt(2)]/3 = 8.714045 | |||||||||||||||||||||||||||||||||||||
Surface | 18+2 sqrt(3) = 21.464102 | |||||||||||||||||||||||||||||||||||||
General of army | (is itself convex) | |||||||||||||||||||||||||||||||||||||
Colonel of regiment | (is itself locally convex – other uniform polyhedral members: socco sroh – other edge facetings) | |||||||||||||||||||||||||||||||||||||
Dual | sladid (old: sit) | |||||||||||||||||||||||||||||||||||||
Dihedral angles |
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Face vector | 24, 48, 26 | |||||||||||||||||||||||||||||||||||||
Confer |
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External links |
As abstract polytope sirco is isomorphic to querco, thereby replacing prograde triangles by retrograde ones.
The right image shows where the rhombi part of its name comes from: in fact it also can be seen as a rectified version of the rhombi-dodecahedron.
When looking more into classes of isogonal variants, then this polyhedron also could be addressed as a rectified cuboctahedron. However true rectification would not produce squares there. In fact it rather would produce reco (x3o4q) instead.
This polyhedron allows for 2 square-prismatic edge facetings tepcus and topfu.
There also is a gyrated axial stack too: squacu + op + gyro squacu = esquigybcu (J37), one of the Johnson solids. In fact, both have the same vertex figure all over. But sirco features full cubical symmetry, while esquigybcu features 4-fold antiprismatic symmetry only, thereby dividing the according vertex set into 2 classes.
The snub derivational process features 2 operations, first the mere alternated faceting, here wrt. a to be alternated edge of girco as its starting figure, which then would result in xwX wXx Xxw&#zh (where X = x+2q = q+w), a pyritohedral gyrated sirco variant, and a secondary edge rescaling to all unit sizes only.
Incidence matrix according to Dynkin symbol
x3o4x . . . | 24 | 2 2 | 1 2 1 ------+----+-------+------- x . . | 2 | 24 * | 1 1 0 . . x | 2 | * 24 | 0 1 1 ------+----+-------+------- x3o . | 3 | 3 0 | 8 * * x . x | 4 | 2 2 | * 12 * . o4x | 4 | 0 4 | * * 6 snubbed forms: β3o4x, x3o4s, x3o4s (as mere faceting), β3o4β
x4/3o3/2x . . . | 24 | 2 2 | 1 2 1 ----------+----+-------+------- x . . | 2 | 24 * | 1 1 0 . . x | 2 | * 24 | 0 1 1 ----------+----+-------+------- x4/3o . | 4 | 4 0 | 6 * * x . x | 4 | 2 2 | * 12 * . o3/2x | 3 | 0 3 | * * 8
s3s4x demi( . . . ) | 24 | 1 2 1 | 1 1 2 --------------+----+----------+------- demi( . . x ) | 2 | 12 * * | 0 1 1 sefa( s3s . ) | 2 | * 24 * | 1 0 1 sefa( . s4x ) | 2 | * * 12 | 0 1 1 --------------+----+----------+------- s3s . ♦ 3 | 0 3 0 | 8 * * . s4x ♦ 4 | 2 0 2 | * 6 * sefa( s3s4x ) | 4 | 1 2 1 | * * 12 starting figure: x3x4x
xxxx4oxxo&#xt → outer heights = 1/sqrt(2) = 0.707107 inner height = 1 ({4} || pseudo {8} || pseudo {8} || {4}) o...4o... | 4 * * * | 2 2 0 0 0 0 0 0 0 | 1 2 1 0 0 0 0 0 .o..4.o.. | * 8 * * | 0 1 1 1 1 0 0 0 0 | 0 1 1 1 1 0 0 0 ..o.4..o. | * * 8 * | 0 0 0 0 1 1 1 1 0 | 0 0 0 1 1 1 1 0 ...o4...o | * * * 4 | 0 0 0 0 0 0 0 2 2 | 0 0 0 0 0 2 1 1 -------------+---------+-------------------+---------------- x... .... | 2 0 0 0 | 4 * * * * * * * * | 1 1 0 0 0 0 0 0 oo..4oo..&#x | 1 1 0 0 | * 8 * * * * * * * | 0 1 1 0 0 0 0 0 .x.. .... | 0 2 0 0 | * * 4 * * * * * * | 0 1 0 1 0 0 0 0 .... .x.. | 0 2 0 0 | * * * 4 * * * * * | 0 0 1 0 1 0 0 0 .oo.4.oo.&#x | 0 1 1 0 | * * * * 8 * * * * | 0 0 0 1 1 0 0 0 ..x. .... | 0 0 2 0 | * * * * * 4 * * * | 0 0 0 1 0 1 0 0 .... ..x. | 0 0 2 0 | * * * * * * 4 * * | 0 0 0 0 1 0 1 0 ..oo4..oo&#x | 0 0 1 1 | * * * * * * * 8 * | 0 0 0 0 0 1 1 0 ...x .... | 0 0 0 2 | * * * * * * * * 4 | 0 0 0 0 0 1 0 1 -------------+---------+-------------------+---------------- x...4o... | 4 0 0 0 | 4 0 0 0 0 0 0 0 0 | 1 * * * * * * * xx.. ....&#x | 2 2 0 0 | 1 2 1 0 0 0 0 0 0 | * 4 * * * * * * .... ox..&#x | 1 2 0 0 | 0 2 0 1 0 0 0 0 0 | * * 4 * * * * * .xx. ....&#x | 0 2 2 0 | 0 0 1 0 2 1 0 0 0 | * * * 4 * * * * .... .xx.&#x | 0 2 2 0 | 0 0 0 1 2 0 1 0 0 | * * * * 4 * * * ..xx ....&#x | 0 0 2 2 | 0 0 0 0 0 1 0 2 1 | * * * * * 4 * * .... ..xo&#x | 0 0 2 1 | 0 0 0 0 0 0 1 2 0 | * * * * * * 4 * ...x4...o | 0 0 0 4 | 0 0 0 0 0 0 0 0 4 | * * * * * * * 1
or o...4o... & | 8 * | 2 2 0 0 0 | 1 2 1 0 0 .o..4.o.. & | * 16 | 0 1 1 1 1 | 0 1 1 1 1 ---------------+------+------------+---------- x... .... & | 2 0 | 8 * * * * | 1 1 0 0 0 oo..4oo..&#x & | 1 1 | * 16 * * * | 0 1 1 0 0 .x.. .... & | 0 2 | * * 8 * * | 0 1 0 1 0 .... .x.. & | 0 2 | * * * 8 * | 0 0 1 0 1 .oo.4.oo.&#x & | 0 2 | * * * * 8 | 0 0 0 1 1 ---------------+------+------------+---------- x...4o... & | 4 0 | 4 0 0 0 0 | 2 * * * * xx.. ....&#x & | 2 2 | 1 2 1 0 0 | * 8 * * * .... ox..&#x & | 1 2 | 0 2 0 1 0 | * * 8 * * .xx. ....&#x | 0 4 | 0 0 2 0 2 | * * * 4 * .... .xx.&#x | 0 4 | 0 0 0 2 2 | * * * * 4
xxwoqo3oqowxx&#xt → height(1,2) = height(2,3) = height(4,5) = height(5,6) = 1/sqrt(3) = 0.577350 height(3,4) = [sqrt(2)-1]/sqrt(3) = 0.239146 o.....3o..... | 3 * * * * * | 2 2 0 0 0 0 0 0 0 0 | 1 2 1 0 0 0 0 0 0 .o....3.o.... | * 6 * * * * | 0 1 1 1 1 0 0 0 0 0 | 0 1 1 1 1 0 0 0 0 ..o...3..o... | * * 3 * * * | 0 0 0 2 0 2 0 0 0 0 | 0 0 1 0 2 1 0 0 0 ...o..3...o.. | * * * 3 * * | 0 0 0 0 2 0 2 0 0 0 | 0 0 0 1 2 0 1 0 0 ....o.3....o. | * * * * 6 * | 0 0 0 0 0 1 1 1 1 0 | 0 0 0 0 1 1 1 1 0 .....o3.....o | * * * * * 3 | 0 0 0 0 0 0 0 0 2 2 | 0 0 0 0 0 0 1 2 1 ------------------+-------------+---------------------+------------------ x..... ...... | 2 0 0 0 0 0 | 3 * * * * * * * * * | 1 1 0 0 0 0 0 0 0 oo....3oo....&#x | 1 1 0 0 0 0 | * 6 * * * * * * * * | 0 1 1 0 0 0 0 0 0 .x.... ...... | 0 2 0 0 0 0 | * * 3 * * * * * * * | 0 1 0 1 0 0 0 0 0 .oo...3.oo...&#x | 0 1 1 0 0 0 | * * * 6 * * * * * * | 0 0 1 0 1 0 0 0 0 .o.o..3.o.o..&#x | 0 1 0 1 0 0 | * * * * 6 * * * * * | 0 0 0 1 1 0 0 0 0 ..o.o.3..o.o.&#x | 0 0 1 0 1 0 | * * * * * 6 * * * * | 0 0 0 0 1 1 0 0 0 ...oo.3...oo.&#x | 0 0 0 1 1 0 | * * * * * * 6 * * * | 0 0 0 0 1 0 1 0 0 ...... ....x. | 0 0 0 0 2 0 | * * * * * * * 3 * * | 0 0 0 0 0 1 0 1 0 ....oo3....oo&#x | 0 0 0 0 1 1 | * * * * * * * * 6 * | 0 0 0 0 0 0 1 1 0 ...... .....x | 0 0 0 0 0 2 | * * * * * * * * * 3 | 0 0 0 0 0 0 0 1 1 ------------------+-------------+---------------------+------------------ x.....3o..... | 3 0 0 0 0 0 | 3 0 0 0 0 0 0 0 0 0 | 1 * * * * * * * * xx.... ......&#x | 2 2 0 0 0 0 | 1 2 1 0 0 0 0 0 0 0 | * 3 * * * * * * * ...... oqo...&#xt | 1 2 1 0 0 0 | 0 2 0 2 0 0 0 0 0 0 | * * 3 * * * * * * .x.o.. ......&#x | 0 2 0 1 0 0 | 0 0 1 0 2 0 0 0 0 0 | * * * 3 * * * * * .oooo.3.oooo.&#xr | 0 1 1 1 1 0 | 0 0 0 1 1 1 1 0 0 0 | * * * * 6 * * * * ...... ..o.x.&#x | 0 0 1 0 2 0 | 0 0 0 0 0 2 0 1 0 0 | * * * * * 3 * * * ...oqo ......&#xt | 0 0 0 1 2 1 | 0 0 0 0 0 0 2 0 2 0 | * * * * * * 3 * * ...... ....xx&#x | 0 0 0 0 2 2 | 0 0 0 0 0 0 0 1 2 1 | * * * * * * * 3 * .....o3.....x | 0 0 0 0 0 3 | 0 0 0 0 0 0 0 0 0 3 | * * * * * * * * 1
or o.....3o..... & | 6 * * | 2 2 0 0 0 | 1 2 1 0 0 .o....3.o.... & | * 12 * | 0 1 1 1 1 | 0 1 1 1 1 ..o...3..o... & | * * 6 | 0 0 0 2 2 | 0 0 1 1 2 --------------------+--------+--------------+---------- x..... ...... & | 2 0 0 | 6 * * * * | 1 1 0 0 0 oo....3oo....&#x & | 1 1 0 | * 12 * * * | 0 1 1 0 0 .x.... ...... & | 0 2 0 | * * 6 * * | 0 1 0 1 0 .oo...3.oo...&#x & | 0 1 1 | * * * 12 * | 0 0 1 0 1 .o.o..3.o.o..&#x & | 0 1 1 | * * * * 12 | 0 0 0 1 1 --------------------+--------+--------------+---------- x.....3o..... & | 3 0 0 | 3 0 0 0 0 | 2 * * * * xx.... ......&#x & | 2 2 0 | 1 2 1 0 0 | * 6 * * * ...... oqo...&#xt & | 1 2 1 | 0 2 0 2 0 | * * 6 * * .x.o.. ......&#x & | 0 2 1 | 0 0 1 0 2 | * * * 6 * .oooo.3.oooo.&#xr | 0 2 2 | 0 0 0 2 2 | * * * * 6
qo3xx3oq&#zx → height = 0 (tegum sum of 2 mutually inverted (q,x)-tuts) o.3o.3o. | 12 * | 2 2 0 | 1 1 2 0 .o3.o3.o | * 12 | 0 2 2 | 0 1 2 1 -------------+-------+----------+--------- .. x. .. | 2 0 | 12 * * | 1 0 1 0 oo3oo3oo&#x | 1 1 | * 24 * | 0 1 1 0 .. .x .. | 0 2 | * * 12 | 0 0 1 1 -------------+-------+----------+--------- .. x.3o. | 3 0 | 3 0 0 | 4 * * * qo .. oq&#zx | 2 2 | 0 4 0 | * 6 * * .. xx ..&#x | 2 2 | 1 2 1 | * * 12 * .o3.x .. | 0 3 | 0 0 3 | * * * 4
or o.3o.3o. & | 24 | 2 2 | 1 1 2 ---------------+----+-------+------- .. x. .. & | 2 | 24 * | 1 0 1 oo3oo3oo&#x | 2 | * 24 | 0 1 1 ---------------+----+-------+------- .. x.3o. & | 3 | 3 0 | 8 * * qo .. oq&#zx | 4 | 0 4 | * 6 * .. xx ..&#x | 4 | 2 2 | * * 12
wx xx4ox&#zx → height = 0 (tegum sum of (w,x,x)-cube and ortho op) o. o.4o. | 8 * | 2 2 0 0 0 | 1 2 1 0 0 .o .o4.o | * 16 | 0 1 1 1 1 | 0 1 1 1 1 ------------+------+------------+---------- .. x. .. | 2 0 | 8 * * * * | 1 1 0 0 0 oo oo4oo&#x | 1 1 | * 16 * * * | 0 1 1 0 0 .x .. .. | 0 2 | * * 8 * * | 0 0 0 1 1 .. .x .. | 0 2 | * * * 8 * | 0 1 0 1 0 .. .. .x | 0 2 | * * * * 8 | 0 0 1 0 1 ------------+------+------------+---------- .. x.4o. | 4 0 | 4 0 0 0 0 | 2 * * * * .. xx ..&#x | 2 2 | 1 2 0 1 0 | * 8 * * * .. .. ox&#x | 1 2 | 0 2 0 0 1 | * * 8 * * .x .x .. | 0 4 | 0 0 2 2 0 | * * * 4 * .x .. .x | 0 4 | 0 0 2 0 2 | * * * * 4
wxx2xwx2xxw&#zx → all heights = 0 (tegum sum of 3 mutually orthogonal (w,x,x)-cubes) o..2o..2o.. | 8 * * | 1 1 1 1 0 0 0 0 0 | 1 1 1 1 0 0 0 .o.2.o.2.o. | * 8 * | 0 0 1 0 1 1 1 0 0 | 0 1 0 1 1 1 0 ..o2..o2..o | * * 8 | 0 0 0 1 0 0 1 1 1 | 0 0 1 1 0 1 1 ---------------+-------+-------------------+-------------- ... x.. ... | 2 0 0 | 4 * * * * * * * * | 1 0 1 0 0 0 0 ... ... x.. | 2 0 0 | * 4 * * * * * * * | 1 1 0 0 0 0 0 oo.2oo.2oo.&#x | 1 1 0 | * * 8 * * * * * * | 0 1 0 1 0 0 0 o.o2o.o2o.o&#x | 1 0 1 | * * * 8 * * * * * | 0 0 1 1 0 0 0 .x. ... ... | 0 2 0 | * * * * 4 * * * * | 0 0 0 0 1 1 0 ... ... .x. | 0 2 0 | * * * * * 4 * * * | 0 1 0 0 1 0 0 .oo2.oo2.oo&#x | 0 1 1 | * * * * * * 8 * * | 0 0 0 1 0 1 0 ..x ... ... | 0 0 2 | * * * * * * * 4 * | 0 0 0 0 0 1 1 ... ..x ... | 0 0 2 | * * * * * * * * 4 | 0 0 1 0 0 0 1 ---------------+-------+-------------------+-------------- ... x..2x.. | 4 0 0 | 2 2 0 0 0 0 0 0 0 | 2 * * * * * * ... ... xx.&#x | 2 2 0 | 0 1 2 0 0 1 0 0 0 | * 4 * * * * * ... x.x ...&#x | 2 0 2 | 1 0 0 2 0 0 0 0 1 | * * 4 * * * * ooo2ooo2ooo&#x | 1 1 1 | 0 0 1 1 0 0 1 0 0 | * * * 8 * * * .x. ... .x. | 0 4 0 | 0 0 0 0 2 2 0 0 0 | * * * * 2 * * .xx ... ...&#x | 0 2 2 | 0 0 0 0 1 0 2 1 0 | * * * * * 4 * ..x2..x ... | 0 0 4 | 0 0 0 0 0 0 0 2 2 | * * * * * * 2
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