Acronym ek
Name diacosipentacontahexazetton,
octacross8),
aeroyott(id),
octaexal antiprism,
vertex figure of vee,
(one of the) Delone cell(s) of lattice E8,
Gosset polytope 51,1,
lattice C8 contact polytope (span of its small roots),
equatorial cross-section of vertex-first vee
Circumradius 1/sqrt(2) = 0.707107
Inradius 1/4 = 0.25
Lace city
in approx. ASCII-art
 ©  
    o    		-- o3o3o3o3o3o4o (point)
         
o   g   o		-- x3o3o3o3o3o4o (zee)
         
    o    		-- o3o3o3o3o3o4o (point)

where:
o = o3o3o3o3o4o (point)
g = x3o3o3o3o4o (gee)
o3o3o3o3o3o   x3o3o3o3o3o   
                            
                            
                            
                            
                            
                            
   o3o3o3o3o3x   o3o3o3o3o3o
x o3o3o3o3o   o x3o3o3o3o  
                           
                           
                           
                           
                           
                           
                           
  o o3o3o3o3x   x o3o3o3o3o
x3o o3o3o3o   o3o x3o3o3o 
                          
                          
                          
                          
                          
                          
                          
                          
 o3o o3o3o3x   o3x o3o3o3o
x3o3o o3o3o   o3o3o x3o3o
                         
                         
                         
                         
                         
                         
                         
                         
                         
o3o3o o3o3x   o3o3x o3o3o
Coordinates (1/sqrt(2), 0, 0, 0, 0, 0, 0, 0)   & all permutations, all changes of sign
Volume 1/2520 = 0.00039683
Surface 4/315 = 0.012698
Rel. Roundness 105 π4/65536 = 15.606620 %
Dual octo
Dihedral angles
  • at hop between oca and oca:   arccos(-3/4) = 138.590378°
Face vector 16, 112, 448, 1120, 1792, 1792, 1024, 256
Confer
more general:
xPo3o...o3o4o  
related segmentozetta:
zeepy   geeasc  
general polytopal classes:
Wythoffian polyzetta   orthoplex   noble polytopes   segmentozetta   fundamental lace prisms   Coxeter-Elte-Gosset polytopes  
analogs:
regular orthoplex On  
External
links
wikipedia   polytopewiki

Incidence matrix according to Dynkin symbol

x3o3o3o3o3o3o4o

. . . . . . . . | 16   14 |  84 |  280 |  560 |  672 |  448 | 128
----------------+----+-----+-----+------+------+------+------+----
x . . . . . . . |  2 | 112   12 |   60 |  160 |  240 |  192 |  64
----------------+----+-----+-----+------+------+------+------+----
x3o . . . . . . |  3 |   3 | 448    10 |   40 |   80 |   80 |  32
----------------+----+-----+-----+------+------+------+------+----
x3o3o . . . . .   4 |   6 |   4 | 1120     8 |   24 |   32 |  16
----------------+----+-----+-----+------+------+------+------+----
x3o3o3o . . . .   5 |  10 |  10 |    5 | 1792     6 |   12 |   8
----------------+----+-----+-----+------+------+------+------+----
x3o3o3o3o . . .   6 |  15 |  20 |   15 |    6 | 1792 |    4 |   4
----------------+----+-----+-----+------+------+------+------+----
x3o3o3o3o3o . .   7 |  21 |  35 |   35 |   21 |    7 | 1024 |   2
----------------+----+-----+-----+------+------+------+------+----
x3o3o3o3o3o3o .   8 |  28 |  56 |   70 |   56 |   28 |    8 | 256

x3o3o3o3o3o3o4/3o

. . . . . . .   . | 16   14 |  84 |  280 |  560 |  672 |  448 | 128
------------------+----+-----+-----+------+------+------+------+----
x . . . . . .   . |  2 | 112   12 |   60 |  160 |  240 |  192 |  64
------------------+----+-----+-----+------+------+------+------+----
x3o . . . . .   . |  3 |   3 | 448    10 |   40 |   80 |   80 |  32
------------------+----+-----+-----+------+------+------+------+----
x3o3o . . . .   .   4 |   6 |   4 | 1120     8 |   24 |   32 |  16
------------------+----+-----+-----+------+------+------+------+----
x3o3o3o . . .   .   5 |  10 |  10 |    5 | 1792     6 |   12 |   8
------------------+----+-----+-----+------+------+------+------+----
x3o3o3o3o . .   .   6 |  15 |  20 |   15 |    6 | 1792 |    4 |   4
------------------+----+-----+-----+------+------+------+------+----
x3o3o3o3o3o .   .   7 |  21 |  35 |   35 |   21 |    7 | 1024 |   2
------------------+----+-----+-----+------+------+------+------+----
x3o3o3o3o3o3o   .   8 |  28 |  56 |   70 |   56 |   28 |    8 | 256

o3o3o *b3o3o3o3o3x

. . .    . . . . . | 16   14 |  84 |  280 |  560 |  672 |  448 |  64  64
-------------------+----+-----+-----+------+------+------+------+--------
. . .    . . . . x |  2 | 112   12 |   60 |  160 |  240 |  192 |  32  32
-------------------+----+-----+-----+------+------+------+------+--------
. . .    . . . o3x |  3 |   3 | 448    10 |   40 |   80 |   80 |  16  16
-------------------+----+-----+-----+------+------+------+------+--------
. . .    . . o3o3x   4 |   6 |   4 | 1120     8 |   24 |   32 |   8   8
-------------------+----+-----+-----+------+------+------+------+--------
. . .    . o3o3o3x   5 |  10 |  10 |    5 | 1792     6 |   12 |   4   4
-------------------+----+-----+-----+------+------+------+------+--------
. . .    o3o3o3o3x   6 |  15 |  20 |   15 |    6 | 1792 |    4 |   2   2
-------------------+----+-----+-----+------+------+------+------+--------
. o . *b3o3o3o3o3x   7 |  21 |  35 |   35 |   21 |    7 | 1024 |   1   1
-------------------+----+-----+-----+------+------+------+------+--------
o3o . *b3o3o3o3o3x   8 |  28 |  56 |   70 |   56 |   28 |    7 | 128   *
. o3o *b3o3o3o3o3x   8 |  28 |  56 |   70 |   56 |   28 |    7 |   * 128

xo3oo3oo3oo3oo3oo3ox&#x   → height = 1/2
(oca || dual oca)

o.3o.3o.3o.3o.3o.3o.    & | 16   7  7 |  21  63 |  35 140 105 |  35 175  350 | 21 126 315 210 |  7  49 147 245 | 1  8 28  56 35
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x. .. .. .. .. .. ..    & |  2 | 56  *    6   6 |  15  30  15 |  20  60   80 | 15  60 105  60 |  6  30  66  90 | 1  6 16  26 15
oo3oo3oo3oo3oo3oo3oo&#x   |  2 |  * 56    0  12 |   0  30  30 |   0  40  120 |  0  30 120  90 |  0  12  60 120 | 0  2 12  30 20
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x.3o. .. .. .. .. ..    & |  3 |  3  0 | 112   *    5   5   0 |  10  20   10 | 10  30  30  10 |  5  20  30  25 | 1  5 10  11  5
xo .. .. .. .. .. ..&#x & |  3 |  1  2 |   * 336    0   5   5 |   0  10   30 |  0  10  40  30 |  0   5  25  50 | 0  1  6  15 10
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x.3o.3o. .. .. .. ..    &   4 |  6  0 |   4   0 | 140   *   *    4   4    0 |  6  12   6   0 |  4  12  12   4 | 1  4  6   4  1
xo3oo .. .. .. .. ..&#x &   4 |  3  3 |   1   3 |   * 560   *    0   4    4 |  0   6  12   6 |  0   4  12  16 | 0  1  4   7  4
xo .. .. .. .. .. ox&#x     4 |  2  4 |   0   4 |   *   * 420    0   0    8 |  0   0  12  12 |  0   0   8  24 | 0  0  2   8  6
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x.3o.3o.3o. .. .. ..    &   5 | 10  0 |  10   0 |   5   0   0 | 112   *    *   3   3   0   0 |  3   6   3   0 | 1  3  3   1  0
xo3oo3oo .. .. .. ..&#x &   5 |  6  4 |   4   6 |   1   4   0 |   * 560    *   0   3   3   0 |  0   3   6   3 | 0  1  3   3  1
xo3oo .. .. .. .. ox&#x &   5 |  4  6 |   1   9 |   0   2   3 |   *   * 1120   0   0   3   3 |  0   0   3   9 | 0  0  1   4  3
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x.3o.3o.3o.3o. .. ..    &   6 | 15  0 |  20   0 |  15   0   0 |   6   0    0 | 56   *   *   * |  2   2   0   0 | 1  2  1   0  0
xo3oo3oo3oo .. .. ..&#x &   6 | 10  5 |  10  10 |   5  10   0 |   1   5    0 |  * 336   *   * |  0   2   2   0 | 0  1  2   1  0
xo3oo3oo .. .. .. ox&#x &   6 |  7  8 |   4  16 |   1   8   6 |   0   2    4 |  *   * 840   * |  0   0   2   2 | 0  0  1   2  1
xo3oo .. .. .. oo3ox&#x     6 |  6  9 |   2  18 |   0   6   9 |   0   0    6 |  *   *   * 560 |  0   0   0   4 | 0  0  0   2  2
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x.3o.3o.3o.3o.3o. ..    &   7 | 21  0 |  35   0 |  35   0   0 |  21   0    0 |  7   0   0   0 | 16   *   *   * | 1  1  0   0  0
xo3oo3oo3oo3oo .. ..&#x &   7 | 15  6 |  20  15 |  15  20   0 |   6  15    0 |  1   6   0   0 |  * 112   *   * | 0  1  1   0  0
xo3oo3oo3oo .. .. ox&#x &   7 | 11 10 |  10  25 |   5  20  10 |   1  10   10 |  0   2   5   0 |  *   * 336   * | 0  0  1   1  0
xo3oo3oo .. .. oo3ox&#x &   7 |  9 12 |   5  30 |   1  16  18 |   0   3   18 |  0   0   3   4 |  *   *   * 560 | 0  0  0   1  1
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x.3o.3o.3o.3o.3o.3o.    &   8 | 28  0 |  56   0 |  70   0   0 |  56   0    0 | 28   0   0   0 |  8   0   0   0 | 2  *  *   *  *
xo3oo3oo3oo3oo3oo ..&#x &   8 | 21  7 |  35  21 |  35  35   0 |  21  35    0 |  7  21   0   0 |  1   7   0   0 | * 16  *   *  *
xo3oo3oo3oo3oo .. ox&#x &   8 | 16 12 |  20  36 |  15  40  15 |   6  30   20 |  1  12  15   0 |  0   2   6   0 | *  * 56   *  *
xo3oo3oo3oo .. oo3ox&#x &   8 | 13 15 |  11  45 |   5  35  30 |   1  15   40 |  0   3  15  10 |  0   0   3   5 | *  *  * 112  *
xo3oo3oo .. oo3oo3ox&#x     8 | 12 16 |   8  48 |   2  32  36 |   0   8   48 |  0   0  12  16 |  0   0   0   8 | *  *  *   * 70

oqo xoo3ooo3ooo3ooo3ooo3oox&#xt   → both heights = 1/sqrt(14) = 0.267261
(hop || perp pseudo q-line || dual hop)

... 

© 2004-2024
top of page