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
 
 © ©
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
Vertex layers
LayerSymmetrySubsymmetries
 o3o5oo3o .o . o. o5o
1x3o5ox3o .
{3} first
x . o
edge first
. o5o
vertex first
2o3f .o . f. x5o
vertex figure
3f3o .f . x. o5x
4o3x .
opposite {3}
o . f. o5o
opposite vertex
5 x . o
opposite edge
 
Lace city
in approx. ASCII-art
 o   o 
   f   
x     x
   f   
 o   o 
Coordinates (τ/2, 1/2, 0)   & even permutations, all changes of sign
where τ = (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
  • between {3} and {3}:   arccos(-sqrt(5)/3) = 138.189685°
Confer
Grünbaumian relatives:
2ike   cid   ike+2gad   2ike+gad   ike+3gad   3ike+gad   3ike+3gad   4ike+gad   5ike+gad   2ike+4gad   4ike+2gad  
related Johnson solids:
peppy   gyepip   mibdi   teddi   pedpy   snadow   snisquap  
facetings:
peppy   gyepip   pap   mibdi   teddi   scuffi   bitdi  
blends:
2 pap blend  
compounds:
siddo   sne  
variations:
cube-dim-doe  
general polytopal classes:
deltahedra   regular   noble polytopes   expanded kaleido-facetings   subsymmetrical diminishings  
External
links
hedrondude   wikipedia   WikiChoron   mathworld   Polyedergarten   quickfur
©

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.


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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